Plea for Diagonals and Telescopers of Rational Functions

Abstract

This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational or algebraic functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form $P/Q^k$ and $1/Q$, are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of the generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple ${}_{2}F_1$ hypergeometric functions associated with elliptic curves, or the (differentially algebraic) lambda-extension of correlation of the Ising model.

1. Introduction: Plea for a Computer Algebra Experimental Mathematics Learn by Example Approach

A paper in honor of Professor Richard Kerner must be a paper on theoretical physics, mathematical physics, physical mathematics, applied mathematics, applicable mathematics or even experimental mathematics [1]. These different domains have large overlaps and, quite often, their differences or shades are slightly irrelevant, only corresponding to social membership to different “mathematical tribes”. This computer algebra paper will actually be a plea for diagonals and telescopers of rational (or algebraic) functions and for creative telescoping, with a computer algebra experimental mathematics learn-by-examples approach.

1.1. Honor, Pride, and Prejudice

The “Journal of Mathematical Physics” defines mathematical physics as “the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories”. An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Mathematical physics clearly raises the question of the watershed between mathematics and physics (especially in France…). Does “Mirror Symmetry” [2,3,4,5], which is a relationship between geometric objects called Calabi–Yau manifolds, belong to algebraic geometry or theoretical physics? Does “Special Relativity” belong to physics or mathematics, Einstein or Poincaré? “Einstein was reluctant to acknowledge that the Michelson-Morley experiment had a significant influence on his road to special relativity” [6]. In fact, “once Maxwell’s equations are properly understood mathematically, special relativity is an inevitable consequence” [6]. Physical mathematics is sometimes viewed with suspicion by both physicists and mathematicians. On the one hand, mathematicians regard it as deficient, for lack of proper mathematical rigor. In the years since this “mathematical physics debate” erupted [7], there have been many spectacular successes scored by physical mathematics, thanks to the “unreasonable effectiveness” of physics in the mathematical sciences. Dyson famously proclaimed: “As a working physicist, I am actualy aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce”. This “divorce” is particularly serious in France because of the overwhelmingly leading figure of Alexander Grothendieck and the huge influence of the Bourbaki group, which raises the question of rigor versus creativity (“We should not confuse rigor with rigor mortis”, Isadore Singer, see [6]). Recalling Pierre Cartier [8], the Bourbaki group has been criticized by several mathematicians, including its own former members, for a variety of reasons. “ Criticisms have included the choice of presentation of certain topics within the Éléments [9] at the expense of others, dislike of the method of presentation for given topics, dislike of the group’s working style, and a perceived elitist mentality around Bourbaki’s project and its books, especially during the collective’s most productive years in the 1950s and 1960s. There is essentially no analysis beyond the foundations: nothing about partial differential equations, nothing about probability. There is also nothing about combinatorics, nothing about algebraic topology [10], nothing about concrete geometry. Anything connected with mathematical physics is totally absent from Bourbaki’s text.” Dieudonné (founding member), later, regretted that Bourbaki’s success had contributed to snobbery regarding pure mathematics in France, at the expense of applied mathematics [11,12]. In an interview (to Marian Schmidt in 1990), he said: "It is possible to say that there was no serious applied mathematics in France for forty years after Poincaré. There was even a snobbery for pure mathematics. When one noticed a talented student, one would tell him “You should do pure math”. On the other hand, one would advise a mediocre student to do applied mathematics while thinking, “It’s all that he can do! ”. Apart from french mathematicians (when in doubt, blame the French), this snobbery regarding pure mathematics met with harsh criticism from Vladimir Arnold in his deliciously polemical paper [13] “Sur l’éducation mathématique”.

Quantum groups emerged from one (Yang–Baxter integrable) explicit example, namely, Quantum Toda, and not from an ex-nihilo abstract, formal construction of a noncommutative algebra formalism, and other $C^{★}$-algebras, dressed with coassociative coproducts. In theoretical physics, we get used to the emergence of modular forms [14] and sometimes automorphic forms [15] like Shimura forms [16]. If a physicist asks a mathematician for more information on these structures he will probably only receive the academical Poincaré upper half-plane definition and formalism, which will be totally and utterly useless to him, and he will not recognize the representation of modular forms and Shimura forms, which naturally emerges in physics [16,17] in terms of pullbacked ${}_{2}F_1$ hypergeometric functions. In theoretical physics, we are flooded by elliptic curves, K3 surfaces, and Calabi–Yau manifolds [3,18,19,20,21,22,23]. If a physicist tries to discuss with a mathematician the elliptic curve he has just discovered (when he has even calculated the j-invariant, or the Hauptmodul, of this elliptic curve…), he might be severely rebuked that he has absolutely no right to talk of an elliptic curve because an elliptic curve must have a “specified point”, or will be seen with suspicion because his elliptic curve does not correspond to the complete intersection of quadrics [24] framework mathematicians like to consider in their theorems. Along this (slightly polemical…) line, pure mathematicians will, often, refuse to provide a representation of their formalism; in particular, they will refuse to provide examples. If a physicist, eager to understand a mathematical concept, asks for an example of an algebraic variety, an example of a holonomic function, or an example of functor, some mathematicians will, maliciously, reply: a point, the constant function and the oblivion functor. In such a frustrating “dialogue of the deaf” between physicists and mathematicians, mathematical physics is probably the perfect for being criticized by physicists as too abstract, or too mathematical, and also by mathematicians for a lack of rigor and a lack of mathematical proofs.

Jean-Louis Verdier performed his thesis under the direction of Alexandre Grothendieck. He was a member of the Bourbaki group. He passed away in August 1989. At this stage, one of us (JMM) would like to seize the opportunity of this experimental mathematics paper in honor of Professor Richard Kerner, to express his deep regrets for the numerous fruitful conversations with Jean-Louis Verdier and his very generous pedagogical explanations. A discussion with him was not flooded with “Derived Categories” or “p-adic cohomology” but with simple examples and representations of the mathematical concepts. A really good mathematician can provide examples; he is not afraid, or ashamed, to provide examples and representations. For Jean-Louis Verdier, mathematics was not an obfuscation contest.

This paper is an experimental mathematics [1] paper with a learn-by-example approach: we obtain puzzling exact results from computer algebra (Maple, Mathematica), and we hope mathematicians will be interested to provide proofs of these results, in a proper framework. Furthermore, these exact results, which are useful for physics, raise a lot of fascinating new questions at the crossroad of different domains of mathematics.

1.2. Diagonal of Rational Functions, Creative Telescoping, Birational Transformations, and Effective Algebtraic Geometry

Diagonals of rational functions (or diagonals of algebraic functions) have been shown to emerge naturally [25] for $n$-fold integrals in physics (corresponding to solutions of linear differential operators of quite high order [26]); field theory; and enumerative combinatorics [27,28], and have been seen as “Periods” [29,30,31] of algebraic varieties (corresponding to the denominators of these rational functions). The fact that diagonals of rational or algebraic functions occur frequently in physics explains many unexpected mathematical properties encountered in physics that are far more obvious from a physics viewpoint. Physicists are clearly very interested to see if the critical exponents of the three-dimensional Ising model are or are not rational numbers. In contrast, since many lattice Green functions in any dimension [32] are diagonals of rational functions, their critical exponents are necessarily rational numbers in any dimension. Accordingly, the linear differential operators, annihilating these “Periods”, are globally nilpotent [33], and, consequently, the critical exponents of all the (regular) singular points of these operators are necessarily rational numbers (Katz theorem states that globally nilpotent linear differential operators are fuchsian with rational exponents; see, for instance, [34]). These $n$-fold integrals are also a globally bounded [25,35] series, which means that they can be recast into a (finite radius of convergence) series with integer coefficients. Furthermore, these series, with integer coefficients, reduce modulo every prime to algebraic functions. The calculation of the linear differential operators annihilating these $n$-fold integrals of algebraic functions can be systematically performed using the creative telescoping method [36,37,38], which corresponds, essentially, to successive differential algebra eliminations, which are blind to the cycles over which one performs the $n$-fold integrals. At first sight one expects the analysis of these $n$-fold integrals to require, as in the S-matrix theory [39], a lot of complex analysis of several complex variables, but one quickly discovers, with creative telescoping, that one needs differential algebra, possibly algebraic geometry [40], because of the crucial role of an algebraic variety, and, surprisingly, one finds out almost “arithmetical” properties (like in the Grothendieck–Katz p-curvature conjecture, which is a local–global principle for linear ordinary differential equations, related to differential Galois theory). More experimentally, this time, one finds out that almost all the diagonals of rational or algebraic functions, corresponding or not to physics, are annihilated by linear differential operators that are homomorphic to their adjoint, and, consequently, their differential Galois groups are (or are a subgroup of) selected $Sp(n,\mathbb{C})$ symplectic or $SO(n,\mathbb{C})$ orthogonal groups [41,42]. More generally, one finds out that the telescopers of almost all the rational or algebraic functions are also homomorphic to their adjoint [41]. A physicist, already surprised to see the emergence of all these mathematical concepts in his backyard, will have the prejudice that these selected differential Galois groups are probably a consequence of some “sampling bias”, these diagonals and telescopers being, in fact, related to (Yang–Baxter) integrable models, like the ${\chi }^{\left(n\right)}$ components of the susceptibility of the Ising model [26], or beyond, Calabi–Yau manifolds, Mirror Symmetries, Picard–Fuchs systems, and other theory “integrable” in some way (Yang-Mills…). In contrast, a mathematician will have the prejudice that this is nothing but the Poincaré duality [43] since we have a canonical algebraic variety for all these diagonals or telescopers [40]. Experimentally, it is quite hard to find telescopers or linear differential operators that are not homomorphic to their adjoint, i.e., that do not have selected symplectic, or orthogonal, differential Galois groups [41,42]. Christol conjectured [44,45] that every D-finite globally bounded series is the diagonal of a rational function. If one considers Christol’s conjecture [44,45,46,47,48], one can seek for ${}_{n}F_{n-1}$ hypergeometric series with integer coefficients that are candidates to be counter-examples to Christol’s conjecture [44,45,46,47]. Among these candidates a sub-set has actually been seen [48] to be diagonals of rational or algebraic functions like ${}_{3}F_2([2/9,5/9,8/9],[2/3,1],x)$ or ${}_{3}F_2([1/9,4/9,7/9],[1/3,1],x)$. The fact that the others, like the original example of G. Christol, ${}_{3}F_2([1/9,4/9,5/9],[1/3,1],3^{6}x)$, are or are not diagonals of rational or algebraic functions remains an open question. It turns out that the linear differential operators of these ${}_{n}F_{n-1}$ candidates precisely provide such rare examples of linear differential operators (annihilating diagonals of rational or algebraic functions) that are not homomorphic to their adjoint. The existence of such examples (curiously related to Christol’s conjecture…) shows that seeing the emergence of such selected differential Galois groups [41] for diagonals of rational or algebraic functions cannot simply be seen as some consequence of the Poincaré duality. The Poincaré duality works for any algebraic variety: the diagonal of any rational or algebraic function should always yield “self-dual” linear differential operators in the sense that they are homomorphic to their adjoint. This is not the case. Could it be that the physicist’s prejudice is right and that, trying to be generic in our computer algebra experiments, we were, in fact, just exploring diagonals of selected subsets of rational or algebraic functions related to some kind of “integrable” physics?

Like Monsieur Jourdain (in “Le Bourgeois Gentilhomme”, Molière) speaking “prose” without noticing himself, physicists often perform some fundamental mathematics when they work on their $n$-fold integrals without noticing these $n$-fold integrals are, in fact, diagonals of rational or algebraic functions. In fact, diagonals of rational or algebraic functions, and more generally telescopers, are a perfect subject of analysis in mathematical physics: they are, essentially, not well-known by mathematicians and by physicists (even if physicists speak “diagonal” without noticing…), and even when these concepts are superficially known, they are not taken seriously by mathematicians, probably because the definition is so simple, and the calculations are just “computer algebra”, which is not highly regarded in the “mathematical food chain”. This is in contrast to the fact that almost every calculation of a diagonal of a rational or algebraic function or of the calculation of a telescoper yields interesting, remarkable, and sometimes even puzzling exact results, providing answers in physics and mathematics but also raising new interesting questions that could be called “speculative mathematics”.

In a learn-by-example approach, we are going to address the previous questions of “duality-breaking” of some telescopers of rational or algebraic functions, and we will also sketch some remarkable birational symmetries [24,49] of the diagonals and telescopers of rational or algebraic functions.

2. Definition of the Diagonals of Rational or Algebraic functions: Definition of Telescopers

The purpose of this paper is not to provide an introduction to creative telescoping [36,37,38,50,51,52,53]. The purpose of this paper is, rather, to provide many (non-trivial) pedagogical examples of “telescopers” by extensively using Chyzak’s algorithm [53] or Koutschan’s semi-algorithm [54] “HolonomicFunctions” package [54]. Koutschan’s package [54] corresponds to a semi-algorithm because the termination is not proven. For the examples displayed in this paper, Koutschan’s package [54] turns out to be more user-friendly and also more efficient.

Creative telescoping [36,37,38,50,51,52,53] has become popular in computer algebra in the last twenty years. It is a methodology to deal with (parametrized) n-fold integrals, or with symbolic sums, yielding linear differential/recurrence equations. By the “telescoper” of a rational function, say $R(x,y,z)$, we here refer to one of the output of the creative telescoping program [54] (the other outputs being the so-called “certificates”). The telescoper $T$ is a linear differential operator that is satisfied by the diagonal $Diag\left(R\right)$, as well as other solutions. These other solutions correspond to “periods” [29,30,31] of algebraic varieties over non-vanishing cycles [55].

The reader interested in the connection between all these notions can read the thesis of Pierre Lairez [56] (see also [31]).

2.1. Definition

The diagonal of the rational function $R$ dependent on (for example) three variables is obtained by expanding $R$ around the origin

$$\begin{array}{ccc}\hfill & & R(x,y,z)=\sum _m\sum _n\sum _l{a}_{m,n,l}·x^m{y}^n{z}^l,\hfill \end{array}$$

(1)

and keeping only the terms such that $m=n=l$. The diagonal reads, with $p=xyz$,

$$\begin{array}{ccc}\hfill & & Diag\left(R(x,y,z)\right)=\sum _m{a}_{m,m,m}·p^m.\hfill \end{array}$$

(2)

In order to avoid a proliferation of variables, the variable $p$, which the diagonal (2) depends on, is, in the following, simply denoted as $x$ (see below (3)). Extracting these diagonal terms essentially amounts to finding constant terms [57] in several complex variable expansions, i.e., it amounts to performing a residue calculation in several complex variable expansions

$$\begin{array}{ccc}\hfill & & Diag\left(R(x,y,z)\right)={\int }_{\mathcal{C}}\frac{1}{yz}·R\left(\frac{x}{yz},y,z\right)·dydz\hfill \\ \hfill & & =\frac{1}{2i\pi }\int \frac{1}{2i\pi }\int \sum _m\sum _n\sum _l{a}_{m,n,l}·x^m{y}^{n-m}{z}^{l-m}·\frac{dy}{y}\frac{dz}{z}=\sum _m{a}_{m,m,m}·x^m,\hfill \end{array}$$

(3)

or equivalently

$$\begin{array}{ccc}\hfill & & Diag\left(R(x,y,z)\right)={\int }_{\mathcal{C}}\frac{1}{yz}·R\left(\frac{x}{y},\frac{y}{z},z\right)·dydz,\hfill \end{array}$$

(4)

where $\mathcal{C}$ denotes a vanishing cycle [55], where ${\int }_{\mathcal{C}}$ is a symbolic notation for the $n$-fold integral with the well-suited pre-factors and where the diagonal (4) is seen as a function of the remaining variable $x$. This is the very reason why diagonals of rational or algebraic functions can be interpreted as $n$-fold integrals [25]. More generally, with $n$ variables, one can write the diagonal of a rational function of $n$-variables as the residue in $n-1$ variables $x_2,\cdots ,x_n$:

$$\begin{array}{ccc}\hfill & & Diag\left(R(x_1,x_2,\cdots ,x_n)\right)\hfill \\ \hfill & & =\frac{1}{2i\pi }\int \cdots \frac{1}{2i\pi }\int \frac{1}{x_2\cdots x_n}·R\left(\frac{x_1}{x_2\cdots x_n},x_2,\cdots ,x_n\right)·d{x}_2\cdots d{x}_n.\hfill \end{array}$$

(5)

If the definition of the diagonal of a rational or algebraic function is very simple, it does not mean that calculating such a diagonal is simple. By “calculating”, we mean finding that the series, corresponding to the diagonal, is the series expansion of some known special function [58,59,60,61] (an algebraic function [62], a pullbacked ${}_{2}F_1$ hypergeometric function that turns out to be a modular form [16,63], a ${}_{n}F_{n-1}$ hypergeometric function, a Heun function [64], etc.). Most of the time, this involves, since diagonals of rational or algebraic functions are selected (Fuchsian [26], G-nilpotent operators, globally bounded series [35]) D-finite functions, finding the linear differential operator annihilating the diagonal series, even if we are not able to “solve” this linear differential equation. Finding this linear differential operator can be performed by first obtaining the large series expansion of the diagonal and then finding, by a “guessing” approach, the linear differential operator or obtaining the linear differential operator from a more global differential algebra approach, called creative telescoping.

2.2. Telescopers

For pedagogical reasons, let us sketch creative telescoping [36,37,38,50,51,52,53] in the case of a rational function of three variables. The “telescoper” of a rational function, say $R(x,y,z)$, applied not to the rational function $R(x,y,z)$ but to the transformed rational function $\widehat{R}=R(x/y,y/z,z)/\left(yz\right)$, is a linear differential operator $T$ in $x$ and $\partial x$, such that

$$\begin{array}{ccc}\hfill & & T·\left(\frac{1}{yz}·R\left(\frac{x}{y},\frac{y}{z},z\right)\right)+\frac{\partial U}{\partial y}+\frac{\partial V}{\partial z}=0,\hfill \end{array}$$

(6)

where $U$, $V$ are rational functions in $x,y,z$ called “certificates”. These rational functions are often quite large rational functions. This equation is called the telescoping equation. Extracting the diagonal of a rational function amounts to calculating residues in several complex variables, namely,

$$\begin{array}{ccc}\hfill & & Diag\left(R(x,y,z)\right)={\int }_{\mathcal{C}}\frac{1}{yz}·R\left(\frac{x}{y},\frac{y}{z},z\right),\hfill \end{array}$$

(7)

where the cycle $\mathcal{C}$ is a vanishing cycle [55]. By performing the previous integration over a cycle $\mathcal{C}$ on the LHS of the telescoping Equation (6), one will obtain (with the reasonable assumption that the linear differential operator $T$ commutes with the integration)

$$\begin{array}{ccc}\hfill & & T·Diag\left(R(x,y,z)\right)+{\int }_{\mathcal{C}}\left(\frac{\partial U}{\partial y}+\frac{\partial V}{\partial z}\right)=0.\hfill \end{array}$$

(8)

Again, (with reasonable assumptions) one can expect the second term in (8) to be equal to zero, thus yielding the equation

$$\begin{array}{ccc}\hfill & & T·Diag\left(R(x,y,z)\right)=T·{\int }_{\mathcal{C}}\frac{1}{yz}·R\left(\frac{x}{y},\frac{y}{z},z\right)=0.\hfill \end{array}$$

(9)

In other words, the telescoper $T$ represents a linear differential operator annihilating the diagonal $Diag\left(R\right)$. For the calculation of a diagonal, the cycle $\mathcal{C}$ has to be a vanishing cycle (residue calculation). Note that the creative telescoping calculations giving as an output the telescoper $T$ and the two “certificates” $U$ and $V$ essentially amount to performing differential algebra calculations (similar to integration by part for several complex variables). Since these creative telescoping calculations are differential algebra eliminations, they are totally and utterly blind to the cycle $\mathcal{C}$. Consequently, even if one performs an integration over a non-vanishing cycle, the telescoper $T$ will also be such that

$$\begin{array}{ccc}\hfill & & T·\mathcal{P}=0\mathrm{where}:\mathcal{P}={\int }_{\mathcal{C}}\frac{1}{yz}·R\left(\frac{x}{y},\frac{y}{z},z\right),\hfill \end{array}$$

(10)

this integral being not necessarily equal to the diagonal $Diag\left(R\right(x,y,z\left)\right)$ (which could be, for instance, equal to zero). Equation (10) means that the telescoper annihilates all the periods $\mathcal{P}$.

To sum-up: In order to calculate the diagonal of a rational function one can try, in a very down-to-earth way, to obtain large enough series expansions of this diagonal from multi-series expansion and then try a guessing approach to obtain the linear differential operator annihilating the diagonal of a rational function, or one can perform the creative telescoping approach that will provide this telescoper even if the diagonal is zero, or even if it cannot be nicely defined because the rational function does not have a multi-Taylor expansion: in this case, the telescoper annihilates periods corresponding to all the cycles, in particular non-vanishing cycles.

2.3. Diagonals versus Telescopers: Vanishing Cycles versus Non-Vanishing Cycles

2.3.1. Diagonals versus Telescopers: A First Example

Let us first consider the following rational function of three variables:

$$\begin{array}{ccc}\hfill & & R(x,y,z)=\frac{1}{-x-y-z^2}.\hfill \end{array}$$

(11)

This rational function does not have a multi-Taylor expansion; thus, we cannot define the diagonal of the rational function. This rational function has, however, a telescoper that is a linear differential operator of order one, namely, $5\theta +2$, where $\theta =x{D}_x=xd/dx$ is the homogeneous derivative. Let us now consider a slightly more general rational function

$$\begin{array}{ccc}\hfill & & R(x,y,z)=\frac{1}{\alpha -x-y-z^2}.\hfill \end{array}$$

(12)

This rational function (12) has a multi-Taylor expansion, and one can, thus, obtain the first terms of the diagonal of this rational function (12):

$$\begin{array}{ccc}\hfill & & Diag\left(R(x,y,z)\right)=\frac{1}{\alpha }+\frac{30}{{\alpha }^6}·x^2+\frac{3150}{{\alpha }^{11}}·x^4+\frac{420420}{{\alpha }^{16}}·x^6+\cdots \hfill \end{array}$$

(13)

The $\alpha$-dependent rational function (12) has an order-four $\alpha$-dependent telescoper $L_4\left(\alpha \right)$

$$\begin{array}{ccc}\hfill & & x^2·L_4\left(\alpha \right)=-5·x^2·(5\theta +2)·(5\theta +4)·(5\theta +6)·(5\theta +8)\hfill \\ \hfill & & +16·{\alpha }^5·{\theta }^2·{(\theta -1)}^2,\hfill \end{array}$$

(14)

which has the following ${}_{4}F_3$ hypergeometric function solution

$$\begin{array}{ccc}\hfill & & \frac{1}{\alpha }·{}_{4}F_3\left([\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}],[\frac{1}{2},\frac{1}{2},1],\frac{3125}{16{\alpha }^5}·x^2\right).\hfill \end{array}$$

(15)

The series expansion of this ${}_{4}F_3$ hypergeometric function (15) is in agreement with the series expansion (13). In the $\alpha \to 0$ limit, the order-four $\alpha$-dependent telescoper $L_4\left(\alpha \right)$ becomes the direct-sum:

$$\begin{array}{ccc}\hfill & & -5·x^4·\left((5\theta +2)\oplus (5\theta +4)\oplus (5\theta +6)\oplus (5\theta +8)\right).\hfill \end{array}$$

(16)

We thus see, in this $\alpha \to 0$ limit, that one recovers, among the different factors in (16), the order-one telescoper of the rational function (11), namely, $5\theta +2$. This first example is a bit too simple, so let us consider another example.

