Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights

Authors:
Attila Máté, Paul Nevai and Thomas Zaslavsky

Journal:
Trans. Amer. Math. Soc. **287** (1985), 495-505

MSC:
Primary 42C05; Secondary 05A15

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768722-7

MathSciNet review:
768722

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the th polynomial orthonormal with respect to the weight where is an even integer. G. Freud conjectured and Al. Magnus proved that, writing , the expression has a limit as . It is shown that this expression has an asymptotic expansion in terms of negative even powers of . In the course of this, a combinatorial enumeration problem concerning one-dimensional lattice walk is solved and its relationship to a combinatorial identity of J. L. W. V. Jensen is explored.

**[1]**G. Freud,*Orthogonal polynomials*, Pergamon Press, Oxford and New York, 1966.**[2]**-,*On the greatest zero of an orthogonal polynomial*, Acta Sci. Math. Szeged**24**(1973), 91-97.**[3]**Géza Freud,*On the coefficients in the recursion formulae of orthogonal polynomials*, Proc. Roy. Irish Acad. Sect. A**76**(1976), no. 1, 1–6. MR**0419895****[4]**H. W. Gould,*Generalization of a theorem of Jensen concerning convolutions*, Duke Math. J.**27**(1960), 71–76. MR**0114766****[5]**H. W. Gould and J. Kaucký,*Evaluation of a class of binomial coefficient summations*, J. Combinatorial Theory**1**(1966), 233–247. MR**0210607****[6]**Peter Henrici,*Applied and computational complex analysis. Vol. 2*, Wiley Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR**0453984****[7]**J. L. W. V. Jensen,*Sur une identité d’Abel et sur d’autres formules analogues*, Acta Math.**26**(1902), no. 1, 307–318 (French). MR**1554966**, https://doi.org/10.1007/BF02415499**[8]**Charles Jordan,*Calculus of finite differences*, Third Edition. Introduction by Harry C. Carver, Chelsea Publishing Co., New York, 1965. MR**0183987****[9]**Donald E. Knuth,*The art of computer programming*, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR**0378456****[10]**John S. Lew and Donald A. Quarles Jr.,*Nonnegative solutions of a nonlinear recurrence*, J. Approx. Theory**38**(1983), no. 4, 357–379. MR**711463**, https://doi.org/10.1016/0021-9045(83)90154-5**[11]**Attila Máté and Paul Nevai,*Asymptotics for solutions of smooth recurrence equations*, Proc. Amer. Math. Soc.**93**(1985), no. 3, 423–429. MR**773995**, https://doi.org/10.1090/S0002-9939-1985-0773995-6**[12]**H. N. Mhaskar and E. B. Saff,*Extremal problems for polynomials with exponential weights*, Trans. Amer. Math. Soc.**285**(1984), no. 1, 203–234. MR**748838**, https://doi.org/10.1090/S0002-9947-1984-0748838-0**[13]**Paul G. Nevai,*Orthogonal polynomials*, Mem. Amer. Math. Soc.**18**(1979), no. 213, v+185. MR**519926**, https://doi.org/10.1090/memo/0213**[14]**-,*Orthogonal polynomials associated with*, Canadian Math. Soc. Conference Proc.**3**(1983), 263-285.**[15]**-,*Asymptotics for orthogonal polynomials associated with*, SIAM J. Math. Anal.**15**(1984).**[16]**E. A. Rakhmanov,*Asymptotic properties of orthogonal polynomials on the real axis*, Mat. Sb. (N.S.)**119(161)**(1982), no. 2, 163–203, 303 (Russian). MR**675192****[17]**G. Szegö,*Orthogonal polynomials*, 4th ed., Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, R.I., 1978.**[18]**A. Zygmund,*Trigonometric series: Vols. I, II*, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR**0236587****[19]**Al. