Szehr, Oleg; Zarouf, Rachid On the asymptotic behavior of Jacobi polynomials with first varying parameter. (English) Zbl 1517.33004 J. Approx. Theory 277, Article ID 105702, 31 p. (2022). For real \(\alpha, \beta\) let \(P_n^{(\alpha,\beta)}(x)=\sum_{\mu=0}^{n}\binom{n+\alpha}{n-\mu} \binom{n+\beta}{\mu}\left(\frac{x-1}{2}\right)^{\mu} \left(\frac{x+1}{2}\right)^{n-\mu}\), \(x\in(-1,1)\), denote the Jacobi polynomial of degree \(n\).The Authors study Jacobi polynomials with first varying parameter, that is, the polynomials \(P_n^{(an+\alpha,\beta)}(1-2\lambda^2)\) with \(a>-1\) and \(\lambda\in(0,1)\). They give “a new representation of Jacobi polynomials in terms of two integrals” and use it to study the asymptotic behaviour of \(\lambda^{an}P_n^{(an+\alpha,\beta)}(1-2\lambda^2)\) depending on the value of a parameter \(a\). Reviewer: Małgorzata Michalska (Lublin) Cited in 1 Document MSC: 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 30E15 Asymptotic representations in the complex plane Keywords:Jacobi polynomials; integral representation; method of stationary phase; Laplace’s method PDFBibTeX XMLCite \textit{O. Szehr} and \textit{R. Zarouf}, J. Approx. Theory 277, Article ID 105702, 31 p. (2022; Zbl 1517.33004) Full Text: DOI arXiv References: [1] Bender, C. M.; Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory (1978), Springer · Zbl 0417.34001 [2] Bleistein, N.; Handelsman, R. A., Asymptotic Expansions of Integrals (1986), Dover Publications, Inc.: Dover Publications, Inc. New York [3] Blyudze, M. Y.; Shimorin, S. M., Estimates of the norms of powers of functions in certain Banach space, J. Math. Sci., 80, 4, 1880-1891 (1996) · Zbl 0860.46012 [4] A. Borichev, K. 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