Representations for the first associated q-classical orthogonal polynomials

摘要

Let {pk(x;q)} be any system of the q-classical orthogonal polynomials, and let ϱ be the corresponding weight function, satisfying the q-difference equation Dq(σϱ)=τϱ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {pk(1)(x;q)} be associated polynomials of the polynomials {pk(x;q)}. Explicit forms of the coefficients bn,k and cn,k in the expansions pn−1(1)(x;q)=∑k=0n−1bn,kϑk(x),pn−1(1)(x;q)=∑k=0n−1cn,kpk(x;q)are given in terms of basic hypergeometric functions. Here ϑk(x) equals xk 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 pn−1(1)(x;q) in a special form, recurrence relations (in k) for bn,k and cn,k are obtained in terms of σ and τ.