Plancherel–Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems
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We study the Plancherel–Rotach asymptotics of four families of orthogonal polynomials: the Chen–Ismail polynomials, the Berg–Letessier–Valent polynomials, and the Conrad–Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems, and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems.
KeywordsAsymptotics The Chen–Ismail polynomials The Berg–Letessier–Valent polynomials The Conrad–Flajolet polynomials Turning points Difference equation technique Indeterminate moment problems Nevanlinna functions Asymptotics of zeros Plancherel–Rotach asymptotics
Mathematics Subject Classification (2010)41A60 33C47 30E05
We would like to thank Professor Doron Lubinsky for his helpful comments and discussions. We are also grateful to the editors and referees for their valuable suggestions and extremely careful reading of the manuscript. Dan Dai was partially supported by a grant from City University of Hong Kong (Project No. 7002883) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 100910). Mourad E.H. Ismail was partially supported by NPST Program of King Saud University (Project No. 10-MAT1293-02) and by the DSFP at King Saud University in Riyadh, and by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1014111).
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