Issue 
4open
Volume 2, 2019
Difference & Differential Equations and Applications



Article Number  4  
Number of page(s)  9  
Section  Mathematics  Applied Mathematics  
DOI  https://doi.org/10.1051/fopen/2019004  
Published online  25 April 2019 
Research Article
On Sheffer polynomial families
^{1}
Academia Militar – Departamento de Ciências Exactas e Engenharia, Av. Conde Castro Guimarães, 2720113, Amadora, Portugal
^{2}
International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186, Roma, Italia
^{*} Corresponding author: paoloemilioricci@gmail.com
Received:
16
October
2018
Accepted:
22
February
2019
Attention is focused to particular families of Sheffer polynomials which are different from the classical ones because they satisfy nonstandard differential equations, including some of fractional type. In particular Sheffer polynomial families are considered whose characteristic elements are based on powers or exponential functions.
Key words: Sheffer polynomial families / Generating functions / Monomiality principle / Fractional derivative equations / Combinatorial analysis
© S. Pinelas & P.E. Ricci, Published by EDP Sciences, 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
In recent articles [1, 2], new sets of Sheffer [3] and Brenke [4] polynomials, based on higher order Bell numbers [5–9], have been studied. Furthermore, several integer sequences [10] associated with the considered polynomials sets both of exponential [11, 12] and logarithmic type have been introduced [2].
We recall that the exponential and logarithmic polynomials have been recently studied even in the multidimensional case [13–15].
It is worth to note that the Sheffer family includes a plenty of unusual polynomials, which satisfy nonstandard 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.
2 Sheffer polynomials
We start recalling the particular meaning of the term set in the framework of polynomial theory.
Definition 2.1. A polynomial family {P _{n} (x)}_{(n ≥ 0)} is called a polynomial set iff , deg P _{n} = n.
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:
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,
Actually, in what follows, if H(t) is a polynomial of degree m, we have found that , where [⋅] denotes the integral part.
In general, we are dealing with a Sheffer polynomial set iff the condition 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:
Denoting by f ^{−1}(t) the compositional inverse of f(t) (i.e., such that f(f ^{−1}) (t)) = (f ^{−1}) (f(t)) = t), the exponential generating function of the sequence {s _{n} (x)} is given by
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 [19]. New Eulertype Sheffer polynomials have been recently introduced in [20].
3 Power based Sheffer polynomials
In this section we derive Sheffer polynomial families assuming the following basic functions:
(7) p and q positive integer numbers, so that,
(8)and the generating function is:
Therefore, denoting by H ^{−1}(t) the compositional inverse of H(t), we have:
Definition 3.1. We recall that a polynomial set {p _{n} (x)} is called quasimonomial if and only if there exist two operators and such that,
(11) is called the derivative operator and 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 quasimonomial 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:
(12)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} ≠ 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:
Theorem 3.3. The derivative and multiplication operators of the Sheffer polynomial family defined by the generating function (9) are given by,
3.1 Differential equation
As a consequence of the monomiality principle, the factorization method gives the differential equation satisfied by the quasimonomial polynomials {p _{n} (x)} in the form:
In the present case, we have the result:
Theorem 3.5. The Sheffer polynomials satisfy the differential equation
4 Some particular examples
We show in this Section some particular example.
4.1 Case
The ordinary differential equation is:
The first few polynomials are as follows:
Further values can be easily achieved by using Wolfram Alpha^{©}.
Remark 4.1. Note that there is a link of the polynomials with the classical H _{ n }(x). In fact, let
Since G _{2}(x, −t) = G _{2}(x, t), it follows that , and
Therefore, putting t ^{2} = 2is (i the imaginary unit), we find,
In a similar way, when (N integer number), we can find a link between the GouldHopper polynomials defined by the generating function,
and the Sheffer polynomials defined by,
In fact, owing the symmetry of these polynomials, we have,
and therefore, putting τ − t ^{ N }, we find:
so that , as a polynomial of degree n, can be expressed in terms of the GouldHopper polynomials by:
Remark 4.2. 1. The equation (16) is an ordinary differential equation of order 2, because (p + 1)/q = 4/2 = 2. This happens, in general, if and only if (N integer number).
2. Note that, owing to the fractional derivative operator , the polynomial degree increases slowly, actually the degree of is equal to (2n)/2 = n.
4.2 Case
The fractional differential equation is:
The first few polynomials are as follows:
If n ≡ 0, (mod 3):
If n ≡ 1, (mod 3):
If n ≡ 2, (mod 3):
Further values can be easily achieved by using Wolfram Alpha^{©}.
Remark 4.3. 1. The equation (17) is a fractional differential equation.
2. Note that, owing to the fractional derivative operator , the degree of the polynomial is equal to:

