On Sheffer polynomial families

Attention is focused to particular families of Sheffer polynomials which are different from the classical ones because they satisfy non-standard differential equations, including some of fractional type. In particular Sheffer polynomial families are considered whose characteristic elements are based on powers or exponential functions.

We recall that the exponential and logarithmic polynomials have been recently studied even in the multidimensional case [13][14][15].
It is worth to note that the Sheffer family includes a plenty of unusual polynomials, which satisfy non-standard differential equations. In this article we focus our attention on Sheffer polynomial families whose characteristic elements are based on powers or exponential functions, deriving the relevant differential equations, which are frequently of fractional type.

Sheffer polynomials
We start recalling the particular meaning of the term set in the framework of polynomial theory. In what follows, we are dealing with polynomial families that, in several cases, don't satisfy the above condition. The Sheffer polynomial families {s n (x)} are introduced [3] by means of the exponential generating function [16] of the type: where, A t ð Þ ¼ P 1 n¼0 a n t n n! ; ða 0 6 ¼ 0Þ; Remark 2.2. It is well known [4,17] that there is a natural link between the function H(t) and the degree of polynomials s n (x) in expansion (1). Namely, deg s n ¼ n iff ; in equation 2 ð Þ; h 1 6 ¼ 0: Actually, in what follows, if H(t) is a polynomial of degree m, we have found that deg s n ½ n m , where ½Á denotes the integral part. In general, we are dealing with a Sheffer polynomial set iff the condition h 1 6 ¼ 0 is satisfied. According to a different characterization (see [18], 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: g n t n n! ; ðg 0 6 ¼ 0Þ; Denoting by f À1 (t) the compositional inverse of f(t) (i.e., such that f(f À1 ) (t)) = (f À1 ) (f(t)) = 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 {r 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 [19]. New Euler-type Sheffer polynomials have been recently introduced in [20].

Power based Sheffer polynomials
In this section we derive Sheffer polynomial families assuming the following basic functions: p and q positive integer numbers, so that, and the generating function is: Therefore, denoting by H À1 (t) the compositional inverse of H(t), we have: P is called the derivative operator andM the multiplication operator, as they act in the same way of classical operators on monomials.
This definition traces back to a paper by J.F. Steffensen [21], recently improved by G. Dattoli [22] and widely used in several applications (see e.g., [23,24] and the references therein). Y. Ben Cheikh [25] proved that every polynomial set is quasi-monomial under the action of suitable derivative and multiplication operators. In particular, if the considered polynomial set is Sheffer, the Corollary 3.2 in the same article ensure that the derivative and multiplication operator are given by: where prime denotes the derivative with respect to t, and D x the derivative with respect to x.

Remark 3.2.
It is worth to note that the above mentioned result (Corollary 3.2 in [25]), given for polynomial sets, never uses in proof the condition h 1 6 ¼ 0. Therefore, it can be applied even to polynomials defined by Sheffer generating functions (1), i.e. to Sheffer polynomial families. According to the above equations (8), (10), (12), we have the result: The derivative and multiplication operators of the Sheffer polynomial family defined by the generating function (9) are given by,P

Differential equation
As a consequence of the monomiality principle, the factorization method gives the differential equation satisfied by the quasi-monomial polynomials {p n (x)} in the form:MP In the present case, we have the result: Theorem 3.5. The Sheffer polynomialsq n ðp; q; xÞ f gsatisfy the differential equation

Some particular examples
We show in this Section some particular example.
The ordinary differential equation is: The first fewq n ð3; 2; xÞ polynomials are as follows: Further values can be easily achieved by using Wolfram Alpha Ó .

A particular family of Sheffer polynomials
In this Section we consider a particular family of Sheffer polynomials defined by generating functions of the type: and H(t) is an invertible function. Therefore, we find, Putting, as before, f(t) = H À1 (t), according to the recalled result by Y. Ben Cheikh [25], the derivative and multiplication operators for the relevant polynomials s n are given by:P so that we have the result: Theorem 5.1. The Sheffer polynomials {s n (x)} satisfy the differential equation 6 Particular examples Note that the particular Sheffer polynomials of Section 5 only depend on H(t).

A power based example, linked to Section 3
We assume, (q a positive integer number), so that, According to the above results, the derivative and multiplication operators for the quasi-monomialss n ðx; qÞ are given by, and the relevant differential equation writes, which is a fractional derivative equation.

Case
The generating function is, n ðx; 3Þ t n n! ; and the fractional differential equation (25) writes, The first fews n ðx; 3Þ polynomials are as follows: Further values can be easily achieved by using Wolfram Alpha Ó . Note the symmetry of the above scheme, according to which the degree of the polynomials n ðx; 3Þ is equal to: n/3 if n 0, (mod 3), (nÀ4)/3 if n 1, (mod 3), and n ! 4, (nÀ8)/3 if n 2, (mod 3), and n ! 8.

An exponential based example
We assume, so that, H 0 ðtÞ ¼ e t ; H À1 ðtÞ ¼ logðt þ 1Þ; Gðt; xÞ ¼ exp e t À t þ xðe t À 1Þ According to the above results, the derivative and multiplication operators for the quasi-monomialsr n ðxÞ are given by, and the relevant differential equation writes, which is an infinite order differential equation reducing to an equation of order n when applied to a polynomial of degree n.
The first fewr n ðxÞ polynomials are as follows: Further values can be easily achieved by using Wolfram Alpha Ó . (that is the valuesr n ð1Þ), has a combinatorial character, since it appears in the Encyclopedia of integer sequences under A217924 -Row sequence of table A217537, a n ð Þ :¼ P k j¼0 P n k¼0 n kÀj 2 j ðÀ1Þ ðkÀjÞ Stirling 2 ðn À k þ j; jÞ ; -Vladimir Kruchinin, Feb 28, 2015. Furthermore, the sequence: 1; 1; 4; 11; 41; 162; 715; . . . ; (that is the valuesr n ð0Þ; n ! 2Þ appears in the Encyclopedia of integer sequences under A000296 -Set partitions without singletons: number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.

A mixed-type (power-exp) Sheffer polynomial family
We assume in this Section: (q a positive integer number), so that, Gðt; xÞ ¼ exp e t À 1 þ xt q ½ ¼ P 1 n¼0q n ðx; qÞ t n n! : ð32Þ According to the above results, the derivative and multiplication operators for the quasi-monomialsq n ðx; qÞ are given by, ð33Þ and the relevant differential equation writes, that is, which is an infinite order differential equation reducing to an equation of order qnÀ1 when it is applied to a polynomial of degree n.
The first fewq n ðx; 2Þ polynomials are as follows: Further values can be easily achieved by using Wolfram Alpha Ó . (that is the valuesq n ð0; 2Þ), has a combinatorial character, since it appears in the Encyclopedia of integer sequences under A000110 -Bell or exponential numbers: number of ways to partition a set of n labeled elements.

Conclusion
We have introduced "unusual families" of Sheffer polynomials, namely they satisfy non-standard differential equations, including some of fractional type.
In some case, we have noticed connections with particular integer sequences, since the polynomialin suitable pointsexhibit a combinatorial character. In this article have considered Sheffer polynomials based on powers or exponential functions. In subsequent papers further families will be introduced, by using different basic elements.

Author contributions
The authors claim to have contributed equally and significantly in this paper. Both authors read and approved the final manuscript.