Explicit Formulas for Meixner Polynomials

There are many authors who have studied polynomials and their properties (see [1–10]). The polynomials are applied in many areas ofmathematics, for instance, continued fractions, operator theory, analytic functions, interpolation, approximation theory, numerical analysis, electrostatics, statistical quantummechanics, special functions, number theory, combinatorics, stochastic processes, sorting, and data compression. The research area of obtaining explicit formulas for polynomials has received much attention from Srivastava [11, 12], Cenkci [13], Boyadzhiev [14], and Kruchinin [15–17]. The main purpose of this paper is to obtain explicit formulas for the Meixner polynomials of the first and second kinds. In this paper we use a method based on a notion of composita, which was presented in [18].

The main purpose of this paper is to obtain explicit formulas for the Meixner polynomials of the first and second kinds.
In this paper we use a method based on a notion of composita, which was presented in [18].Definition 1. Suppose () = ∑ >0 ()  is the generating function, in which there is no free term (0) = 0. From this generating function we can write the following condition: The expression (, ) is composita [19] and it is denoted by  Δ (, ).Below we show some required rules and operations with compositae.

Bessel Polynomials
Krall and Frink [20] considered a new class of polynomials.Since the polynomials connected with the Bessel function, they called them the Bessel polynomials.The explicit formula for the Bessel polynomials is Then Carlitz [21] defined a class of polynomials associated with the Bessel polynomials by The   () is defined by the explicit formula [22] ) and by the following generating function: Using the notion of composita, we can obtain an explicit formula (9) from the generating function (10).

Meixner Polynomials of the First Kind
The Meixner polynomials of the first kind are defined by the following recurrence relation [23,24]: where Using (16), we obtain the first few Meixner polynomials of the first kind: The Meixner polynomials of the first kind are defined by the following generating function [22]: Using the notion of composita, we can obtain an explicit formula   (; , ) from the generating function (19).
First, we represent the generating function (19) as a product of generating functions () (), where the functions () and () are expanded by binomial theorem: Coefficients of the generating functions () = ∑ ∞ =0 ()  and () = ∑ ∞ =0 ()  are, respectively, given as follows: Then, using (6), we obtain a new explicit formula for the Meixner polynomials of the first kind:

Meixner Polynomials of the Second Kind
The Meixner polynomials of the second kind are defined by the following recurrence relation [23,24]: Using ( 23 The Meixner polynomials of the second kind are defined by the following generating function [22]: Using the notion of composita, we can obtain an explicit formula   (; , ) from the generating function (26).
We represent the generating function (26) as a product of generating functions () (), where Next we represent () as a composition of generating functions  1 ( 2 ()) and we expand  1 () by binomial theorem: ) .