Hlawka, Edmund An asymptotic formula for the Laguerre polynomials. (Eine asymptotische Formel der Laguerreschen Polynome.) (German) Zbl 0015.02001 Monatsh. Math. Phys. 42, 275-278 (1935). Also reviewed in JFM 61.0390.03 by W. Hahn (Berlin).Using the differential equation of the Laguerre polynomials \(L_n(x)\) defined by \[ (1-z)^{-1}\exp\left(-\frac{xz}{1-z}\right)=\sum_{n=0}^\infty L_n(x)x^n \] the author obtains the formula \[ e^{-x/2}L_n(x)=J_0(2\sqrt{nx})=(n^{-3/4}), \] where \(x>0\) is bounded and \(J_0\) denotes Bessel’s function of the first kind. – Cf. the paper of E. M. Wright in J. Lond. Math. Soc. 7, 256–262 (1932; Zbl 0005.35102). Reviewer: G. Szegö (St. Louis, Mo.) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page MSC: 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Keywords:Laguerre polynomials; Special functions Citations:Zbl 0005.35102; JFM 58.0306.02; Zbl 0010.02303; JFM 60.0300.02; JFM 61.0390.03 PDFBibTeX XMLCite \textit{E. Hlawka}, Monatsh. Math. Phys. 42, 275--278 (1935; Zbl 0015.02001) Full Text: DOI References: [1] L. Fejér, Comptes rendus,147 (1908), S. 1040; Math. és term. tud. Ért.27 (1909), S. 1–33. [2] O. Perron, Arch. der Math. u. Phys. (3)22 (1914), S. 329–340. Journal für die r. u. a. Math.151 (1921), S. 63–78. [3] G. Szegö, Math. Zeitschr.25 (1926), S. 99–102. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.