Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-15T01:54:15.173Z Has data issue: false hasContentIssue false
  • English
  • Français

Linearization of the Product of Jacobi Polynomials. II

Published online by Cambridge University Press:  20 November 2018

George Gasper
George Gasper*
Affiliation:
University of Toronto, Toronto, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let [3, p. 170, (16)]

(1.1)

denote the Jacobi polynomial of order (α, β), α, β > – 1, and let g(k, m, n; α, β) be denned by

(1.2)

where Rn(α, β)(x) = Pn(α, β)(x)/Pn(α, β)(1). It is well known [1; 2; 4; 5; 6] that the harmonic analysis of Jacobi polynomials depends, at crucial points, on the answers to the following two questions.

Question 1. For which (α, β) do we have

(1.3)

Question 2. For which (α, β) do we have

(1.4)

where G depends only on (α, β)?

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 582 - 593
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Askey, R. and Hirschman, I. I. Jr., Weighted quadratic norms and ultraspherical polynomials. I, Trans. Amer. Math. Soc. 91 (1959), 294–313.Google Scholar
2. Askey, R. and Wainger, S., A dual convolution structure for Jacobi polynomials, pp. 25-36 in Orthogonal expansions and their continuous analogues, Proc. Conf., Edwardsville, Illinois, 1967 (Southern Illinois Univ. Press, Carbondale, Illinois, 1968).Google Scholar
3. Erdélyi, A., Higher transcendental functions, Vol. 2 (McGraw-Hill, New York, 1953).Google Scholar
4. Gasper, G., Linearization of the product of Jacobi polynomials. I, Can. J. Math. 22 (1970), 171–175.Google Scholar
5. Hirschman, I. I. Jr., Harmonic analysis and ultraspherical polynomials, Symposium on Harmonic Analysis and Related Integral Transforms, Cornell University, 1956.Google Scholar
6. Hylleraas, E. A., Linearization of products of Jacobi polynomials, Math. Scand. 10 (1962), 189–200.Google Scholar
7. Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23 (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar