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Recently, Casadei [4] provided an explicit formula for statistical marginal model in terms of Poisson-Gamma mixture. This model involving certain polynomials which play the key role in reference analysis of the signal and background model in counting experiments. The principal object of this paper is to present a natural further step toward the mathematical properties concerning this polynomials. We Ąrst obtain explicit representations for these polynomials in form of the Laguerre polynomials and the conĆuent hyper-geometric function and then based on these representations we derive a number of useful properties including generating functions, recurrence relations, differential equation, Rodrigues formula, Ąnite sums and integral transforms. [ABSTRACT FROM AUTHOR]

In this manuscript, we present the generalized hypergeometric function of the type F r r , r > 2 and extension of the K Laguerre polynomial for the K extended Laguerre polynomials A r , n , k α x . Additionally, we describe the K generating function, K recurrence relations, and K S Rodrigues formula. [ABSTRACT FROM AUTHOR]

We develop a new method for the construction of generalized classical polynomials (mainly of Hermite polynomials in the sense of A. Krall, J. Koekoek, R. Koekoek, H. Bavinck, L. Littlejohn, et al.). We construct a differential operator of infinite order whose eigenfunctions are polynomials of this kind. For the generalized Hermite polynomials, we analyze various characteristics typical of the classical orthogonal polynomials (such as orthogonality, generalized Rodrigues formula, three-term recurrence relation, and generating function). The universality of the proposed method is demonstrated by constructing generalized Legendre and Chebyshev polynomials of the first kind. [ABSTRACT FROM AUTHOR]

In this paper, we mainly concerned with an alternate form of the generating functions for a certain class of combinatorial numbers and polynomials. We give matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type formula, recurrence relation and derivative formula for the aforementioned combinatorial numbers. Besides, we present some plots of the generating functions for these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, but the two-variable Hermite polynomials and the number of cyclic derangements. We also present some applications of these relationships. By applying Laplace transform and Mellin transform respectively to the aforementioned functions, we give not only an infinite series representation, but also an interpolation function of these combinatorial numbers. We also provide a contour integral representation of these combinatorial numbers. In addition, we construct exponential generating functions for a new family of numbers arising from the linear combination of the numbers of cyclic derangements in the wreath product of the finite cyclic group and the symmetric group of permutations of a set. Finally, we analyse the aforementioned functions in probabilistic and asymptotic manners, and we give some of their relationships with not only the Laplace distribution, but also the standard normal distribution. Then, we provide an asymptotic power series representation of the aforementioned exponential generating functions. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to define and investigate more general multiple Charlier polynomials on the linear lattice ω N = { 0 , ω , 2 ω , ... } , ω ∈ R . We call these polynomials ω-multiple Charlier polynomials. Some of their properties, such as the raising operator, the Rodrigues formula, an explicit representation and a generating function are obtained. Also an (r + 1) th order difference equation is given. As an example we consider the case ω = 3 2 and define 3 2 -multiple Charlier polynomials. It is also mentioned that, in the case ω = 1 , the obtained results coincide with the existing results of multiple Charlier polynomials. [ABSTRACT FROM AUTHOR]

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to extend the Hurwitz–Lerch zeta function Φ δ , ς ; γ (ξ , s , υ ; p) involving the extension of the beta function (Choi et al. in Honam Math. J. 36(2):357–385, 2014). We also study the basic properties of this extended Hurwitz–Lerch zeta function which comprises various integral formulas, a derivative formula, the Mellin transform, and the generating relation. The fractional kinetic equation for an extended Hurwitz–Lerch zeta function is also obtained from an application point of view. Furthermore, we obtain certain interesting relations in the form of particular cases. [ABSTRACT FROM AUTHOR]

The holomorphic extensions F α (i z) , z ∈ ℂ d , of the Fourier transforms F α ≔ f ̂ α of a sequence of continuous real functions f α (x) , x ∈ R d , α ∈ N 0 d , with exponential decay at infinity, generates a sequence of polynomials Q β , β ∈ N 0 d , of degree | β | , that are biorthogonal to the distributional derivatives μ α ≔ (− 1) | α | f α (α) , α ∈ N 0 d. The generating function is a generalized Taylor series expansion of the Fourier-Laplace kernel e x ⋅ z in terms of μ ̂ α (i z) = z α F α (i z). Sufficient conditions on F α (i z) are given for the series expansion of holomorphic functions in terms of z α F α (i z) , α ∈ N 0 d , from which the generalized generating function is derived. In the case μ α = ω P α , where ω is a weight function and P α is a polynomial of degree | α | , Q α is a constant multiple of P α and for n = 1 , 2 , ... , the set { Q α : α ∈ N 0 d , | α | = n } , suitably normalized, is an orthonormal basis for the space of orthogonal polynomials of degree n associated with P α. [ABSTRACT FROM AUTHOR]