2.3.2. Diagonals versus Telescopers: A Second Example

Let us now consider the rational function of three variables:

$$\begin{array}{ccc}\hfill & & R(x,y,z)=\frac{1}{-x-y-z-x^{5}y}.\hfill \end{array}$$

(17)

This rational function has a telescoper $L_4$, which is a linear differential operator of order four, which reads:

$$\begin{array}{ccc}\hfill & & L_4=-(800000x^5-27)·x^4{D}_x^4-(11200000x^5+27)·x^3{D}_x^3\hfill \\ \hfill & & -15·(2800000x^5-1)·x^2{D}_x^2-60·(700000x^5-1)·x{D}_x\hfill \\ \hfill & & -12·(437500x^5+9),\hfill \end{array}$$

(18)

or, introducing the homogeneous derivative $\theta =x{D}_x$,

$$\begin{array}{ccc}\hfill & & L_4=-50000·x^5·(2\theta +7)(2\theta +5)(2\theta +3)(2\theta +1)\hfill \\ \hfill & & +3·(3\theta +1)(3\theta -4){(\theta -3)}^2.\hfill \end{array}$$

(19)

The rational function (17) does not have a multi-Taylor expansion. We have a problem to define the diagonal of the rational function (17). The analytic solutions of (18) or (19) are thus just “Periods” of the rational function (17), i.e., integrals over a non-vanishing cycle of the rational function (17). A solution of (18) or (19) is, for instance, the hypergeometric function:

$$\begin{array}{ccc}\hfill & & x^3·{}_{4}F_3\left([\frac{7}{10},\frac{9}{10},\frac{11}{10},\frac{13}{10}],[1,\frac{4}{3},\frac{5}{3}],\frac{800000}{27}·x^5\right).\hfill \end{array}$$

(20)

If one finds that the concept of diagonal is easier to understand, compared to“Periods” over non-vanishing cycles that are not really defined (we just know they exist), such a result may look a bit too abstract and thus slightly frustrating. In fact, one can recover some contact with the easier concept of diagonals, performing some kind of “desingularization”. Let us consider the more general $\alpha$-dependent rational function of three variables:

$$\begin{array}{ccc}\hfill & & R(x,y,z)=\frac{1}{\alpha -x-y-z-x^{5}y}.\hfill \end{array}$$

(21)

It has a telescoper that is a linear differential operator of order four $M_4\left(\alpha \right)$. The first terms of the diagonal of that rational function (21) can easily be calculated. We have calculated this order-four linear differential operator $M_4\left(\alpha \right)$. It is a bit too large to be given here. However, one remarks that this $\alpha$-dependent order-four linear differential operator $M_4\left(\alpha \right)$ is actually related to the previous order-four linear differential operator $L_4$, in the $\alpha \to 0$ limit:

$$\begin{array}{ccc}\hfill & & M_4\left(0\right)=-675000000x^{11}·L_4.\hfill \end{array}$$

(22)

To sum-up: The telescoper corresponding to “Periods” over a non-vanishing cycles can be obtained from a one-parameter telescoper having clear-cut diagonal solutions (“Periods” over a vanishing cycle).

2.4. The Devil Is in the Detail: The Number of Variables

Let us consider the diagonal of the following rational function of four variables

$$\begin{array}{ccc}\hfill & & \frac{1}{1-\alpha x-y-z-\beta ·xu}.\hfill \end{array}$$

(23)

Its telescoper is, for any value of $\alpha$, and for $\beta \ne 0$, the order-two linear differential operator

$$\begin{array}{ccc}\hfill & & L_2=(1-27\beta ·x)·x{D}_x^2+(1-54\beta ·x)·D_x-6\beta ,\hfill \end{array}$$

(24)

which has the following hypergeometric ${}_{2}F_1$ solution:

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right],27\beta ·x\right).\hfill \end{array}$$

(25)

Recalling the definition of the diagonal of a rational function based on multi-Taylor expansion, it is easy to see, based on this almost trivial example, that the various powers of the product $t=xyzu$ that the diagonal extracts require the occurrence of the variable $u$, which only occurs in the denominator of (23), through the product $xu$, automatically yielding the occurrence of the variable $x$. Consequently, any further occurrence of the variable $x$, from the $-\alpha x$ monomial in the denominator of (23), is excluded. This explains why the diagonal of (23) is actually blind to the $-\alpha x$ term. In other words, the diagonal of the four variables rational function (23) is in fact the diagonal of a rational function of three variables: $y$, $z$, and the product $xu$.

Remark 1.

To take into account this problem, we will introduce the concept of an “effective number” of variables. In the previous example, the number of variables is four but the “effective number” of variables is three.

2.5. Understanding the Complexity of the Diagonal of a Rational Function

2.5.1. Order of the Linear Differential Operator and Number of Variables

The simplest example of a diagonal of rational function of $n$ variables corresponds to the diagonal of the rational function

$$\begin{array}{ccc}\hfill & & \frac{1}{1-x_1-x_2-x_3\cdots -x_n}.\hfill \end{array}$$

(26)

The diagonal of (26) is annihilated by an order-$(n-1)$ linear differential operator with a ${}_{n-1}F_{n-2}$ hypergeometric solution

$$\begin{array}{ccc}\hfill & & {}_{n-1}F_{n-2}\left([\frac{1}{n},\frac{2}{n},\frac{3}{n},\cdots ,\frac{n-1}{n}],[1,1,\cdots ,1],n^n·x\right).\hfill \end{array}$$

(27)

This simple example may provide the prejudice that, for a given globally bounded series (36), the number of variables of the rational function is related to the (minimal) order of the linear differential operator annihilating the series. One should note, however, for the class of the above example, that the corresponding linear differential operator has the Maximally Unipotent Monodromy property (MUM), which means that all its indicial exponents (at the origin) of the operator are equal (see, for instance, [22,32]).

This result is reminiscent of the well-known ${}_{4}F_3([1/5,2/5,3/5,4/5],[1,1,1],x)$ Candelas et al. hypergeometric series emerging in [3] for a particular Calabi–Yau manifold. The simplest Calabi–Yau series (see, for instance, [18]) are ${}_{4}F_3$ hypergeometric series like ${}_{4}F_2([1/2,1/2,1/2,1/2],[1,1,1],x)$ or ${}_{4}F_2([1/5,2/5,3/5,4/5],[1,1,1],x)$ (see equation 3.11 in [3]).

Let us recall that Calabi-operators [22], annihilating the Calabi–Yau series [18], are (self-adjoint) order-four linear differential operators that have the Maximally Unipotent Monodromy property (MUM) at $x=0$: if one considers their formal series expansions at $x=0$, among the four formal series expansions, one is analytic (it actually corresponds to our diagonals of rational functions); another one is a formal series with a $ln{\left(x\right)}^1$, another one is a formal series with a $ln{\left(x\right)}^2$, and the last one is a formal series with a $ln{\left(x\right)}^3$. Accordingly, with ((26) yielding (27)), one would expect that the diagonal of rational function representation of a Calabi–Yau series (the solution of an order-four linear differential operator) should require at least five variables for the rational function.

2.5.2. Order of the Linear Differential Operator and Degree in the Variables

Let us now consider the diagonal of the following rational function of three variables

$$\begin{array}{ccc}\hfill & & \frac{1}{1-x-\alpha y-z^2},\hfill \end{array}$$

(28)

whose diagonal writes as a simple ${}_{4}F_3$ hypergeometric solution:

$$\begin{array}{ccc}\hfill & & {}_{4}F_3\left([\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}],[1,\frac{1}{2},\frac{1}{2}],\frac{5^5}{2^4}·{\alpha }^2·x^2\right).\hfill \end{array}$$

(29)

In contrast with example (26), here, we just need, for the rational function, three variables, instead of the expected five variables. Note, however, that the order-four linear differential operator $L_4$, annihilating this hypergeometric solution (29), does not have MUM. As usual, this order-four linear differential operator is homomorphic to it adjoint with a very simple order-two intertwiner:

$$\begin{array}{ccc}\hfill & & L_4·\left(x{D}_x^2+D_x\right)=\left(x{D}_x^2+D_x\right)·adjoint\left(L_4\right).\hfill \end{array}$$

(30)

One thus expects [42] this order-four linear differential operator $L_4$ to have a symplectic differential Galois group included in $Sp(4,\mathbb{C})$. Actually, the exterior square of this order-four operator $L_4$ has a simple rational function solution [42], namely, $1/x/(5^5·x^2-2^4)$.

Let us now consider the diagonal of the following rational function of three variables:

$$\begin{array}{ccc}\hfill & & \frac{1}{1-x-\alpha y-z^3}.\hfill \end{array}$$

(31)

The linear differential operator annihilating this diagonal is an order-six linear differential operator with a quite simple ${}_{6}F_5$ hypergeometric solution:

$$\begin{array}{ccc}\hfill & & {}_{6}F_5\left([\frac{1}{7},\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}],[1,\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3}],\frac{7^7}{3^6}·{\alpha }^3·x^3\right).\hfill \end{array}$$

(32)

Let us restrict ourselves to $\alpha =1$. The order-six linear differential operator, annihilating the diagonal of (31), does not have MUM. One has three series analytic at $x=0$: one of the form $x·(1+2377375/6561x^3+\cdots )$, one of the form $x^2·(1+16509584/32805x^3+\cdots )$, and the third one being the diagonal of the rational function which is the expansion of (32):

$$\begin{array}{ccc}\hfill & & 1+140x^3+84084x^6+64664600x^9+55367594100x^{12}+50356110752640x^{15}\hfill \\ & & +47606217704845800x^{18}+46236665756994672960x^{21}+\cdots \hfill \end{array}$$

(33)

One also has three other formal series solutions with a $ln{\left(x\right)}^1$ but no $ln{\left(x\right)}^2$ or $ln{\left(x\right)}^3$.

As usual, this order-six linear differential operator is homomorphic to its adjoint with a very simple order-four intertwiner:

$$\begin{array}{ccc}\hfill & & L_6·\left(x^2{D}_x^4+4x{D}_x^3+2D_x^2\right)=\left(x^2{D}_x^4+4x{D}_x^3+2D_x^2\right)·adjoint\left(L_6\right).\hfill \end{array}$$

(34)

One expects [42] this order-six linear differential operator $L_6$ to have a symplectic differential Galois group included in $Sp(6,\mathbb{C})$. Actually, the exterior square of this order-six linear differential operator $L_6$ has a simple rational function solution [42], namely, $1/x/(7^7·x^3-3^6)$.

Remark 2.

This result can be generalized. Let us consider the rational function:

$$\begin{array}{ccc}\hfill & & \frac{1}{1-x-y-z^n}.\hfill \end{array}$$

(35)

The linear differential operator $L_{2n}^{\left(1\right)}$, annihilating this diagonal, is an order-$\left(2n\right)$ linear differential operator with a quite simple ${}_{2n}F_{2n-1}$ hypergeometric solution:

$$\begin{array}{ccc}\hfill & & {}_{2n}F_{2n-1}([\frac{1}{2n+1},\frac{2}{2n+1},\frac{3}{2n+1},\cdots ,\frac{2n}{2n+1}],\hfill \\ & & [1,\frac{1}{n},\frac{1}{n},\frac{2}{n},\frac{2}{n},\cdots ,\frac{n-1}{n},\frac{n-1}{n}],\frac{{(2n+1)}^{(2n+1)}}{n^{2n}}·x^n).\hfill \end{array}$$

(36)

Let us also consider the linear differential operators $L_{2n}^{\left(m\right)}$ annihilating the diagonal of the rational function:

$$\begin{array}{ccc}\hfill & & {\left(\frac{1}{1-x-y-z^n}\right)}^m.\hfill \end{array}$$

(37)

One finds (using the Homomorphisms command in Maple) the following homomorphisms between successive linear differential operators $L_{2n}^{\left(m\right)}$:

$$\begin{array}{ccc}\hfill & & Homomorphisms\left(L_{2n}^{\left(m\right)},L_{2n}^{(m+1)}\right)=(2n+1)·x·D_x+m·n.\hfill \end{array}$$

(38)

In other words, one has the relations:

$$\begin{array}{ccc}\hfill & & L_{2n}^{(m+1)}·\left((2n+1)·\theta +m·n\right)=Z_1\left(m\right)·L_{2n}^{\left(m\right)},\hfill \end{array}$$

(39)

where $Z_1\left(m\right)$ is an order-one linear differential operator. The linear differential operator $L_{2n}^{\left(1\right)}$ is simply homomorphic to its adjoint:

$$\begin{array}{ccc}\hfill & & Homomorphisms\left(adjoint\left(L_{2n}^{\left(1\right)}\right),L_{2n}^{\left(1\right)}\right)=\hfill \\ \hfill & & \frac{1}{x^{n-1}}·{\theta }^2·{(\theta -1)}^2·{(\theta -2)}^2·{(\theta -3)}^2\cdots {\left(\theta -(n-2)\right)}^2.\hfill \end{array}$$

(40)

Remark 3.

