Journal article
On a structure formula for classical qorthogonal polynomials
Publication Details
Authors:  Koepf, W.; Schmersau, D.

Journal:  Journal of Computational and Applied Mathematics 
Abstract
The classical orthogonal polynomials are given as the polynomial solutions Pn(x) of the differential equation sigma (x)y " (x) + tau (x)y ' (x) + lambda (n)y(x) = 0, where sigma (x) turns out to be a polynomial of at most second degree and tau (x) is a polynomial of first degree. In a similar way, the classical discrete orthogonal polynomials are the polynomial solutions of the difference equation sigma (x)Delta dely(x) + tau (x)Deltay(x) + lambda (n)y(x) = 0, where Deltay(x) = y(x + 1)  y(x) and dely(x) = y(x)  y(x  1) denote the forward and backward difference operators, respectively. Finally, the classical qorthogonal polynomials of the Hahn tableau are the polynomial solutions of the qdifference equation sigma (x)D(q)D(1/q)y(x) + tau (x)D(q)y(x) + lambda (q,n)y(x) = 0 where Dqf(x) = f(qx)  f(x)/(q1)x, q not equal 1 denotes the qdifference operator. We show by a purely algebraic deduction  without using the orthogonality of the families considered  that a structure formula of the type sigma (x)D1/qPn(x) = alpha Pn(n+1)(x) + beta Pn(n)(x) + gamma Pn(n1)(x) (n is an element of 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 sigma (x)delP(n)(x) = alpha Pn(n+1)(x) + beta Pn(n)(x) + gamma Pn(n1)(x) (n is an element of N) and sigma (x)Pn' (x) = alpha Pn(n+1)(x) + beta Pn(n)(x) + gamma Pn(n1)(x) (n is an element ofN). Whereas the latter formulas are wellknown, their previous deduction used the orthogonality property. Hence our approach is also of interest in these cases. (C) 2001 Elsevier Science B.V. All rights reserved.