Differential Equations for Classical and Non-Classical Polynomial Sets : A Survey †

By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown.


Introduction
In this survey article, a uniform method is presented for constructing the differential equations satisfied by several sets of classical and non classical polynomials.This has been done by starting from the basic elements of the relevant generating functions, using the monomiality principle by G. Dattoli [1] and a general result by Y. Ben Cheikh [2].Of course, the polynomials considered in this paper are only examples for showing that the method works, but obviously this technique can be theoretically extended to every polynomial set.This method has been recently applied in several articles (see [3][4][5][6][7][8][9]), which include works in collaboration with several authors.The most outstanding of them is Prof.Dr. Hari M. Srivastava, to whom this article is dedicated.
The derived differential equations are generally of infinite order, but they reduce to finite order when applied to polynomials.
It is worth noting that the differential equations for Sheffer polynomial sets have been studied even with different methods (see [10][11][12][13]), but here we use only elements directly connected with the theory of polynomials.
We start recalling, in Section 2, the definitions relevant to Sheffer polynomials, the G. Dattoli monomiality principle, and a general result by Y. Ben Cheikh.
The classical polynomial sets, considered in Section 3, are the Bernoulli, Euler, Genocchi and Mittag-Leffler polynomials.In Section 4, we show some new polynomial sets derived from non-classical generating functions.

Sheffer Polynomials
The Sheffer polynomials {s n (x)} are introduced [14] by means of the exponential generating function [15] of the type: where According to a different characterization (see [16], p. 18), the same polynomial sequence can be defined by means of the pair (g(t), f (t)), where g(t) is an invertible series and f (t) is a delta series: Denoting by f −1 (t) the compositional inverse of f (t , the exponential generating function of the sequence {s n (x)} is given by so that When g(t) ≡ 1, the Sheffer sequence corresponding to the pair (1, f (t)) is called the associated Sheffer sequence {σ n (x)} for f (t), and its exponential generating function is given by A list of known Sheffer polynomial sequences and their associated ones can be found in [17].

Shift Operators and Differential Equation
We recall that a polynomial set {p n (x)} is called quasi-monomial if and only if there exist two operators P and M such that P is called the derivative operator and M the multiplication operator, as they act in the same way as classical operators on monomials.
This definition traces back to a paper by J.F. Steffensen [18] recently improved by G. Dattoli and widely used in several applications [19,20].
Y. Ben Cheikh proved that every polynomial set is quasi-monomial under the action of suitable derivative and multiplication operators.In particular, in the same article, the following result is proved, as a particular case of Corollary 3.2 in [2]: ) denote a Sheffer polynomial set, defined by the generating function where and Denoting, as before, by f (t) the compositional inverse of H(t), the Sheffer polynomial set {p n (x)} is quasi-monomial under the action of the operators where prime denotes the ordinary derivatives with respect to t.
Furthermore, according to the monomiality principle, the quasi-monomial polynomials {p n (x)} satisfy the differential equation

Bernoulli Polynomials
The Bernoulli polynomials are defined by the generating function so that where b k are the Bernoulli numbers.

Differential Equation of the B k (x)
Note that, recalling that B n (1) = (−1) n b n , the following expansion holds: The shift operators for the Bernoulli polynomials are given by Therefore, by using the factorization method, we find Theorem 2. The Bernoulli polynomials {B n (x)} satisfy the differential equation that is or, in equivalent form: Proof.It is sufficient to expand in series the operator (17).Equation (19) follows because, for any fixed n, the series expansion in Equation ( 18) reduces to a finite sum when applied to a polynomial of degree n.

Euler Polynomials
The Euler polynomials are defined by the generating function so that where e k are the Euler numbers.

Differential Equation of the E k (x)
Note that the following expansion holds: where c 0 = −1/2, and c n = e n /2.The shift operators for the Euler polynomials are given by Therefore, by using the factorization method, we find Theorem 3. The Euler polynomials {E n (x)} satisfy the differential equation that is or, in equivalent form: Proof.It is sufficient to expand in series the operator (24).Equation (26) follows because, for any fixed n, the series expansion in Equation (25) reduces to a finite sum when applied to a polynomial of degree n.