With the previous, rather simple, examples we see that the order of the linear differential operator annihilating the diagonal of a rational function is not related to the number of variables of the rational function (or even to the number of “effective” variables see Section 2.4). Furthermore, a given globally bounded series can be seen to be the diagonals of an infinite number of rational functions of a certain number of variables, but also, in the same time, of other infinite number of rational functions with a different number of variables. For a given globally bounded series, we can find the (minimal order) linear differential operator annihilating this series. Having this (minimal order) linear differential operator, the question is: can we find the minimal number of variables necessary to see this globally bounded series as the diagonal of a rational function of that number of variables? We will address these questions in a forthcoming paper.

3. Diagonals of Rational Functions: Should We Only Consider Rational Functions of the Form 1/Q?

With $P$ and $Q$ multivariate polynomials (with $Q\left(0\right)\ne 0$), the diagonals of the rational functions $P/Q^k$ are, for fixed polynomial $Q$, and for arbitrary integer $k$, a finite dimensional vectorial space related, as shown by Christol [44,45], to the de Rham cohomology (we are thankful to P. Lairez for having clarified this point). There are so many cohomologies in mathematics. For non-mathematicians, let us just say that the introduction of a cohomology often amounts to seeing that “something” you expect, at first sight, to be infinite, for instance, the number of solutions of a system of PDE’s (partial differential equations), is in fact a finite set (for instance for D-finite systems of PDE’s). For physicists, not familiar with de Rham cohomology, let us just say that this can be seen as a consequence of the fact that these $P/Q^k$ rational functions are solutions of D-finite systems, which means that these systems of PDE’s have a finite set of solutions of the form $P/Q^k$. Being in such a “finite box” will force the telescopers of the diagonals of $P/Q^k$ and $1/Q$ to be related (by homomorphisms). This requires one to find a “cyclic vector” in mathematicians wording.

Experimentally, if one considers the (minimal order) linear differential operators for the diagonal of $P/Q^k$ and for the diagonal of $1/Q$, these two linear differential operators are actually homomorphic. Note that this experimental result, valid for diagonals (i.e., integrals over vanishing cycles), is no longer valid for telescopers of rational functions with analytic solutions corresponding to “periods”, $n$-fold integrals, over non-vanishing cycles. In this case, we have a slight generalization of that homomorphism between telescopers $P/Q^k$ and telescopers $1/Q$, which will be described in the sequel (see Section 5.2 below).

It is true that the analysis of lattice Green functions (LGF) [65,66,67,68] in physics naturally yields to diagonals of rational functions in the form $R=1/Q$, where $Q$ is a polynomial. However, the other $n$-fold integrals, emerging in physics, are much more complex (for instance, the ${\chi }^{\left(n\right)}$ terms of the susceptibility of the two-dimensional Ising model [26]). The lattice Green functions [32,65,66,67,68,69] and some Occam’s razor simplicity argument are not sufficient to justify a bias of studying, quite systematically, rational functions of the form $R=1/Q$ (as we often do). In fact, these de Rham cohomology arguments are the reason why, for diagonals (and diagonals only), one can focus only on rational functions in the form $R=1/Q$, but since these arguments may look too esoteric for physicists, let us, in a learn-by-example, pedagogical approach, provide examples showing that telescopers of rational functions in the form $R=1/Q^k$ are homomorphic to telescopers of rational functions in the form $R=1/Q$ and then that telescopers of rational functions in the form $R=P/Q$ are homomorphic to telescopers of rational functions in the form $R=1/Q$.

3.1. Diagonals of Rational Functions: $R=1/Q^k$ Reducing to $1/Q$

Let us denote $Q$ the polynomial:

$$\begin{array}{ccc}\hfill & & Q=1+xy+yz+zx+3·(x^2+y^2+z^2).\hfill \end{array}$$

(41)

Let us denote $L_4^{\left(n\right)}$ the telescopers of $Diag(1/Q^n)$

$$\begin{array}{ccc}\hfill & & L_4^{\left(n\right)}·Diag\left(\frac{1}{Q^n}\right)=0.\hfill \end{array}$$

(42)

One remarks that these telescopers are all of order four. One actually finds the following homomorphisms between successive telescopers (42)

$$\begin{array}{ccc}\hfill & & Homomorphisms\left(L_4^{\left(n\right)},L_4^{(n+1)}\right)=3x·D_x+2n.\hfill \end{array}$$

(43)

In other words, one has the relations:

$$\begin{array}{ccc}\hfill & & L_4^{(n+1)}·(3\theta +2n)=Z_1\left(n\right)·L_4^{\left(n\right)},\hfill \end{array}$$

(44)

where $Z_1\left(n\right)$ is an order-one linear differential operator, with the intertwining relation (44) yielding:

$$\begin{array}{ccc}\hfill & & L_4^{(n+1)}·(3\theta +2n)\cdots (3\theta +6)·(3\theta +4)·(3\theta +2)\hfill \\ \hfill & & =Z_1\left(n\right)\cdots Z_1\left(3\right)·Z_1\left(2\right)·Z_1\left(1\right)·L_4^{\left(1\right)}.\hfill \end{array}$$

(45)

One deduces:

$$\begin{array}{ccc}\hfill & & 2^n·n!·Diag\left(\frac{1}{Q^{n+1}}\right)\hfill \\ \hfill & & =(3\theta +2n)\cdots (3\theta +6)·(3\theta +4)·(3\theta +2)·Diag\left(\frac{1}{Q}\right).\hfill \end{array}$$

(46)

In other words, the diagonal of $1/Q^{n+1}$ can be simply deduced from the diagonal of $1/Q$.

Remark 4.

The product $(3\theta +2n)\cdots (3\theta +6)·(3\theta +4)·(3\theta +2)$ in the intertwining relation (45) is in fact a direct sum:

$$\begin{array}{ccc}\hfill & & (3\theta +6)·(3\theta +4)·(3\theta +2)\hfill \\ \hfill & & =27x^3·LCLM\left((3\theta +6),(3\theta +4),(3\theta +2)\right).\hfill \end{array}$$

(47)

One has, for instance, the relations:

$$\begin{array}{ccc}\hfill & & 2·Diag\left(\frac{1}{Q^2}\right)=(3\theta +2)·Diag\left(\frac{1}{Q}\right)\hfill \\ \hfill & & 8·Diag\left(\frac{1}{Q^3}\right)=(3\theta +4)·(3\theta +2)·Diag\left(\frac{1}{Q}\right)\hfill \\ \hfill & & 48·Diag\left(\frac{1}{Q^4}\right)=(3\theta +6)·(3\theta +4)·(3\theta +2)·Diag\left(\frac{1}{Q}\right)\hfill \\ \hfill & & 384·Diag\left(\frac{1}{Q^5}\right)=(3\theta +8)·(3\theta +6)·(3\theta +4)·(3\theta +2)·Diag\left(\frac{1}{Q}\right).\hfill \end{array}$$

(48)

Of course, since the telescoper of $Diag\left(\frac{1}{Q}\right)$ is an order-four linear differential operator, the order-$(k-1)$ product in front of $Diag\left(\frac{1}{Q}\right)$ in (48) can be, for $Diag\left(\frac{1}{Q^k}\right)$, reduced to an order-three linear differential operator (the simple products $(3\theta +2·(k-1\left)\right)\cdots (3\theta +4)·(3\theta +2)$ in (48) being taken “modulo” $L_4$, for $k\ge 5$).

3.2. Diagonals of Rational Functions: $R=P/Q$ Reducing to $1/Q$

Experimentally, one finds, quite often, that the telescoper of a rational function of the form $R=P/Q$ and the telescoper of the simple rational function $1/Q$ with its numerator normalized to $1$ are homomorphic. The intertwiner $M$ occurring in the homomorphisms of these two telescopers yields a relation of the form

$$\begin{array}{c}\hfill Diag\left(\frac{P}{Q}\right)=M·Diag\left(\frac{1}{Q}\right),\end{array}$$

(49)

yielding the prejudice that the diagonals of the rational functions of the form $P/Q$ should reduce to the “simplest” diagonal, namely, $Diag(1/Q)$. In fact, things are slightly more subtle, as will be seen below. In fact, one is looking for a cyclic vector, and the cyclic vector is not necessarily $Diag(1/Q)$ (see relation (58) and (59) below).

Sticking with the polynomial (41), one has

$$\begin{array}{c}\hfill L_4^{\left(1\right)}·Diag\left(\frac{1}{Q}\right)=0,\end{array}$$

(50)

and considering the diagonal of $xy/Q$, one obtains an order-five differential operator with unique factorization:

$$\begin{array}{c}\hfill L_4^{\left(xy\right)}·D_x·Diag\left(\frac{xy}{Q}\right)=0.\end{array}$$

(51)

The homomorphisms between $L_4^{\left(1\right)}$ and $L_4^{\left(xy\right)}$ amount to seeking for linear differential operators that map the solutions of one differential operator into the other. These relations are

$$\begin{array}{c}\hfill L_4^{\left(xy\right)}·Q_3=K_3·L_4^{\left(1\right)},\end{array}$$

(52)

and

$$\begin{array}{c}\hfill L_4^{\left(1\right)}·J_3=P_3·L_4^{\left(xy\right)},\end{array}$$

(53)

where the intertwiners $Q_3$, $K_3$, $J_3$, and $P_3$ are linear differential operators of order three.

Note that the above two relations show [23] that the linear differential operator $J_3·Q_3$ (resp. $Q_3·J_3$) leaves the solutions of $L_4^{\left(1\right)}$ (resp. $L_4^{\left(xy\right)}$) unchanged,

$$\begin{array}{c}\hfill J_3·Q_3·Diag\left(\frac{1}{Q}\right)=Diag\left(\frac{1}{Q}\right)\\ \hfill =1-195x^2+135225x^4-143647728x^6+182699446545x^8\\ \hfill -252437965534755x^{10}+364803972334074000x^{12}+\cdots \end{array}$$

(54)

and:

$$\begin{array}{ccc}\hfill & & Q_3·J_3·D_x·Diag\left(\frac{xy}{Q}\right)=D_x·Diag\left(\frac{xy}{Q}\right)\hfill \\ & & =16x-38400x^3+71593536x^5-126120445440x^7\hfill \end{array}$$

(55)

$$\begin{array}{ccc}& & +218901889206000x^9-378463218115207680x^{11}+\cdots \hfill \end{array}$$

(56)

Equivalently, the adjoint of $P_3·K_3$ (resp. the adjoint of $K_3·P_3$) leaves the solutions of the adjoint of $L_4$ (resp. the adjoint of $L_4^{\left(xy\right)}$) unchanged.

Introducing the differential operator $D_x$ on both sides of (53), and using (51), one obtains:

$$\begin{array}{c}\hfill L_4^{\left(1\right)}·J_3·D_x·Diag\left(\frac{xy}{Q}\right)=P_3·(L_4^{\left(xy\right)}·D_x)·Diag\left(\frac{xy}{Q}\right).\end{array}$$

(57)

The RHS of (57) cancels and, therefore, the LHS of (57), according to (50), leads to

$$\begin{array}{ccc}\hfill & & Diag\left(\frac{1}{Q}\right)=J_3·D_x·Diag\left(\frac{xy}{Q}\right).\hfill \end{array}$$

(58)

Also, acting by both sides of (52) on $Diag(1/Q)$, using (50) and keeping (51) in mind leads to:

$$\begin{array}{ccc}\hfill & & D_x·Diag\left(\frac{xy}{Q}\right)=Q_3·Diag\left(\frac{1}{Q}\right).\hfill \end{array}$$

(59)

With these relations, we see that the derivative of the diagonal of $xy/Q$ simply reduces to the diagonal of $1/Q$, but the diagonal of $xy/Q$ does not simply reduce to the diagonal of $1/Q$. Here, $1/Q$ is not the “cyclic vector”.

