On the properties of special functions on the linear-type lattices

Abstract

We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices: the linear type lattices. In particular, using the integral representation of the solutions we obtain several difference-recurrence relations for such functions. Finally, applications to q-classical polynomials are given. 2000 Mathematics Subject Classification 33D15, 33D45 1 Γ(s − z + µ) Γ(s − z) , ν ∈ R,

On the Complex Difference Equation of Hypergeometric Type on Non-uniform Lattices

“…To our best knowledge, the adjoint difference equation of Eq. (7) for the case of uniform lattices such as x(s) = s and x(s) = q s has already been obtained in [27,28,29].…”

“…On the other hand, many researchers like R.Álvarez-Nodarse, K. L. Cardoso, I. Area, E. Godoy, A. Ronveaux, A. Zarzo, E. Hille, and E. L. Ince [27,28,29,30,31] studied particular solutions for the adjoint differential…”

Recurrence Relations of the Hypergeometric-type functions on the quadratic-type lattices

“…This paper is motivated by the work done by R. Álvarez-Nodarse et al [4,5,6]. In fact, in [5], the authors considered the continuous case and obtained some recurrence relations for the Jacobi, Laguerre and Hermite polynomials in addition to the difference analogues of hypergeometric functions on the linear lattices x(s) = s to apply the theory for the Hahn, Meixner, Charlier and Kravchuk polynomials.…”

“…In fact, in [5], the authors considered the continuous case and obtained some recurrence relations for the Jacobi, Laguerre and Hermite polynomials in addition to the difference analogues of hypergeometric functions on the linear lattices x(s) = s to apply the theory for the Hahn, Meixner, Charlier and Kravchuk polynomials. In [6], the authors studied the difference analogues of hypergeometric functions on the linear-type lattices, and later applied the theory to the q-polynomials on q-linear lattices x(s) = c 1 q s + c 2 while considering the big q-Jacobi, Alternative q-Charlier polynomials as applications. For the quadratic case, there are only a few known recurrence relations (see the results by Suslov in [19,20]).…”

“…For the quadratic case, there are only a few known recurrence relations (see the results by Suslov in [19,20]). As such, the main aim of the present paper is to extend the results of [6] to the general quadratic-type lattice and develope constructive approach for the recurrence relations of such functions. Here we go further and consider the recurrence relations for the functions on the quadratic-type lattices and apply the theory to the q-Racah and dual Hahn polynomials, thus expanding the results of the papers [4,5,6].…”