
We derive a family of reproducing kernels for the q-Jacobi polynomials Φ n ( α , β ) ( x ) = 2 Φ 1 ( q − n , q n − 1 + β ; q α ; q , q x ) \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 Φ n ( α , β ) ( x ) \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.
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Classical hypergeometric functions, \({}_2F_1\), Other special functions
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Classical hypergeometric functions, \({}_2F_1\), Other special functions
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