4. Diagonals of Algebraic Functions

4.1. Diagonals of Algebraic Functions: A First Example

Let us consider the algebraic functions:

$$\begin{array}{ccc}\hfill & & A(x,y)=\frac{1}{{\left(1-\alpha ·(x+y)\right)}^{1/n}}n=2,3,\cdots \hfill \end{array}$$

(60)

The telescopers of these algebraic functions are order-two linear differential operators with the simple ${}_{2}F_1$ hypergeometric solution:

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{2n},\frac{n+1}{2n}],\left[1\right],4·{\alpha }^2·x\right)\hfill \\ \hfill & & =1+\frac{n+1}{n^2}{\alpha }^{2}x+\frac{(1+n)·(1+2n)·(1+3n)}{4·n^4}{\alpha }^4{x}^2+\cdots \hfill \end{array}$$

(61)

Note that, among these ${}_{2}F_1$ hypergeometric functions, the $n=2$, $n=3$, $n=4$, and $n=6$ cases correspond to modular forms (see Appendix B in [16]).

These hypergeometric series can be seen to be, as they should, the diagonals of the algebraic functions (60). In particular, for $n=2$, one obtains:

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{4},\frac{3}{4}],\left[1\right],4·{\alpha }^2·x\right)\hfill \\ \hfill & & ={\left(\frac{1}{1-3{\alpha }^{2}x}\right)}^{1/4}·{}_{2}F_1\left([\frac{1}{12},\frac{5}{12}],\left[1\right],\frac{27}{4}·\frac{{\alpha }^4·x^2·(1-4{\alpha }^{2}x)}{{(1-3{\alpha }^{2}x)}^3}\right)\hfill \\ \hfill & & =1+\frac{3}{4}{\alpha }^{2}x+\frac{105}{64}{\alpha }^4{x}^2+\frac{1155}{256}{\alpha }^6{x}^3+\frac{225225}{16384}{\alpha }^8{x}^4+\cdots \hfill \end{array}$$

(62)

For $n=2$, it is natural to associate the denominator of (60) with the algebraic surface

$$\begin{array}{ccc}\hfill & & z^2=1-\alpha ·(x+y),\hfill \end{array}$$

(63)

and, following ideas developed in [40], since calculating the diagonal of the function (60) for $n=2$ amounts, in the multi-Taylor expansion, to extracting the terms depending only on the product $p=xy$, take the intersection of the algebraic surface (63) with the surface $p=xy$. This amounts, for instance, to eliminating $y=p/x$ in (63), thus obtaining the algebraic curve

$$\begin{array}{ccc}\hfill & & -\alpha ·x^2-x{z}^2-\alpha ·p+x=0,\hfill \end{array}$$

(64)

which turns out to be an elliptic curve (genus-one). Calculating the j-invariant of the elliptic curve (64), one deduces the following Hauptmodul

$$\begin{array}{ccc}\hfill & & \mathcal{H}=\frac{1728}{j}=\frac{27}{4}·\frac{{\alpha }^4·p^2·(1-4{\alpha }^{2}p)}{{(1-3{\alpha }^{2}p)}^3},\hfill \end{array}$$

(65)

which is actually the Hauptmodul pullback in (62). This example gives some hope that the effective algebraic geometry approach of diagonals of rational functions, detailed in [40], could also work with diagonals of algebraic functions.

For $n\ne 2$, it is tempting to associate the denominator of (60) with the algebraic surface

$$\begin{array}{ccc}\hfill & & z^n=1-\alpha ·(x+y),\hfill \end{array}$$

(66)

and after the elimination $y=p/x$ in (63), the algebraic curve

$$\begin{array}{ccc}\hfill & & -\alpha ·x^2-x{z}^n-\alpha ·p+x=0,\hfill \end{array}$$

(67)

but such algebraic curves turn out to be of the genus $g=n-1$. Understanding the emergence of modular forms for the $n=3$, $n=4$, and $n=6$ subcases of (61) from, respectively, genus $2$, $3$, and $5$ algebraic curves is an open (and challenging) problem.

Remark 5.

From the definition of the diagonals of rational or algebraic functions, it is straightforward to see that the diagonals of the algebraic functions (60) are series of the variable ${\alpha }^{2}x$. Consequently, the previous calculations for a particular value of $\alpha$ are sufficient to recover the previous results valid for arbitrary $\alpha$. For that reason, we will, in the next example, take specific values of the parameters.

4.2. Diagonals of Algebraic Functions: A Second Example

Let us consider the algebraic functions:

$$\begin{array}{ccc}\hfill & & A(x,y)=\frac{1}{{\left(1-3·(x+y)+5·(x^2+y^2)\right)}^{1/n}},n=2,3,\cdots \hfill \end{array}$$

(68)

For $n=2$, the telescoper of the algebraic function (68) is an order-two linear differential operator with the pullbacked ${}_{2}F_1$ hypergeometric solution:

$$\begin{array}{ccc}\hfill & & \frac{1}{{(1-30x)}^{1/2}}·{}_{2}F_1\left([\frac{1}{4},\frac{3}{4}],\left[1\right],-\frac{4·(11-200x)·x}{{(1-30x)}^2}\right)\hfill \\ \hfill & & =\frac{1}{{(1-27x+300x^2)}^{1/4}}\hfill \\ \hfill & & \times {}_{2}F_1\left([\frac{1}{12},\frac{5}{12}],\left[1\right],\frac{27}{4}·\frac{x^2·{(11-200x)}^2·(1-16x+100x^2)}{{(1-27x+300x^2)}^3}\right)\hfill \\ \hfill & & =1+\frac{27}{4}x+\frac{4305}{64}{x}^2+\frac{199395}{256}{x}^3+\frac{167040825}{16384}{x}^4+\cdots \hfill \end{array}$$

(69)

With multi-Taylor series expansion, it is straightforward to see that the hypergeometric series is actually the diagonal of the algebraic function (68) for $n=2$.

As in the previous subsection, we introduce the algebraic surface

$$\begin{array}{ccc}\hfill & & z^2=1-3·(x+y)+5·(x^2+y^2),\hfill \end{array}$$

(70)

and, again, eliminate $y=p/x$ in (70), thus obtaining the algebraic curve

$$\begin{array}{ccc}\hfill & & 5x^4-x^2{z}^2-3x^3+5p^2-3px+x^2=0,\hfill \end{array}$$

(71)

which turns out to be an elliptic curve (genus-one). Calculating the j-invariant of the elliptic curve (71), one deduces the following Hauptmodul:

$$\begin{array}{ccc}\hfill & & \mathcal{H}=\frac{1728}{j}=\frac{27}{4}·\frac{p^2·{(11-200p)}^2·(1-16p+100p^2)}{{(1-27p+300p^2)}^3},\hfill \end{array}$$

(72)

which is actually the Hauptmodul pullback in (69). Again, this last example gives some hope that the effective algebraic geometry approach of diagonals of rational functions, detailed in [40], could also work with diagonals of algebraic functions. For $n\ne 2$, it is tempting to introduce the algebraic surface

$$\begin{array}{ccc}\hfill & & z^n=1-3·(x+y)+5·(x^2+y^2),\hfill \end{array}$$

(73)

and, again, eliminate $y=p/x$ in (70), thus obtaining the algebraic curve

$$\begin{array}{ccc}\hfill & & 5x^4-x^2{z}^n-3x^3+5p^2-3px+x^2=0,\hfill \end{array}$$

(74)

which is an algebraic curve of genus $g=2n-3$ for $n$ even, and $g=2n-2$ for $n$ odd. For $n=3$ (genus $4$), the telescoper of the algebraic function (68) is an (irreducible) order-three linear differential operator that is not homomorphic to its adjoint. The interpretation of such non-self-dual order-three linear differential operators from these higher genus algebraic curves is a totally open problem.

5. Understanding the Emergence of Selected Differential Galois Groups for Diagonals of Rational Functions

Experimentally, one finds that almost all the linear differential operators annihilating the diagonal of a rational or algebraic function are homomorphic to their adjoint [41]. For instance, recalling an example in [41]

$$\begin{array}{ccc}\hfill & & {}_{4}F_3\left([\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3}],[\frac{1}{2},1,1],\frac{729}{4}·x\right)=Diag\left(\frac{1}{1-(1+u)·(x+y+z)}\right)\hfill \\ \hfill & & =1+18x+1350x^2+\cdots \hfill \end{array}$$

(75)

we find the corresponding order-four linear differential operator

$$\begin{array}{ccc}\hfill & & x·L_4=2·x·{(3\theta +2)}^2·{(3\theta +1)}^2-81·{\theta }^3·(2\theta -1),\hfill \end{array}$$

(76)

which can be seen to be non-trivially homomorphic to its adjoint:

$$\begin{array}{ccc}\hfill & & L_4·\left(\theta +\frac{1}{2}\right)=\left(\theta +\frac{1}{2}\right)·adjoint\left(L_4\right).\hfill \end{array}$$

(77)

Beyond diagonals of rational or algebraic functions, one also finds experimentally that almost all the telescopers of rational or algebraic functions are homomorphic to their adjoint. This homomorphism to the adjoint property is so systematic that, following a mathematician’s prejudice, one can imagine that this is nothing but the Poincaré duality. The Poincaré duality [43] works for any algebraic variety: the diagonal of any rational or algebraic function should yield self-dual linear differential operators in the sense that they are homomorophic to their adjoint. This is not the case. It turns out that the linear differential operators of some ${}_{n}F_{n-1}$, candidates to rule-out Christol’s conjecture [44,45,48], precisely provide such rare examples of linear differential operators annihilating the diagonal of rational or algebraic functions that are not homomorphic to their adjoint. Among these candidates, a large set has been seen to actually be composed of diagonals of rational or algebraic functions [48,70].

5.1. A Recall on Christol’s Conjecture

Let us recall one of the ${}_{3}F_2$ hypergeometric candidates introduced to rule out Christol’s conjecture:

$$\begin{array}{ccc}\hfill & & {}_{3}F_2\left([\frac{2}{9},\frac{5}{9},\frac{8}{9}],[\frac{2}{3},1],27·x\right)\hfill \\ \hfill & & =1+\frac{40}{9}·x+\frac{5236}{81}·x^2+\frac{7827820}{6561}·x^3+\frac{1444588600}{59049}·x^4+\cdots \hfill \end{array}$$

(78)

It is a globally bounded series (change $x\to 3^3·x$ to obtain a series with integer coefficients). In fact, it actually corresponds [48] to the diagonal of the algebraic function:

$$\begin{array}{ccc}\hfill & & \frac{{(1-y-z)}^{1/3}}{1-x-y-z}.\hfill \end{array}$$

(79)

The telescoper of the algebraic function (79) is the order-three linear differential operator, which has (78) as a solution. This order-three linear differential operator is not homomorphic to its adjoint. We have a $SL(3,\mathbb{C})$ differential Galois group.

Other similar examples are, for instance,

$$\begin{array}{ccc}\hfill & & {}_{3}F_2\left([\frac{1}{9},\frac{4}{9},\frac{7}{9}],[\frac{2}{3},1],27·x\right)=Diag\left(\frac{{(1-y-2z)}^{2/3}}{1-x-y-z}\right),\hfill \end{array}$$

(80)

or

$$\begin{array}{ccc}\hfill & & {}_{3}F_2\left([\frac{2}{9},\frac{5}{9},\frac{8}{9}],[\frac{5}{6},1],27·x\right)=Diag\left(\frac{{(1-y-2z)}^{1/3}}{1-x-y-z}\right),\hfill \end{array}$$

(81)

or even the ${}_{4}F_3$ hypergeometric function:

$$\begin{array}{ccc}\hfill & & {}_{4}F_3\left([\frac{2}{9},\frac{5}{9},\frac{8}{9},\frac{1}{2}],[\frac{1}{3},\frac{5}{6},1],27·x\right)=Diag\left(\frac{{(1-x)}^{1/3}}{1-x-y-z}\right).\hfill \end{array}$$

(82)

Again, these three diagonals, (80), (81), and (82), are solutions of telescopers that are not homomorphic to their adjoint.

