{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,3]],"date-time":"2026-09-03T01:48:49Z","timestamp":1788400129111,"version":"build-2803163510"},"reference-count":46,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs $(a,z)$ one has the large-n asymptotic behavior $L_n^{(-a)}(-z) \\sim C(a,z) n^{-a\/2-1\/4} e^{2\\sqrt{nz}}$. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an \u201cexp-arc\u201d method giving a natural bridge between converging series and effective asymptotics. (A corrected version of this paper has been appended to the originally posted pdf).<\/jats:p>","DOI":"10.1137\/07068031x","type":"journal-article","created":{"date-parts":[[2008,10,17]],"date-time":"2008-10-17T18:04:21Z","timestamp":1224266661000},"page":"3285-3312","source":"Crossref","is-referenced-by-count":32,"title":["Effective Laguerre Asymptotics"],"prefix":"10.1137","volume":"46","author":[{"given":"David","family":"Borwein","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonathan M.","family":"Borwein","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Richard E.","family":"Crandall","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,10,17]]},"reference":[{"key":"R1","unstructured":"M. Ambramowitz and I. Stegun,\n                      Handbook of Mathematical Functions\n                      , NBS (now NIST), Washington, D.C., 1965."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/39\/40\/001"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2007.10.003"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/s11139-006-0243-3"},{"key":"R5","unstructured":"J. M. Borwein and P. B. Borwein,\n                      Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity\n                      , John Wiley, New York, 1987. Paperback edition 1998."},{"key":"R6","doi-asserted-by":"crossref","unstructured":"J. Borwein and M. Chamberland,\n                      Integer powers of Arcsin\n                      , Int. J. Math. Math. Sci., 2007, Article ID 19381, http:\/\/www.hindawi.com\/GetArticle\/aspx?doi=10.1155\/2007\/19381.","DOI":"10.1155\/2007\/19381"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1080\/10586458.2004.10504540"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1080\/10586458.2004.10504541"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(98)00147-2"},{"key":"R10","unstructured":"K. Chao and R. Plymen,\n                      \n                        A new bound for the smallest\n                        x\n                        with $\\pi(x)>{\\rm li}(x)$\n                      \n                      , arXiv:math\/0509312v5."},{"key":"R11","unstructured":"R. Crandall,\n                      \n                        Unified algorithms for polylogarithm,\n                        L\n                        -series, and zeta variants\n                      \n                      , http:\/\/www.reed.edu\/~crandall, 2008, preprint."},{"key":"R12","unstructured":"R. Crandall,\n                      Alternatives to the Riemann\u2013Siegel formula\n                      , http:\/\/www.reed.edu\/~crandall, 2008, preprint."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1063\/1.449483"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1119\/1.13650"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1006\/jath.2002.3725"},{"key":"R16","first-page":"235","volume":"24","author":"Erd\u00e9lyi A.","year":"1960","journal-title":"J. Indian Math. 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Thron,\n                      Oval convergence regions for continued fractions $K(a_n\/1)$\n                      , preprint, 1986.","DOI":"10.1007\/BFb0075935"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF03320986"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.4310\/MAA.1999.v6.n2.a1"},{"key":"R24","unstructured":"L. Lorentzen and H. Waadeland,\n                      Continued Fractions with Applications\n                      , North-Holland, Amsterdam, 1992."},{"key":"R25","first-page":"293","volume":"1","author":"Love E.","year":"1997","journal-title":"J. Inequal. Appl."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/1012044"},{"key":"R27","unstructured":"F. Olver,\n                      Asymptotics and Special Functions\n                      , Academic Press, New York, 1974."},{"key":"R28","unstructured":"R. B. 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Math., to appear."},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2004.1307"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2005.02.024"},{"key":"R35","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1515\/crll.1921.151.63","volume":"151","author":"Perron O.","year":"1921","journal-title":"Journal f\u00fcr die reine und angewandte Math"},{"key":"R36","unstructured":"G. Szeg\u00f6,\n                      Orthogonal Polynomials\n                      , Amer. Math. Soc. Colloq. Publ., 4th ed., 23, American Mathematical Society, Providence, RI, 1975."},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1002\/sapm1993893233"},{"key":"R38","unstructured":"N. Temme,\n                      private communication\n                      , 2006."},{"key":"R39","doi-asserted-by":"crossref","unstructured":"S. Thangavelu,\n                      Lectures on Hermite and Laguerre Expansions\n                      , Math. 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