Results on the associated Jacobi and Gegenbauer polynomials
In this paper the author first proves that the coefficients \(b_{nk}^{(\alpha, \beta)}\) which occur in the expansion of associated Jacobi polynomials \(P_ n^{(\alpha, \beta)} (x;c)\) due to \textit{J. Wimp} [Can. J. Math. 39, 983-1000 (1987; Zbl 0643.33009)] in terms of classical Jacobi polynomials \(P_ n^{(\alpha, \beta)}(x)\) obey a second order recurrence relation which yields the explicit forms for \(b_{nk}^{(\alpha, \beta)}\) in the special cases (i) \(\alpha = \beta\), and (ii) \(\alpha = 1/2\) or \(\beta = 1/2\). The explicit form of \(b_{nk}^{\alpha, \beta)}\) when \(\alpha = \beta\) has been used to give a generalization of the corresponding classical result for associated Gegenbauer polynomials.
scientific article; zbMATH DE number 540458