These examples are taken in a list of 116 potential counter-examples constructed in 2011 by Bostan et al. [25]. Note that, more recently, 38 cases in that list of 116 have actually been found to be diagonals of algebraic functions [70]. The two relations (80) and (81) can be generalized [70,71] as follows:

$$\begin{array}{ccc}\hfill & & {}_{4}F_3([\frac{1-(R+S)}{3},\frac{2-(R+S)}{3},\frac{3-(R+S)}{3},\frac{1-S}{2}],\hfill \\ \hfill & & [\frac{1-(R+S)}{2},\frac{2-(R+S)}{2},1],27·x)\hfill \\ \hfill & & =Diag\left(\frac{{(1-x)}^R·{(1-x-2y)}^S}{1-x-y-z}\right),\hfill \end{array}$$

(83)

where $R$ and $S$ are rational numbers. These diagonals are annihilated by the order-four linear differential operator:

$$\begin{array}{ccc}\hfill & & 2·x·(S-1-2\theta )·(S+R-3\theta )·(S+R-1-3\theta )·(S+R-2-3\theta )\hfill \\ \hfill & & -{\theta }^2·(S+R+1-2\theta )·(S+R-2\theta ).\hfill \end{array}$$

(84)

This order-four linear differential operator is not homomorphic to its adjoint. Other more involved similar relations can be found in section 2.1 of chapter 2 of [70].

Experimentally, we found, after quite systematic calculations of thousands of telescopers of rational or algebraic functions, that the telescopers are (almost always) homomorphic to their adjoint, or if they are not irreducible, that each of the factors of these telescopers are homomorphic to their adjoint. Such previous examples like (78), (79), or (80) and (81), curiously related to Christol’s conjecture, provide rare examples of diagonals of algebraic functions such that their corresponding telescopers are not homomorphic to their adjoint. We have similar results with the algebraic function:

$$\begin{array}{ccc}\hfill & & \frac{x^{1/3}}{1-x-y-z}.\hfill \end{array}$$

(85)

In order to understand this “duality-breaking” (the telescoper is not self-adjoint up to homomorphisms), it is tempting to introduce the (algebraic) function:

$$\begin{array}{ccc}\hfill & & \frac{1}{1-x-y-z-\alpha ·x^{1/3}}.\hfill \end{array}$$

(86)

However, in order to avoid the introduction of rational functions of $n$-th roots of variables, we will introduce the diagonal of the following rational function:

$$\begin{array}{ccc}\hfill & & \frac{1}{1-x^3-y^3-z^3-\alpha ·x}.\hfill \end{array}$$

(87)

5.2. Understanding the Emergence of Selected Differential Galois Groups for Almost All the Diagonal of Rational Functions

The linear differential operator annihilating the diagonal of the rational function (87) is a (quite large) order-eight linear differential operator $L_8\left(\alpha \right)$, depending on the parameter $\alpha$, which is homomorphic to its adjoint with an order-six intertwiner. This order-eight linear differential operator $L_8\left(\alpha \right)$ is irreducible except at $\alpha =0$. For $\alpha =1$, $\alpha =2$, and $\alpha =3$ the order-eight linear differential operator $L_8\left(\alpha \right)$ is homomorphic to its adjoint with an order-six intertwiner. The differential Galois group should, thus, be included in $Sp(8,\mathbb{C})$. This is confirmed when calculating [42] the exterior square of $L_8\left(\alpha \right)$. This exterior square has a rational function solution $P_a/x/Q_a$, where the polynomials $P_a$ and $Q_a$ read:

$$\begin{array}{ccc}\hfill & & P_a=(4{\alpha }^3-27)·(20{\alpha }^3-81)+18·(-6561-891{\alpha }^3+500{\alpha }^6)·x^3+1594323x^6,\hfill \\ & & Q_a=387420489x^9-531441·(81+100{\alpha }^3)·x^6\hfill \\ & & +(1594323-2972133{\alpha }^3+729000{\alpha }^6-50000{\alpha }^9)·x^3-27·{(4{\alpha }^3-27)}^2.\hfill \end{array}$$

(88)

Let us now take the $\alpha \to 0$ limit of the order-eight linear differential operator $L_8\left(\alpha \right)$. In this limit, the order-eight linear differential operator just becomes the direct-sum

$$\begin{array}{ccc}\hfill & & L_2\oplus L_3\oplus M_3,\hfill \end{array}$$

(89)

where the order-two linear differential operator $L_2$ has the ${}_{2}F_1$ hypergeometric solution

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right],27x^3\right),\hfill \end{array}$$

(90)

where the order-three linear differential operator $L_3$ has the ${}_{3}F_2$ hypergeometric function solution

$$\begin{array}{ccc}\hfill & & {}_{3}F_2\left([\frac{5}{9},\frac{8}{9},\frac{11}{9}],[\frac{2}{3},1],27x^3\right),\hfill \end{array}$$

(91)

and where the order-three linear differential operator $M_3$ has the ${}_{3}F_2$ hypergeometric function solution:

$$\begin{array}{ccc}\hfill & & {}_{3}F_2\left([\frac{7}{9},\frac{10}{9},\frac{13}{9}],[\frac{1}{3},1],27x^3\right).\hfill \end{array}$$

(92)

These two order-three linear differential operators, similarly to the previous example (78), are not homomorphic to their adjoint: they break the self-adjoint duality (up to homomorphisms of operators) and thus have a $SL(3,\mathbb{C})$ differential Galois group.

These two hypergeometric series are exactly on the same footing as (78): they are globally bounded series (just change $x^3\to 3^3{x}^3$ in order to obtain a series with integer coefficients), and their respective order-three linear differential operators are not homomorphic to their adjoint, their differential Galois group being $SL(3,\mathbb{C})$. Let us note, however, that the order-three linear differential operator $L_3$ is actually homomorphic to the adjoint of $M_3$, and of course the order-three linear differential operators $M_3$ is homomorphic to the adjoint of $L_3$.

If, from an algebraic geometry perspective [40], one sees the fact that all our linear differential operators, annihilating diagonals of rational functions, are homomorphic to their adjoint as the differential algebra expression of the Poincaré duality on the algebraic varieties corresponding to the denominators of our rational functions [40], the fact that this Poincaré duality is broken for $L_3$ or $M_3$ is, in fact, restored in the bigger picture (87) with the linear differential order-eight operator. In the $\alpha \to 0$ limit, we see that these two linear differential operators breaking the duality actually emerge in a dual pair, thus restoring the duality. For instance, if one focuses on $L_6=L_3\oplus M_3$ in (89), one finds easily that this order-six linear differential operator is homomorphic to its adjoint. Its exterior square has the following rational function solution:

$$\begin{array}{ccc}\hfill & & \frac{4+621x^3}{{(1-27x^3)}^3·x}.\hfill \end{array}$$

(93)

Since these calculations are in the $\alpha \to 0$ limit, let us expand, in $\alpha$, the rational function (87):

$$\begin{array}{ccc}\hfill & & \frac{1}{1-x^3-y^3-z^3-\alpha ·x}=\frac{1}{1-x^3-y^3-z^3}+\frac{x}{{(1-x^3-y^3-z^3)}^2}·\alpha \hfill \\ & & +\frac{x^2}{{(1-x^3-y^3-z^3)}^3}·{\alpha }^2+\frac{x^3}{{(1-x^3-y^3-z^3)}^4}·{\alpha }^3\hfill \\ & & +\frac{x^4}{{(1-x^3-y^3-z^3)}^5}·{\alpha }^4+\cdots \hfill \end{array}$$

(94)

The diagonal of a sum is clearly the sum of the diagonals. Thus, the diagonal of the LHS of (94) will be the sum of the various rational function terms in ${\alpha }^n$ in the RHS of (94). The diagonal of the ${\alpha }^1$ term in the $\alpha$-expansion (94)

$$\begin{array}{ccc}\hfill & & \frac{x}{{(1-x^3-y^3-z^3)}^2},\hfill \end{array}$$

(95)

is clearly equal to zero since the diagonal extracts, in the multi-Taylor series, the terms in the product $p=xyz$, or, in this case, the terms in the product $x^3{y}^3{z}^3$. Similarly, the diagonal of the ${\alpha }^2$ term in the $\alpha$-expansion (94)

$$\begin{array}{ccc}\hfill & & \frac{x^2}{{(1-x^3-y^3-z^3)}^3},\hfill \end{array}$$

(96)

is also zero, but the diagonal of the ${\alpha }^3$ term

$$\begin{array}{ccc}\hfill & & \frac{x^3}{{(1-x^3-y^3-z^3)}^4},\hfill \end{array}$$

(97)

is not zero. Actually, this last diagonal reads:

$$\begin{array}{ccc}\hfill & & -\frac{1}{9}·\frac{1+216x^3}{{(1-27x^3)}^3}·x·\frac{d}{dx}{}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right],27x^3\right)\hfill \\ & & -18·\frac{x^3}{{(1-27x^3)}^2}·{}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right],27x^3\right)\hfill \\ & & =-20x^3-1680x^6-92400x^9-4204200x^{12}-171531360x^{15}+\cdots \hfill \end{array}$$

(98)

and is annihilated by an order-two operator $M_2$.

We have a different story with telescopers. Since the telescoper of a sum of rational functions is the direct sum (LCLM) of the telescopers of these rational functions (or at least is a rightdivisor of the LCLM of the telescopers), let us consider the telescopers of the first five terms in the RHS of (94). The telescoper of the first term is, of course, the order-two linear differential operator $L_2$ annihilating the diagonal of this rational function. The telescoper of the second term (in ${\alpha }^1$) is the previous order-three linear differential operator $L_3$. The telescoper of the third term (in ${\alpha }^2$) is exactly the previous $M_3$. The telescoper of the fourth term (in ${\alpha }^3$) is the order-two linear differential operator $M_2$. The telescoper of the sum of the first orders in $\alpha$ in the expansion (94)

$$\begin{array}{ccc}\hfill & & \frac{1}{1-x^3-y^3-z^3}+\frac{x}{{(1-x^3-y^3-z^3)}^2}·\alpha +\frac{x^2}{{(1-x^3-y^3-z^3)}^3}·{\alpha }^2,\hfill \end{array}$$

(99)

is actually the LCLM of the three telescopers $L_2$, $L_3$, and $M_3$ , which is precisely the $\alpha \to 0$ limit of the order-eight linear differential operator.

5.3. Revisiting $1/Q\to P/Q^k$ for Telescopers

The next terms in the $\alpha$-expansion (94), namely, the terms in ${\alpha }^{4+3n}$ with $n=0,1,\cdots$

$$\begin{array}{ccc}\hfill & & \frac{x^{4+3n}}{{(1-x^3-y^3-z^3)}^{5+3n}},\hfill \end{array}$$

(100)

have telescopers homomorphic to the telescoper $L_3$ for (95). Similarly, considering the $\alpha$-expansion (94), namely, the terms in ${\alpha }^{5+3n}$ with $n=0,1,\cdots$

$$\begin{array}{ccc}\hfill & & \frac{x^{5+3n}}{{(1-x^3-y^3-z^3)}^{6+3n}},\hfill \end{array}$$

(101)

have telescopers homomorphic to the telescoper $M_3$ for (96). Finally, the terms in ${\alpha }^{3+3n}$ with $n=0,1,\cdots$

$$\begin{array}{ccc}\hfill & & \frac{x^{3+3n}}{{(1-x^3-y^3-z^3)}^{4+3n}},\hfill \end{array}$$

(102)

have telescopers homomorphic to the telescoper $L_2$, generalizing the result (98) for $n=0$. This last sequence of telescopers can be understood from the ideas sketched in Section 3.1 and Section 3.2 for diagonals (changing, for instance, $(x,y,z)$ into $(x^3,y^3,z^3)$). However, we see that these ideas do not work anymore when we compare the telescopers for (100) (resp. the telescopers for (101)) with the telescopers for (102). These different telescopers are not homomorphic. They correspond to three different sequences of telescopers of a different nature, corresponding to three hypergeometric functions of quite a different nature:

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right],27x^3\right),{}_{3}F_2\left([\frac{7}{9},\frac{10}{9},\frac{13}{9}],[\frac{1}{3},1],27x^3\right),{}_{3}F_2\left([\frac{5}{9},\frac{8}{9},\frac{11}{9}],[\frac{2}{3},1],27x^3\right).\hfill \end{array}$$

Accordingly, similar $\alpha$-dependent examples are sketched in Appendix A.

To sum-up: The ideas sketched in Section 3.1 and Section 3.2 for diagonals can be generalized to telescopers (which may correspond to vanishing cycles, i.e., diagonals), with the caveat that the unique “root” rational function $1/Q$ has to be replaced by a finite set of rational functions ($1/Q_1$, $1/Q_2$, $1/Q_3$ in our previous example).

6. An Infinite Number of Birational Symmetries of the Diagonals and Telescopers

Let us consider the simplest example of the non-trivial diagonal of rational function, namely, the diagonal of the rational function of three variables:

$$\begin{array}{ccc}\hfill & & R(x,y,z)=\frac{1}{1-x-y-z}.\hfill \end{array}$$

(103)

Let us consider the birational transformation $B$:

$$\begin{array}{ccc}\hfill & & B:(x,y,z)⟶\left(x,y·(1+3x+7x^2),\frac{z}{1+3x+7x^2}\right).\hfill \end{array}$$

(104)

It is birational because its compositional inverse is also a rational function:

$$\begin{array}{ccc}\hfill & & (x,y,z)⟶\left(x,\frac{y}{1+3x+7x^2},z·(1+3x+7x^2)\right).\hfill \end{array}$$

(105)

Note that this birational transformation preserves the product $p=xyz$, as well as the neighbourhood of the point $(x,y,z)=(0,0,0)$. This birational transformation is an infinite order transformation. The composition of this transformation $n$ times gives:

$$\begin{array}{ccc}\hfill & & (x,y,z)⟶\left(x,y·{(1+3x+7x^2)}^n,\frac{z}{{(1+3x+7x^2)}^n}\right).\hfill \end{array}$$

(106)

The rational function (103), transformed by the (infinite order) birational transformation (104), reads:

$$\begin{array}{ccc}\hfill & & R_B(x,y,z)=R\left(x,y·(1+3x+7x^2),\frac{z}{1+3x+7x^2}\right)\hfill \\ \hfill & & =\frac{1}{1-x-y·(1+3x+7x^2)-z/(1+3x+7x^2)}.\hfill \end{array}$$

(107)

With the multi-Taylor expansion of (107), one finds easily that the diagonals of (103) and (107) are actually identical.

More generally, let us consider

$$\begin{array}{ccc}\hfill & & B_x:(x,y,z)⟶\left(x,y·Q_1\left(x\right),\frac{z}{Q_1\left(x\right)}\right),\hfill \end{array}$$

(108)

where $Q_1\left(x\right)$ is a rational function (see, however, Section 6.4) with a Taylor expansion such that $Q_1\left(0\right)\ne 0$. One also finds for any such rational function $Q_1\left(x\right)$ that the diagonals of (103) and (107) are actually identical. This can be seen from the multi-Taylor expansion of (107):

$$\begin{array}{ccc}\hfill & & R_B(x,y,z)=\sum _m\sum _n\sum _l{a}_{m,n,l}·x^m·y^n·Q_1{\left(x\right)}^n·z^l·Q_1{\left(x\right)}^{-l}\hfill \\ \hfill & & =\sum _m{a}_{m,m,m}·{\left(xyz\right)}^m+\sum _{(m,n,l)\ne (m,m,m)}{a}_{m,n,l}·x^m·y^n·z^l·Q_1{\left(x\right)}^{n-l}.\hfill \end{array}$$

(109)

The second triple sum can be decomposed into the terms such that $n\ne l$, which cannot contribute to the diagonal (which extracts terms in $p=xyz$ and thus terms in the product $yz$) and the $n=l$ terms (such that the $Q_1{\left(x\right)}^{n-l}$ factor in (109) disappear):

$$\begin{array}{ccc}\hfill & & \sum _{m\ne n}{a}_{m,n,n}·x^m·y^n·z^n.\hfill \end{array}$$

(110)

This last sum (110), which excludes the power of $x$ to be equal to the power of the product $yz$, cannot contribute to the diagonal. We have thus proved that the diagonals of (103) and (107) are equal.

Of course, there is nothing particular with the variable $x$. We can also introduce other birational transformations that single out, respectively, $y$ and $z$:

$$\begin{array}{ccc}\hfill & & B_y:(x,y,z)⟶\left(x·Q_2\left(y\right),y,\frac{z}{Q_2\left(y\right)}\right),\hfill \end{array}$$

(111)

and

$$\begin{array}{ccc}\hfill & & B_z:(x,y,z)⟶\left(x·Q_3\left(z\right),\frac{y}{Q_3\left(z\right)},z\right),\hfill \end{array}$$

(112)

for any rational functions $Q_2\left(x\right)$ and $Q_3\left(x\right)$ with a Taylor expansion such that $Q_2\left(0\right)\ne 0$ and $Q_3\left(0\right)\ne 0$. We can compose these birational transformations (108), (111) and (112), in any order and change the various $Q_1\left(x\right)$, $Q_2\left(x\right)$, and $Q_3\left(x\right)$ at each step. Thus, we obtain quite a large infinite set of birational transformations preserving the product $p=xyz$ and the neighbourhood of the point $(x,y,z)=(0,0,0)$. Since the product $p=xyz$ is preserved, let us eliminate (for instance) the variable $z=p/x/y$. The three previous birational transformations (108), (111), and (112), on the three variables $x,y,z$ become birational transformations depending on a parameter $p$, of only two variables $x,y$:

$$\begin{array}{ccc}\hfill & & {\tilde{B}}_x:(x,y)⟶\left(x,y·Q_1\left(x\right)\right),\hfill \end{array}$$

(113)

$$\begin{array}{ccc}\hfill & & {\tilde{B}}_y:(x,y)⟶\left(x·Q_2\left(y\right),y\right),\hfill \end{array}$$

(114)

and

$$\begin{array}{ccc}\hfill & & {\tilde{B}}_z:(x,y)⟶\left(x·Q_3\left(\frac{p}{xy}\right),y/Q_3\left(\frac{p}{xy}\right)\right).\hfill \end{array}$$

(115)

Composing these birational transformations of two variables (113), (114), and (115), in any order and changing the various $Q_1\left(x\right)$, $Q_2\left(x\right)$, and $Q_3\left(x\right)$ at each step, one obtains that way a quite large subset of the (huge set of) Cremona transformations [49,72].

Remark 6.

Of course there is nothing specific about the particular simple example (103) of the rational function. The previous birational transformations (113), (114), and (115) are symmetries of the diagonals of any rational function of three variables. Furthermore, there is nothing specific about rational function of three variables. We can generalize such birational transformations for the diagonal of the rational function of $n$ variables, for any number of variables $n$.

6.1. Non-Birational Symmetries for Diagonals

6.1.1. Monomial Transformation

Let us consider the (non-birational) monomial transformation:

$$\begin{array}{ccc}\hfill & & M:(x,y,z)⟶\left(x,x^2{y}^2,y{z}^3\right).\hfill \end{array}$$

(116)

By performing this monomial transformation (116) on the rational function (103), one obtains the new rational function:

$$\begin{array}{ccc}\hfill & & R_M(x,y,z)=R\left(x,x^2{y}^2,y{z}^3\right)=\frac{1}{1-x-x^2{y}^2-y{z}^3}.\hfill \end{array}$$

(117)

The calculation of the telescoper of (117) gives an order-two linear differentizal operator that has the ${}_{2}F_1$ hypergeometric series solution:

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right],27x^3\right)=1+6x^3+90x^6+1680x^9+34650x^{12}\hfill \\ \hfill & & +756756x^{15}+17153136x^{18}+\cdots \hfill \end{array}$$

(118)

One verifies easily, that the diagonal of the multi-Taylor expansion of (117), is actually the ${}_{2}F_1$ hypergeometric series (118). This series identifies with the diagonal of (103), where $x$ is changed into $x^3$, by the monomial transformation (116).

6.1.2. Non-Birational Transformation

Let us now consider the non-birational “monomial-like” transformation

$$\begin{array}{ccc}\hfill & & B:(x,y,z)⟶\left(x,x^2{y}^2·(1+3x),\frac{y{z}^3}{1+3x}\right).\hfill \end{array}$$

(119)

By performing this non-birational monomial transformation (119) on the rational function (103), one obtains the new rational function

$$\begin{array}{ccc}\hfill & & R_B(x,y,z)=R\left(x,x^2{y}^2·(1+3x),\frac{y{z}^3}{1+3x}\right)\hfill \\ \hfill & & =\frac{1}{1-x-x^2{y}^2·(1+3x)-y{z}^3/(1+3x)}.\hfill \end{array}$$

(120)

The calculation of the telescoper of (120) gives an order-two linear differential operator that has, again, the ${}_{2}F_1$ hypergeometric series solution:

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right]27x^3\right)=1+6x^3+90x^6+1680x^9\hfill \\ \hfill & & +34650x^{12}+756756x^{15}+17153136x^{18}+\cdots \hfill \end{array}$$

(121)

For the multi-Taylor expansion of (120), one verifies easily that its diagonal is the ${}_{2}F_1$ hypergeometric series (121). This result can be understood from the results on (117) and the diagonal-preservation results on the birational transformations (108), (111), and (112).

Consequently, we have another infinite set of (non-birational) transformations such that the diagonal of a rational function is changed into the diagonal of that rational function where $x$ is changed into $x^N$.

6.2. Birational Symmetries for Telescopers

Recalling the creative telescoping equation (6) and (9), we have verified experimentally, on thousands of examples, that the previous birational transformations generated by (108), (111) and (112), are actually compatible with the creative telescoping equations (6) and (9). Note however, in the birationally transformed creative telescoping equations, that if the telescoper does remain invariant (even if the rational function has not a multi-Taylor expansion), the two “certificates” $U$ and $V$ are transformed in a very involved way (they become quite large rational functions).

6.2.1. Birational Symmetries Not Preserving $(x,y,z)=(0,0,0)$

Let us consider the involutive birational transformation:

$$\begin{array}{ccc}\hfill & & I:(x,y,z)⟶\left(\frac{1}{x},\frac{1}{y},x^2{y}^{2}z\right).\hfill \end{array}$$

(122)

This involutive birational transformation transforms the rational function (103) into:

$$\begin{array}{ccc}\hfill & & R_I(x,y,z)=-\frac{xy}{x^2{y}^{3}z-xy+x+y}.\hfill \end{array}$$

(123)

The calculation of the telescoper of (123) gives the same telescoper as the telescoper of (103), whose diagonal is the hypergeometric series:

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right],27x\right)\hfill \\ \hfill & & ={(1-24x)}^{-1/4}·{}_{2}F_1\left([\frac{1}{12},\frac{5}{12}],\left[1\right],\frac{1728x^3·(1-327x)}{{(1-24x)}^3}\right)\hfill \\ \hfill & & =1+6x+90x^2+1680x^3+34650x^4+756756x^5+17153136x^6+\cdots \hfill \end{array}$$

(124)

The hypergeometric series (124) (which is equal to the diagonal of (103)), is, here, just an analytical solution of the telescoper of (123), that is, a “Period” of (123) but corresponding to a non-vanishing cycle since (123) does not have a multi-Taylor expansion.

