Journal article
On a structure formula for classical q-orthogonal polynomials



Publication Details
Authors:
Koepf, W.; Schmersau, D.
Publication year:
2001
Journal:
Journal of Computational and Applied Mathematics
Pages range:
99-107
Volume number:
136
Start page:
99
End page:
107
ISSN:
0377-0427

Abstract
The classical orthogonal polynomials are given as the polynomial solutions P-n(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 q-orthogonal polynomials of the Hahn tableau are the polynomial solutions of the q-difference 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)/(q-1)x, q not equal 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 sigma (x)D1/qPn(x) = alpha P-n(n+1)(x) + beta P-n(n)(x) + gamma P-n(n-1)(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 P-n(n+1)(x) + beta P-n(n)(x) + gamma P-n(n-1)(x) (n is an element of N) and sigma (x)P-n' (x) = alpha P-n(n+1)(x) + beta P-n(n)(x) + gamma P-n(n-1)(x) (n is an element ofN). Whereas the latter formulas are well-known, 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.


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