Reproducing kernels for 𝑞-Jacobi polynomials

Al-Salam, Waleed A.; Ismail, Mourad E. H.

doi:10.1090/S0002-9939-1977-0454104-5

Reproducing kernels for $q$-Jacobi polynomials
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by Waleed A. Al-Salam and Mourad E. H. Ismail PDF

Proc. Amer. Math. Soc. 67 (1977), 105-110 Request permission

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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