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.
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.
