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Linearization of the Product of Jacobi Polynomials. III

Published online by Cambridge University Press:  20 November 2018

Richard Askey  and
George Gasper
Richard Askey
Affiliation:
Mathematics Centre, Amsterdam, The Netherlands
George Gasper
Affiliation:
University of Wisconsin, Madison, Wisconsin University of Toronto, Toronto, Ontario
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In a series of papers [1; 2; 3; 4] the operation of linearizing the product of two Jacobi polynomials Pn(α, β)(x), α, β > –1, has been investigated and the existence of a natural Banach algebra associated with the linearization coefficients has been proven. This was proven for α + β + 1 ≧ 0 in [3] and for a slightly larger region in [4]. It was shown in [4] that such a Banach algebra does not exist for . The method used in [1; 3; 4] was to prove the non-negativity of the expansion coefficients from which the existence of the Banach algebra easily follows. However, as shown in [4], the coefficients for a subset of can be negative infinitely often and so a different method must be used for these values of α and β.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 332 - 338
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Askey, R., Linearization of the product of orthogonal polynomials, pp. 223-228 in Problems in analysis (Princeton Univ. Press, Princeton, N.J., 1970).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. Conference, Edwardsville, Illinois, 1967 (Southern Illinois Univ. Press, Carbondale, Illinois, 1968).Google Scholar
3. Gasper, G., Linearization of the product of Jacobi polynomials. I, Can. J. Math. 22 (1970), 171–175.Google Scholar
4. Gasper, G., Linearization of the product of Jacobi polynomials. II, Can. J. Math. 22 (1970), 582–593.Google Scholar
5. Gasper, G., Positivity and the convolution structure for Jacobi series, Ann. of Math, (to appear).Google Scholar
6. Igari, S. and Uno, Y., Banach algebra related to the Jacobi polynomials, Töhoku Math. J. 21 (1969), 668–673.Google Scholar
7. Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23 (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar