Reflection identities of harmonic sums of weight four

We consider the reflection identities for harmonic sums at weight four. We decompose a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or positive values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible bilinear set of reflection identities at weight four which present the main result of the paper. We also discuss how other trilinear and quartic reflection identities can be easily constructed from our result with the use of well known shuffle relations for harmonic sums.


I. INTRODUCTION
In this paper we continue discussion of our previous study 1 regarding the reflection identities of harmonic sums, where a product of two harmonic sums of argument z and −1 − z is expressed through a linear combination of other harmonic sums of the same arguments, i.e.
The reflection identities at weight two presented here are not new and were known long time ago in the context of functions related to the Euler Gamma function. To the best of our knowledge they appear the eraliest in Chapter 20 of the book by Nielsen 2 and then were related to the harmonic sums (see eqs.6.11-6.15 of the paper by Blumlein 3 ). At weight three they were recently calculated by the author 1 . This paper deals with weight four.
The harmonic sums have pole singularities at negative integers. The reflection identities present a pole separation for a product of two sums with mixed pole structure. We call those functional relations the reflection identities because the argument of the harmonic sums is reflected with respect to the point z+(−1−z) 2 = − 1 2 . The reflection identities up to weight three were published in our previous study 1 and here we present them at weight four.
The harmonic sums are defined through a nested summation with their argument being the upper limit in the outermost sum [4][5][6][7] S a 1 ,a 2 ,...,a k (n) = n≥i 1 ≥i 2 ≥...≥i k ≥1 sign(a 1 ) i 1 i In this paper we consider the harmonic sums with only real integer values of a i , which build the alphabet of the possible negative and positive indices. In Eq. (2) k is the depth and w = k i=1 |a i | is the weight of the harmonic sum S a 1 ,a 2 ,...,a k (n). The indices of harmonic sums a 1 , a 2 , ..., a k can be either positive or negative integers and label uniquely S a 1 ,a 2 ,...,a k (n) for any given weight. However there is no unique way of building the functional basis for a given weight because the harmonic sums are subject to so called shuffle relations, where a linear combination of S a 1 ,a 2 ,...,a k (n) with the same argument but all possible permutations of indices can be expressed through a non-linear combinations of harmonic sums at lower weight. There is also some freedom in choosing the irreducible minimal set of S a 1 ,a 2 ,...,a k (n) that builds those non-linear combinations. The shuffle relations make a connection between the linear and non-linear combinations of the harmonic sums of the same argument. For example, the shuffle relation at depth two reads S a,b (z) + S b,a (z) = S a (z)S b (z) + S sign(a) sign(b)(|a|+|b|) (z) (3) The shuffle relations of the harmonic sums is closely connected to the shuffle algebra of the harmonic polylogarithms 7 .
There is another type of identity called the duplication identities where a combination of harmonic sums of argument n can be expressed through a harmonic sum of the argument 2n. The duplication identities introduce another freedom in choosing the functional basis.
In this paper we consider the analytic continuation of the harmonic sums to from positive integer values of the argument to the complex plane denoted byS + a 1 ,a 2 ,... (z) (this notation was introduced by Kotikov and Velizhanin 8 ). The analytic continuation is done in terms of the Mellin transform of corresponding Harmonic Polylogarithms and was recently used by Gromov, Levkovich-Maslyuk and Sizov 9,10 and Caron Huot and Herraren 11 for expressing the eigenvalue Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation using the principle of Maximal Transcedentality 12 in super Yang-Mills N = 4 field theory. We plan to use their results together with analysis done by one of the authors and collaborators 16,17 to understand the general structure of the BFKL equation in QCD and beyond.
The Mellin transform allows to make the analytic continuation to the complex plane. For example, consider the harmonic sum The corresponding Mellin transform reads One can see that S −1 (z) on its own is not an analytic function because of the term (−1) z and we impose that we start from even integer values of the argument z. In this case we define its analytic continuation from even positive integers to all positive integers through and thus we can writeS This way we definedS + a 1 ,a 2 ,... (z) using the Mellin transform of ratio function 1 1+x . In more complicated cases of other harmonic sums one includes also Harmonic Polylogarithms on top of the ratio functions, but the general procedure is very similar and largely covered in a number of publications 3,6,13-15 .
It is worth mentioning that there is another analytic continuation for the harmonic sum, from odd positive integer values of the argument, which is different for harmonic sums with at least one negative index and denoted byS − a 1 ,a 2 ,... (z). Both analytic continuations are equally valid. Our goal is to find a closed expression of the BFKL eigenvalue for all possible values of anomalous dimension and conformal spin, so that we follow the notation of Gromov, Levkovich-Maslyuk and Sizov 9 , and useS + (z) throughout the text. For simplicity of presentation in this paper we write everywhere S a 1 ,a 2 ,... (z) instead ofS + a 1 ,a 2 ,... (z). As it was already mentioned there is no unique way in defining a minimal irreducible set of harmonic sums due to the functional relations between them. For example, one can use the shuffle relations and them the minimal irreducible basis would include quadratic terms S a 1 (z)S a 2 (z) in place of either S a 1 ,a 2 (z) or S a 2 ,a 1 (z). It is convenient to use shuffle relations to remove from the minimal basis the harmonic sums with the first index being equal 1, because those are divergent as z → ∞. Then, the remaining harmonic sums give transcendental constants at z → ∞. Most of constants are reducible and one is free to choose an irreducible set of transcendental constants at any given weight. We use the set implemented in the HarmonicSums package. The irreducible constants are given by and and C 3 = π 2 log(2), log 3 (2), ζ 3 as well as where C w stands for a minimal set of irreducible constants at given weight w. There is only one of those at w = 1, two at w = 2, three at w = 3 and five irreducible constants at weight w = 4.
We choose to use a linear minimal set of the harmonic sums to represent our results. In this set we do not apply shuffle relations and thus all the terms of the basis are linear in S a 1 ,a 2 ,... (z). This choice is dictated mostly by a convenience and was also used by Caron Huot and Herraren 11 on which we would like to rely in our future calculations. The minimal linear set of harmonic sums we use is as follows and and as well as A comprehensive discussion on harmonic sums, irreducible constants, functional identities and possible choice of the minimal set of functions at given weight is presented by J. Ablinger 13 . In this paper we focus only at the reflection identities for harmonic sums at weight w = 4 analytically continued from even positive points to complex plane. In the next Section we discuss them in more details along the method we use in our calculations.

