Second structure relation for q-semiclassical polynomials of the Hahn Tableau

Abstract

The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a generalization of the classical ones) we find only in the literature the first structure relation. In this paper, a second structure relation is deduced. In particular, by means of a general finite-type relation between a q-semiclassical polynomial sequence and the sequence of its q-differences such a structure relation is obtained.

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

“…Some recent papers have applied this operator to construct families of orthogonal polynomials as well as to investigate some approximation and optimization problems; cf., e.g., [1,[3][4][5][33][34][35][36]. This operator unifies two well known difference operators.…”

“…3 The functions cos q,ω (z; ·); sin q,ω (z; ·); Cos q,ω (z; ·) and Sin q,ω (z; ·) satisfy the following second order initial value problems:…”

“…(1.3)] Hahn introduced the difference operator D q,ω , that extends both Jackson's q-difference operator, D q , and the difference operator Δ ω ; see definitions below. Hahn's operator has been considered from computational points of view, in particular in the determination of new families of orthogonal polynomials; see, e.g., [2][3][4][5]. Calculi, difference equations and special functions based on the q-difference operator, as well as the difference operator Δ ω , have been considered extensively; see, e.g., the monographs [6][7][8][9][10][11] and the papers [1,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

“…The Hahn difference operator has been used in finding families of orthogonal polynomials as well to determine some approximation problems (see [11][12][13]). The right inverse of the Hahn difference operator was proposed by Aldwoah in 2009 [14,15].…”

On nonlocal Robin boundary value problems for Riemann–Liouville fractional Hahn integrodifference equation

“…The Hahn difference operator has been employed to construct families of orthogonal polynomials and investigate some approximation problems (see [14][15][16] and the references therein).…”