Representations for the first associated q-classical orthogonal polynomials

Abstract

Let {p k(x; q)} be any system of the q-classical orthogonal polynomials, and let ϱ be the corresponding weight function, satisfying the q-difference equation D q ( σϱ)= τϱ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let { p k (1)( x; q)} be associated polynomials of the polynomials {p k(x; q)} . Explicit forms of the coefficients b n, k and c n, k in the expansions p n−1 (1)(x;q)= ∑ k=0 n−1 b n,kϑ k(x), p n−1 (1)(x;q)= ∑ k=0 n−1 c n,kp k(x; q) are given in terms of basic hypergeometric functions. Here ϑ k ( x) equals x k if σ +(0)=0, or ( x; q) k if σ +(1)=0, where σ +( x)≔ σ( x)+( q−1) xτ( x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively. Writing the second-order nonhomogeneous q-difference equation satisfied by p n−1 (1)( x; q) in a special form, recurrence relations (in k) for b n, k and c n, k are obtained in terms of σ and τ.