6.2.2. Birational Symmetries from Collineations

Let us recall Noether’s theorem [49,73,74] on the decomposition [75] of Cremona transformations. Noether’s theorem shows that any Cremona transformation can be seen as the composition [49,75] of collineation transformations and of the Hadamard inverse transformation:

$$\begin{array}{ccc}\hfill & & (x,y)⟶\left(\frac{1}{x},\frac{1}{y}\right).\hfill \end{array}$$

(125)

Let us consider Cremona transformations preserving $(x,y)=(0,0)$:

$$\begin{array}{ccc}\hfill & & (x,y)⟶\left(\frac{x}{1-x+2y},\frac{y}{1-x+2y}\right).\hfill \end{array}$$

(126)

With this theorem in mind, since we have already considered the involutive transformation (122) corresponding to the Hadamard inverse (125), let us just introduce the following birational transformation associated with the collineation (126):

$$\begin{array}{ccc}\hfill & & (x,y,z)⟶\left(\frac{x}{1-x+2y},\frac{y}{1-x+2y},z·{(1-x+2y)}^2\right).\hfill \end{array}$$

(127)

This birational transformation (associated with collineations) is an (infinite order) transformation. It preserves $(x,y,z)=(0,0,0)$ and the product $p=xyz$. Let us perform this birational transformation (127) on the rational function (103). One obtains a new rational function whose telescoper is an order-four linear differential operator $L_4$, which is the product of two order-two linear differential operator $M_2$ and $N_2$: $L_4=M_2·N_2$. The order-two linear differential operator $M_2$ is (non-trivially) homomorphic to the order-two telescoper of the rational function (103). The second order-two linear differential operator $N_2$ corresponds to algebraic functions. For such transformations, associated with collineations, we see that the telescoper is not preserved, we just have a (non-trivial) homomorphism property. The example (127) is revisited in detail in Appendix B.4. More examples of birational symmetries for telescopers, associated with collineations, are given in Appendix B. These examples illustrate the complexity of the homomorphism.

6.3. Algebraic Geometry Comments on These Birational Symmetries

The diagonal of the rational function (103) is the hypergeometric series:

$$\begin{array}{ccc}\hfill & & {}_{2}F_1\left([\frac{1}{3},\frac{2}{3}],\left[1\right],27x\right)\hfill \\ \hfill & & ={(1-24x)}^{-1/4}·{}_{2}F_1\left([\frac{1}{12},\frac{5}{12}],\left[1\right],\frac{1728x^3·(1-327x)}{{(1-24x)}^3}\right)\hfill \\ \hfill & & =1+6x+90x^2+1680x^3+34650x^4+756756x^5+17153136x^6+\cdots \hfill \end{array}$$

(128)

The algebraic curve, associated with the denominator of the rational function (103), is the genus-one algebraic curve (elliptic curve):

$$\begin{array}{ccc}\hfill & & 1-x-y-\frac{p}{xy}=0\mathrm{or}:-x^{2}y-x{y}^2+xy-p=0.\hfill \end{array}$$

(129)

The calculation of its j-invariant gives the following Hauptmodul:

$$\begin{array}{ccc}\hfill & & \mathcal{H}=\frac{1728}{j}=\frac{1728p^3·(1-27p)}{{(1-24p)}^3},\hfill \end{array}$$

(130)

which is exactly the Hauptmodul pullback in (128).

Let us consider the rational function (107); the algebraic curve corresponding to eliminate $z=p/x/y$ in the denominator of (107) reads:

$$\begin{array}{ccc}\hfill & & -49x^5{y}^2-42x^4{y}^2-7x^{4}y-23x^3{y}^2+4x^{3}y-6x^2{y}^2\hfill \\ \hfill & & +2x^{2}y-x{y}^2+xy-p=0.\hfill \end{array}$$

(131)

This algebraic curve is a genus-one algebraic curve (elliptic curve), and the calculation of its j-invariant gives the same Hauptmodul pullback in (128) as the Hauptmodul (130) for (129). This is in agreement with the fact that the diagonals of (103) and (107) are equal. At first sight, the fact that (131) is an elliptic curve is not totally obvious; however, it is a consequence of the fact that (129) and (131) are birationally equivalent elliptic curves (since one obtains one from the other one from a birational transformation). Consequently, they should have the same j-invariant.

This kind of remark will be seen as obvious, or slightly tautological, for an algebraic geometer; however, as far as down-to-earth computer algebra calculations of diagonals of rational functions or telescopers of rational functions are concerned, it becomes more and more spectacular for more complicated birational transformations generated by the composition of birational transformations like (108), (111), and (112).

More generally, the previous birational transformations preserving the product $p=xyz$, $p=xyzu$, … occurring in the diagonals will preserve the algebraic geometry description of the diagonal of rational functions [40]. For instance, the genus-two curves associated with the split Jacobians situation we have encountered in [40] (which corresponds to products of elliptic curves) will be preserved by such birational transformations.

6.4. Diagonal of Transcendental Functions

Generalizing the rationals functions

$$\begin{array}{ccc}\hfill & & R_B(x,y,z)=R\left(x,y·Q_1\left(x\right),\frac{z}{Q_1\left(x\right)}\right)=\frac{1}{1-x-y·Q_1\left(x\right)-z/Q_1\left(x\right)},\hfill \end{array}$$

(132)

deduced from (107), using birational transformations like (108), one can consider beyond transcendental functions like

$$\begin{array}{ccc}\hfill & & R_T(x,y,z)=R\left(x,y·cos\left(x\right),\frac{z}{cos\left(x\right)}\right)=\frac{1}{1-x-y·cos\left(x\right)-z/cos\left(x\right)}.\hfill \end{array}$$

(133)

One verifies easily, from the multi-Taylor expansion of the (simple) transcendental function (133), that its diagonal is actually the same as the one of (103), namely, (128). This is not a surprise since the demonstration of the invariance of the diagonal by birational transformation sketched in Section 6 (see (109)) just requires that $Q_1\left(0\right)\ne 0$ with $Q_1\left(x\right)$ behaving at the origin as a polynomial.

7. Conclusions

Diagonals of rational functions have been shown to emerge naturally for $n$-fold integrals in physics, field theory, and enumerative combinatorics, seen as “Periods” of algebraic varieties (corresponding to the denominators of these rational functions). Of the thousands of examples we have analyzed, corresponding to $n$-fold integrals of theoretical physics (in particular the ${\chi }^{\left(n\right)}$’s of the susceptibility of the Ising model, …) or corresponding to the rather academic diagonal of rational functions, we have seen the emergence of many striking properties, and we want to understand if these remarkable properties are inherited from the “physics”, and, more precisely, the rather “integrable” framework of these examples (Yang–Baxter integrability, 2D Ising models, Calabi–Yau and other mirror symmetries,…), or, on the contrary, if they are a consequence of the remarkable nature of diagonals of rational functions in the most general framework.

This paper is a plea for diagonals of rational or algebraic functions and more generally telescopers of rational or algebraic functions.

• We show that “periods” corresponding to non-vanishing cycles, obtained as solutions of telescopers of rational functions, can sometimes be recovered from diagonals of rational functions corresponding to vanishing cycles, introducing an extra parameter. These two concepts are not that compartmentalized.
• When considering the diagonals of rational functions we have shown that the number of variables of a rational function must, from time to time, be replaced by a notion of “effective number” of variables.
• We have shown that the “complexity” of the diagonals of a rational function, and, for instance, the order of the (minimal order) linear differential operator annihilating this diagonal, is not related to the number of variables or the “effective number” of variables of the rational function. In a forthcoming publication, we will try to understand what is the minimal number of variables necessary to represent a given D-finite globally bounded series as a diagonal of a rational function.
• We have shown that the algebraic geometry approach of the diagonals of rational functions, or of the telescopers of these rational functions, described in [40], can, probably, be generalized to diagonals of algebraic functions, or the telescoper of algebraic functions. These are just preliminary studies, and almost everything remains to be done.
• When studying diagonals of rational functions, our explicit examples enable one to understand why one can actually be restricted to rational functions of the form $1/Q$ provided the polynomial at the denominator is irreducible. The situation where the denominator $Q$ factorizes clearly needs further analysis, which will be displayed in a forthcoming paper. The case of the calculations of telescopers is slightly different: one can (probably), again, be restricted to rational functions of the form $1/Q$ but with a finite set of polynomials $Q$.
• We have shown that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. This remarkable result can, in fact, be generalized to an infinite set of rational transformations, with the diagonals of the transformed rational functions becoming the diagonal of the original rational function where the variable $x$ is changed into $x^n$. These invariance results generalize to telescopers. A more general (infinite) set of birational transformations is shown to correspond to a more convoluted “covariance” property of the telescopers (see Appendix B).
• We provide some examples of diagonals of transcendental functions that can also yield simple ${}_{2}F_1$ hypergeometric functions associated with elliptic curves. The analysis of diagonal of transcendental functions is clearly an interesting new domain to study. Accordingly, we thank one of the referees for his remark of a link to recent preprints of Golyshev et al. [76], where the classical Clausen–Sonin–Gegenbauer formulae are interpreted as special degenerated cases of the more general “multiplication kernel” setting developed by Kontsevich and Odesskii [77] (these formulae can be seen as examples of “diagonal” forms of generating functions for the multiplication kernels).
• Finally, when trying to understand the puzzling fact that telescopers of rational functions are almost always homomorphic to their adjoint and thus have selected symplectic or orthogonal differential Galois groups, we understand a bit better the emergence of curious examples of telescopers that are not homomorphic to their adjoint; this (up to homomorphisms) self-duality-breaking rules out a Poincaré duality interpretation of this quite systematic emergence of operators homomorphic to their adjoint. A “desingularization” of such puzzling cases, corresponding to the introduction of an extra parameter, shows that such operators now occur in dual (adjoint) pairs, thus restoring the duality (homomorphism to the adjoint). The limit when the extra parameter goes to zero is the direct sum of different telescopers corresponding to the first rational function terms of the expansion of the extended rational function in term of this extra parameter. With Section 5.2, we see that the puzzling (non self-adjoint up to homomorphism) order-three linear differential operator $L_3$ with $SL(3,\mathbb{C})$ differential Galois group is better understood as a member of a triplet of three “quarks” (90), (91), and (92), which restores the duality. This may suggest that the quite strange ${}_{3}F_2$ hypergeometric functions (91) or (92) could be related to (90), which has a clear elliptic curve origin. After all, these functions are three periods of the same algebraic variety. The existence of such a relation between hypergeometric functions of a totally and utterly different nature is a challenging open question.
• In Appendix B, the calculations of telescopers of rational functions, associated with very simple collineations, yield quite massive linear differential operators, which factor into an order-two operator associated with an elliptic curve, and a “dressing” of products of factors, which turn out to be direct sums of operators with algebraic function solutions. This occurrence of this “mix” between products and direct sums of a large number of operators (occurring, for instance, for the linear differential operators annihilating the ${\chi }^{\left(n\right)}$ components of the susceptibility of the Ising model [1,26]) will be revisited in a forthcoming paper.

Instead of pursuing one specific mathematical problem, this paper can be seen as a journey into the amazing world of integer sequences and differential equations. With all the examples displayed in this paper, we provide some answers, sometimes involving some plausible scenarios, to many important questions naturally emerging when working on diagonals of rational or algebraic functions, or on telescopers of rational or algebraic functions related, or not related, to problems of physics or enumerative combinatorics. Like any fruitful concept, every answered questions does not “close” the subject but, on the contrary, often raises more new questions than the number of answered questions.

Diagonals of rational or algebraic functions correspond to (globally bounded) series that can be recast into series with integer coefficients and are solutions of linear differential operators. When studying the two dimensional Ising model and its related Painlevé equations, one finds that the $\lambda$-extensions of the correlation functions [78,79] can also produce series with integer coefficients that are differentially algebraic [80] solutions of non-linear differential equations of the Painlevé type. These series are, also, such that their reduction modulo primes give algebraic functions, just like diagonals of rational or algebraic functions (for other examples of differentially algebraic series with integer coefficients see, for instance, [81]).

This paper tries to show that the concept of diagonals of rational or algebraic functions is a remarkably rich and fruitful concept not only providing tools for physics but also bridging, in a quite fascinating way, different domains of mathematics. The case of the diagonal of transcendental functions, or of these $\lambda$-extensions, seems to show that the “unreasonable richness” of diagonals and telescopers may just be the tip of an even more fascinating mathematical “iceberg” of mathematical physics.