New Bell – Sheffer Polynomial Sets

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear.

It is worth noting that exponential and logarithmic polynomials have been recently studied in the multidimensional case [11][12][13].
In this article, new sets of Bell-Sheffer polynomials are considered and some particular cases are analyzed.
It is worth noting that the Sheffer A-type 0 polynomial sets have been also approached with elementary methods of linear algebra (see, e.g., [14][15][16] and the references therein).

Sheffer Polynomials
The Sheffer polynomials {s n (x)} are introduced [3] by means of the exponential generating function [19] of the type: where According to a different characterization (see [20], 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 [21].

New Bell-Sheffer Polynomial Sets
We introduce, for shortness, the following compact notation.Put, by definition: and in a similar way: Remark 2. Note that the coefficients of the Taylor expansion of E 1 (t) are given by the Bell numbers b n = b and, in general, the coefficients of the Taylor expansion of E r (t) are given by the higher order Bell numbers b The higher order Bell numbers, also known as higher order exponential numbers, have been considered in [5,7,22], and used in [2] in the framework of Brenke and Sheffer polynomials.
Remark 3. Note that the coefficients of the Taylor expansion of Λ 0 (t) are given by the logarithmic numbers l and, in general, the coefficients of the Taylor expansion of Λ r−1 (t) are given by the higher order logarithmic numbers l The higher order logarithmic numbers, which are the counterpart of the higher order Bell (exponential) numbers, have been considered in [1], and used there in the framework of new sets of Sheffer polynomials.

The Polynomials E
(1) k (x) Therefore, we consider the Sheffer polynomials, defined through their generating function, by putting

Recurrence Relation for the E
(1) Theorem 1.For any k ≥ 0, the polynomials E (1) k (x) satisfy the following recurrence relation: Proof.Differentiating G(t, x) with respect to t , we have and therefore h (x) so that the recurrence relation ( 8) follows.

Generating Function's PDEs
Theorem 2. The generating function (7) 2 satisfies the homogeneous linear PDEs: Proof.Differentiating G(t, x) with respect to x , we have By taking the ratio between the members of Equations ( 9) and ( 13), we find Equation (10).The other ones easily follows by elementary algebraic manipulations.

Shift Operators
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 of classical operators on monomials.
This definition traces back to a paper by Steffensen [23], recently improved by Dattoli [24] and widely used in several applications.
Ben Cheikh [25] proved that every polynomial set is quasi-monomial under the action of suitable derivative and multiplication operators.In particular, in the same article (Corollary 3.2), the following result is proved.Theorem 3. Let (p n (x)) denote a Boas-Buck polynomial set, i.e., a set defined by the generating function where with ψ(t) not a polynomial, and lastly Let σ ∈ Λ (−) the lowering operator defined by Put Denoting, as before, by f (t) the compositional inverse of H(t), the Boas-Buck polynomial set {p n (x)} is quasi-monomial under the action of the operators where prime denotes the ordinary derivatives with respect to t.
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 ( 16) simply reduces to the derivative operator D x .Furthermore, we have: and, consequently, Theorem 4. The Bell-Sheffer polynomials {E (1) k (x)} are quasi-monomial under the action of the operators 3.5.Differential Equation for the E k (x) According to the results of monomiality principle [24], the quasi-monomial polynomials {p n (x)} satisfy the differential equation In the present case, recalling Equation ( 22), we have Theorem 5.The Bell-Sheffer polynomials {E k (x)} satisfy the differential equation n (x) .
Proof.Equation (22), by using Equation ( 21), becomes n (x) , and, furthermore, for any fixed n, the last series expansion reduces to a finite sum, with upper limit n − 1, when it is applied to a polynomial of degree n because the last not vanishing term (for k = n − 1) contains the derivative of order n.

First Few Values of the E
Here, we show the first few values for the Bell-Sheffer polynomials E k (x), defined by the generating function (7) 2

Iterated Bell-Sheffer Polynomial Sets
Here, we iterate the procedure introduced in Section 3, by considering the Sheffer polynomial sets defined by putting We find: and, consequently, Theorem 6.The Bell-Sheffer polynomials {E k (x)} are quasi-monomial under the action of the operators 4.1.Differential Equation for the E k (x) According to the results of monomiality principle [24,26], the quasi-monomial polynomials {p n (x)} satisfy the differential equation In the present case, recalling Equation ( 19), we have Theorem 7. The Bell-Sheffer polynomials {E k (x)} satisfy the differential equation k (x) Here, we show the first few values for the Bell-Sheffer polynomials E (2) k (x), defined by the generating function ( 7) 2

The General Case
In general, by putting we find: