Asymptotic behaviour of Jacobi polynomials and their zeros (Q2789853)

There are numerous limit relations between orthogonal polynomials. For example R. Koekoek and R. F. Swart\-touw proved the limit relation NEWLINE\[NEWLINE\lim\limits_{\beta \to \infty} P_n^{(\alpha,\beta)} \left( 1-\frac{2x}{\beta}\right)=L_n^{\alpha}(x) NEWLINE\]NEWLINE between the Laguerre polynomials (\(L_n^{(\alpha)}(x)\)) and Jacobi polynomials (\(P_n^{(\alpha,\beta)}(x)\))NEWLINENEWLINE\noindent (see \url{http://arxiv.org/pdf/math/9602214.pdf}; page 48.).NEWLINENEWLINEIn this paper the authors obtain analogues of the above relation. The main result is the representation NEWLINE\[NEWLINEP_n^{(\alpha,\beta + \mu)}\left( 1-\frac{2x}{\beta}\right)=\sum\limits_{j=0}^n R_{n,j}(x) \beta^{-j}NEWLINE\]NEWLINE where \(R_{n,0} (x) = L_n^{(\alpha)}(x), \) and \(R_{n,j}(x)\; (j>1)\) is a suitable linear combination of the polynomials \(x \frac{d}{dx} L_n^{(\alpha)}(x),\;\) \(x^2 \frac{d^2}{dx^k} L_n^{(\alpha)}(x), \dots \;\) The asymptotic behaviour of zeros of Jacobi polynomials and its derivatives as \(\beta \to \infty\) is also given.