Related papers: A note on a series containing the Laguerre polynom…
Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields.
This paper delves into classical multiple orthogonal polynomials with an arbitrary number of weights, including Jacobi-Pi\~neiro, Laguerre of both first and second kinds, as well as multiple orthogonal Hermite polynomials. Novel explicit…
In this paper we introduce a new approach to the concept of multipolynomials and generalize several results of the homogeneous polynomials and symmetric multilinear applications. We also present an abstract approach to the concept of…
Very recently a new series representation of Humbert's double hypergeometric series $\Phi_3$ in series of Gauss's $_2F_1$ function was given by one of us. The aim of this short research note is to provide an alternative proof of the result.…
The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values…
The Stieltjes constants $\gamma_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and…
For any complex parameters $x$ and $\nu$, we provide a new class of linear inversion formulas $T = A(x,\nu) \cdot S \Leftrightarrow S = B(x,\nu) \cdot T$ between sequences $S = (S_n)_{n \in \mathbb{N}^*}$ and $T = (T_n)_{n \in…
The coefficients occurring in summation formulae of the Lubbock type are shown to be generalised Bernoulli polynomials which turn up in subdivision questions such as quantum field theory around a conical singularity and on spherical lunes.…
A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author's previous results on a transformation formula for Milne's multivariate generalization of basic…
According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and…
We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic…
We study the bispectrality of Laguerre type polynomials, which are defined by taking suitable linear combinations of a fixed number of consecutive Laguerre polynomials. These Laguerre type polynomials are eigenfunctions of higher-order…
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…
This paper addresses two primary objectives in the realm of classical multiple orthogonal polynomials with an arbitrary number of weights. Firstly, it establishes new and explicit hypergeometric expressions for type I Hahn multiple…
Using a realization of the q-exponential function as an infinite multiplicative sereis of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and…
Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a…
In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the…
We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental…
Generalized Pl\"ucker numbers are defined to count certain types of tangent lines of generic degree $d$ complex projective hypersurfaces. They can be computed by identifying them as coefficients of GL(2)-equivariant cohomology classes of…
The 3-term recurrence relation for Hermite polynomials was recently generalized to a recurrence relation for Wronskians of Hermite polynomials. In this note, a similar generalization for Laguerre polynomials is obtained.