On a structure formula for classical q-orthogonal polynomials

摘要

The classical orthogonal polynomials are given as the polynomial solutions Pn(x) of the differential equationσ(x)y″(x)+τ(x)y′(x)+λny(x)=0,where σ(x) turns out to be a polynomial of at most second degree and τ(x) is a polynomial of first degree. In a similar way, the classical discrete orthogonal polynomials are the polynomial solutions of the difference equationσ(x)Δ∇y(x)+τ(x)Δy(x)+λny(x)=0,where Δy(x)=y(x+1)−y(x) and ∇y(x)=y(x)−y(x−1) denote the forward and backward difference operators, respectively. Finally, the classical q-orthogonal polynomials of the Hahn tableau are the polynomial solutions of the q-difference equationσ(x)DqD1/qy(x)+τ(x)Dqy(x)+λq,ny(x)=0,whereDqf(x)=f(qx)−f(x)(q−1)x,q≠1,denotes the q-difference operator. We show by a purely algebraic deduction — without using the orthogonality of the families considered — that a structure formula of the typeσ(x)D1/qPn(x)=αnPn+1(x)+βnPn(x)+γnPn−1(x)(n∈N≔{1,2,3,…})is valid. Moreover, our approach does not only prove this assertion, but generates the form of this structure formula. A similar argument applies to the discrete and continuous cases and yieldsσ(x)∇Pn(x)=αnPn+1(x)+βnPn(x)+γnPn−1(x)(n∈N)andσ(x)Pn′(x)=αnPn+1(x)+βnPn(x)+γnPn−1(x)(n∈N).Whereas the latter formulas are well-known, their previous deduction used the orthogonality property. Hence our approach is also of interest in these cases.