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Proceedings of the American Mathematical Society

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Reproducing kernels for $ q$-Jacobi polynomials

Authors: Waleed A. Al-Salam and Mourad E. H. Ismail
Journal: Proc. Amer. Math. Soc. 67 (1977), 105-110
MSC: Primary 33A65
MathSciNet review: 0454104
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Abstract: We derive a family of reproducing kernels for the q-Jacobi polynomials $ \Phi _n^{(\alpha ,\beta )}(x){ = _2}{\Phi _1}({q^{ - n}},{q^{n - 1 + \beta }};{q^\alpha };q,qx)$. This is achieved by proving that the polynomials $ \Phi _n^{(\alpha ,\beta )}(x)$ satisfy a discrete Fredholm integral equation of the second kind with a positive symmetric kernel, then applying Mercer's theorem.

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Keywords: q-Jacobi polynomials, q-Laguerre polynomials, connection relations, bilinear forms, reproducing kernels, discrete integral equations, Mercer's theorem, symmetric kernels
Article copyright: © Copyright 1977 American Mathematical Society