Global Asymptotics for Meixner‐Pollaczek Polynomials with a Varying Parameter. Issue 4 (13th February 2013)

• Record Type: Journal Article
• Authors: Wang, Jun; Qiu, Weiyuan; Wong, R.
• Is Part Of: Studies in applied mathematics. Volume 130:Issue 4(2013)
• Page Start: 345
• Page End: 392
• Publication Date: 2013-02-13
• Subjects: Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/j.1467-9590.2012.00570.x
• Languages: English
• ISSNs: 0022-2526

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>In this paper, we study the uniform asymptotics of the Meixner‐Pollaczek polynomials <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1xrk1mdq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00358711:sapm570:equation:sapm570-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>P</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>;</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> with varying parameter <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1xrk1mc5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00358711:sapm570:equation:sapm570-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:math> as <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1xrk1mbm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00358711:sapm570:equation:sapm570-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math>, where <italic>A</italic> > 0 is a constant. Two asymptotic expansions are obtained, which hold uniformly for <italic>z</italic> in two overlapping regions which together cover the whole complex plane. One involves parabolic cylinder functions, and the other is in terms of elementary functions only. Our approach is based on the steepest descent method for oscillatory Riemann‐Hilbert problems first introduced by Deift and Zhou [1].</p> </abstract>