Generalized little 𝑞-Jacobi polynomials as eigensolutions of higher-order 𝑞-difference operators

Vinet, Luc; Zhedanov, Alexei

doi:10.1090/S0002-9939-01-06047-6

Generalized little $q$-Jacobi polynomials as eigensolutions of higher-order $q$-difference operators
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by Luc Vinet and Alexei Zhedanov PDF

Proc. Amer. Math. Soc. 129 (2001), 1317-1327 Request permission

Abstract:

We consider the polynomials $p_n(x;a,b;M)$ obtained from the little $q$-Jacobi polynomials $p_n(x;a, b)$ by inserting a discrete mass $M$ at $x=0$ in the orthogonality measure. We show that for $a=q^j, \; j=0,1,2,\dots$, the polynomials $p_n(x;a,b;M)$ are eigensolutions of a linear $q$-difference operator of order $2j+4$ with polynomial coefficients. This provides a $q$-analog of results recently obtained for the Krall polynomials.

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