n/3 if n ≡ 0, (mod 3),

(n−4)/3 if n ≡ 1, (mod 3), and n ≥ 4,

(n−2)/3 if n ≡ 2, (mod 3), and n ≥ 2.
5 A particular family of Sheffer polynomials
In this Section we consider a particular family of Sheffer polynomials defined by generating functions of the type:
(18)and H(t) is an invertible function. Therefore, we find,
Putting, as before, f(t) = H ^{−1}(t), according to the recalled result by Y. Ben Cheikh [25], the derivative and multiplication operators for the relevant polynomials τ _{ n } are given by:
(20)so that we have the result:
Theorem 5.1. The Sheffer polynomials {τ _{n} (x)} satisfy the differential equation
6 Particular examples
Note that the particular Sheffer polynomials of Section 5 only depend on H(t).
6.1 A power based example, linked to Section 3
We assume,
(22)(q a positive integer number), so that,
According to the above results, the derivative and multiplication operators for the quasimonomials are given by,
(24)and the relevant differential equation writes,
(25)which is a fractional derivative equation.
6.2 Case q = 1, H(t) = t ^{3}/3
The generating function is,
and the fractional differential equation (25) writes,
The first few polynomials are as follows:
Further values can be easily achieved by using Wolfram Alpha^{©}.
Note the symmetry of the above scheme, according to which the degree of the polynomial is equal to:

n/3 if n ≡ 0, (mod 3),

(n−4)/3 if n ≡ 1, (mod 3), and n ≥ 4,

(n−8)/3 if n ≡ 2, (mod 3), and n ≥ 8.
6.3 An exponential based example
We assume,
According to the above results, the derivative and multiplication operators for the quasimonomials are given by,
(29)and the relevant differential equation writes,
(30)Which is an infinite order differential equation reducing to an equation of order n when applied to a polynomial of degree n.
The first few polynomials are as follows:
Further values can be easily achieved by using Wolfram Alpha^{©}.
Remark 6.1. Note that the sequence:
(that is the values ), has a combinatorial character, since it appears in the Encyclopedia of integer sequences under A217924 – Row sequence of table A217537, ; – Vladimir Kruchinin, Feb 28, 2015.
Furthermore, the sequence:
(that is the values appears in the Encyclopedia of integer sequences under A000296 – Set partitions without singletons: number of partitions of an nset into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.
7 A mixedtype (powerexp) Sheffer polynomial family
We assume in this Section:
(31)(q a positive integer number), so that,
According to the above results, the derivative and multiplication operators for the quasimonomials are given by,
(33)and the relevant differential equation writes,
(34)which is an infinite order differential equation reducing to an equation of order qn−1 when it is applied to a polynomial of degree n.
7.1 Case q = 2, G(t, x) = exp[e ^{ t } − 1 + xt ^{2}]
The first few polynomials are as follows:
Further values can be easily achieved by using Wolfram Alpha^{©}.
Remark 7.1. 1. Note that owing to the fractional derivative operator , the degree of the polynomial is equal to .
A similar phenomenon also applies to q > 2.
2. Note that the sequence:
(that is the values ), 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.
8 Conclusion
We have introduced “unusual families” of Sheffer polynomials, namely they satisfy nonstandard differential equations, including some of fractional type.
In some case, we have noticed connections with particular integer sequences, since the polynomial – in suitable points – exhibit 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.
Acknowledgments
The authors thanks the anonymous referee for his skillful remarks, useful for improving the manuscript.
Compliance with ethical standards
Conflict of interest
The authors declare that they have not received funds from any institution and that they have no conflict of interest.
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Cite this article as: Pinelas S, Ricci P.E 2019. On Sheffer polynomial families. 4open, 2, 4.
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