Genocchi Polynomials
The Genocchi polynomials are defined by the generating function so that where g k are the Genocchi numbers.

Differential Equation of the G k (x)
Note that the following expansion holds: where The shift operators for the Genocchi polynomials are given by so that the Genocchi polynomials satisfy the differential equation Therefore, by using the factorization method, we find Theorem 4. The Genocchi polynomials {G n (x)} satisfy the differential equation or, in equivalent form: Proof.It is sufficient to expand in series the operator (32).Equation (34) follows because, for any fixed n, the series expansion in Equation (33) reduces to a finite sum when applied to a polynomial of degree n.

The Mittag-Leffler Polynomials
We recall that the Mittag-Leffler polynomials [21] are a special case of associated Sheffer polynomials, defined by the generating function Therefore, we have so that, for the Mittag-Leffer polynomials, we find the shift operators: (37)

Differential Equation of the M n (x)
In the present case, according to the identity: we can write so that we have the theorem Theorem 5.The Mittag-Leffler polynomials {M n (x)} satisfy the differential equation where n−1 2 denotes the integral part of (n − 1)/2.
Proof.It is sufficient to expand in series the operator (39).Equation (41) follows because, for any fixed n, the series expansion in Equation ( 40) reduces to a finite sum when applied to a polynomial of degree n.

Euler-Type Polynomials
Here, we introduce a Sheffer polynomial set connected with the classical Euler polynomials. Assuming: we consider the Euler-type polynomials Ẽn (x), defined by the generating function Note that the Euler numbers are recovered, since we have: In what follows, we use the expansions Note that, in our case, we are dealing with a Sheffer polynomial set, so that, since we have ψ(t) = e t , the operator σ defined by Equation ( 6) simply reduces to the derivative operator D x .Furthermore, we have: , so that we have the theorem Theorem 6.The Euler-type polynomial set { Ẽn (x)} is quasi-monomial under the action of the operators (by arcsinh t = log(t + √ t 2 + 1), we denote the inverse of the function sinh t), i.e., There is no problem about the convergence of the above series, since they reduce to finite sums when applied to polynomials.

Differential Equation of the Ẽn (x)
In the present case, we have Theorem 7. The Euler-type polynomials { Ẽn (x)} satisfy the differential equation i.e., Note that, for any fixed n, the Cauchy product of series expansions in Equation (49) reduces to a finite sum, with upper limit n−1

2
, when it is applied to a polynomial of degree n, because the successive addends vanish.

Adjointness for Sheffer Polynomial Sequences
According to the above considerations, Sheffer polynomials are characterized both by the ordered couples (A(t), H(t)), or by (g(t), f (t)).
Definition 1. Adjoint Sheffer polynomials are defined by interchanging the ordered couple (A(t), H(t)) with (g(t), f (t)), when writing the generating function.
Here and in the following the tilde "∼" above the symbol of a polynomial set stands for the adjective "adjoint" (see e.g., [4]).

Adjoint Hahn Polynomials
Assuming: we consider the adjoint Hahn Rn (x), defined by the generating function It is a Sheffer set.
We have: Note that, in this case, we have: so that we have the theorem Theorem 8.The adjoint Hahn polynomial set { Rn (x)} is quasi-monomial under the action of the operators i.e., (54)

Differential Equation of the Rn (x)
In the present case, we have Theorem 9.The Sheffer-type adjoint Hahn polynomials { Rn (x)} satisfy the differential equation Note that, for any fixed n, in Equation (55), a finite sum appears, with upper limit n−1 2 , instead of a complete series expansion since, when this series is applied to a polynomial of degree n, the subsequent addends vanish.
Remark 3. Table of adjoint Hahn numbersR0 5.3.Adjoint Bernoulli Polynomials of the Second Kindwe consider the adjoint Bernoulli polynomials of the second kind { bk (x)}, defined by the generating function Note that, in our case, we are dealing with a Sheffer polynomial set, so that, since we have ψ(t) = e t , the operator σ defined by Equation (6) simply reduces to the derivative operator D x .