On a structure formula for classical q-orthogonal polynomials

摘要

The classical orthogonal polynomials are given as the polynomial solutions P_n(x) of the differential equation σ(x)y"(x) + τ(x)y'(x) + λ_ny(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 equation σ(x)Δ▽y(x) + τ(x)Δy(x) + λ_ny(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 equation σ(x)D_qD_(1/q)y(x) + τ(x)D_qy(x) - λ_(q,n)y(x) = 0, where D_qf(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 type σ(x)D_(1/q)P_n(x) = α_nP_(n+1)(x) + β_nP_n(x) + γ_nP_(n-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 yields σ(x)▽P_n(x) = α_nP_(n+1)(x) + β_nP_n(x) + γ_nP_(n-1)(x) (n ∈ N) and σ(x)P'_n(x) = α_nP_(n+1)(x) + β_nP_n(x) + γ_nP_(n-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.