Constructive Approximation

, Volume 14, Issue 1, pp 113–150

Uniform Asymptotic Expansions for Meixner Polynomials


  • X. -S. Jin
    • Department of Mathematics University of Manitoba Winnipeg Canada, R3T 2N2
  • R. Wong
    • Department of Mathematics City University of Hong Kong Tat Chee Avenue Kowloon Hong Kong

DOI: 10.1007/s003659900066

Cite this article as:
-S. Jin, X. & Wong, R. Constr. Approx. (1998) 14: 113. doi:10.1007/s003659900066


Meixner polynomials mn(x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are \(j(x;\beta,c) = \frac{c^x(\beta)_x}{x!} \qquad \mbox{at}\quad x=0,1,2\ldots.\) Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for mn(nα;β,c) as \(n\to\infty\) . One holds uniformly for \(0 < \epsilon\le \alpha\le 1+a\) , and the other holds uniformly for \(1-b\le \alpha\le M < \infty\) , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases.

Key words. Meixner polynomials Uniform asymptotic expansions, Steepest descent method Parabolic cylinder function. AMS Classification. Primary 41A60, 33C45. <lsiheader> <onlinepub>8 May, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;R.A. DeVore E.B. Saff&lsilt;/a&lsigt; <pdfname>14n1p113.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>

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