II. REFLECTION IDENTITIES
The reflection identities at weight w = 4 are obtained by taking a product of harmonic sums of argument z at weight w = 1 and harmonic sums of argument −1 − z at weight w = 3, i.e. B 1 ⊗B 3 , and also by taking a product of harmonic sums of argument z and −1 − z at weight w = 2, i.e. B 2 ⊗B 2 .
The number of basis harmonic sums in B 1 , B 2 and B 3 is given by Length(B 1 ) = 2, Length(B 2 ) = 6, Length(B 3 ) = 18 (16) so that the number of elements in the products B 1 ⊗B 3 and B 2 ⊗B 2 reads and so that the total number of terms in the expansion ansatz at w = 4 is given by The full expansion ansatz at w = 4 is given by (2), The expansion of the product of two functions of argument z and argument −1 − z we search in terms of two sets of ANZ 4 , one of argument z and another one of argument −1 − z.
The total number of elements in this expression equals 95 × 2 − 5 = 185, where we remove redundant five constants at w = 4 because they are the same for both arguments. We fix the 185 free coefficients using pole expansion of the product s a 1 ,a 2 ,.. and One can see that the reflection identities for harmonic sums with negative indices are more complicated than those with only positive indices and this happens mostly due to appearance of constant ln (2), which originates from sign alternating summation in S −1 (z) absent for positive indices.
In the present paper we consider only bilinear reflection identities expressing a product of two harmonic sums of argument z and −1 − z in terms of a linear combination of other sums of the same arguments. One can consider also trilinear and quartic identities, but whose reducible and form a linear combination of the bilinear identities presented in this paper.
Plugging those together with Eq. (23) into Eq. (25) we get In a similar way one can can build any trilinear or quartic reflection identity using shuffle relations for harmonic sums and the bilinear reflection identities listed in the Appendix of this paper. All possible shuffle relations required for the present discussion are available in the HarmonicSums package by J. Ablinger. Shuffle relation before and after analytic continuation of the harmonic sums to the complex plane are the same.

III. CONCLUSIONS
We discuss the reflection identities for harmonic sums of weight four. There are 57 irreducible bilinear identities listed in the Appendix. All other bilinear reflection identities are easily obtained by a trivial change of argument z ↔ −1 − z. The trilinear and quartic identities for a product of three and four harmonic sums are obtained from the identities listed in the Appendix using the shuffle relations for harmonic sums. In our analysis we use the linear basis for harmonic sums and limit ourselves to harmonic sums analytically continued from even integer values of the argument to the complex plane. The analytic continuation from odd integers is beyond the scope of the present study.
In deriving the reflection identities presented in this paper we We attach a Mathematica notebook with our results.

IV. ACKNOWLEDGEMENTS
We would like to thank Fedor Levkovich-Maslyuk and Mikhail Alfimov for fruitful discussions on details of their calculations of NNLO BFKL eigenvalue 9,10 . We are grateful to Simon Caron-Huot for explaining us the structure of his result on NNLO BFKL eigenvalue and his calculation techniques 11 .
We are indebted to Jochen Bartels for his hospitality and enlightening discussions during our stay at University of Hamburg where this project was initiated.

A. Appendix
Here we list below the irreducible reflection identities at weight w = 4. In all our expressions we used the linear minimal set of harmonic sums given in Eqs. (12)- (15).
The constants are also written in a compact and readable way ln 2 = ln 2 ≃ 0.693147 and