Co-recursive associated Jacobi polynomials
Starting from the three term recurrence relation for a sequence of orthogonal polynomials \[ p_{n+ 2}(x)= (x- \beta_{n+ 1}) p_{n+ 1}(x)- \gamma_{n+ 1} p_n(x)\;(n\geq 0),\;p_0(x)= 1,\;p_1(x)= x- \beta_0 \] with \(\beta_n\in {\mathcal C}\), \(\gamma_n\in {\mathcal C}\backslash \{0\}\), the co-recursive associated polynomials are defined by shifting the index \(n\) to \(n+ c\) in \(\beta_n\), \(\gamma_n\) \((n\geq 0)\) and replacing \(\beta_0\) by \(\beta_0+ \nu\). The author studies the case of the Jacobi polynomials and gives for the co-recursive associated polynomials explicit representations, the orthogonality measure, a fourth order differential equation and he moreover treats 9 limiting cases (including the Laguerre case limit).
scientific article; zbMATH DE number 770347
scientific article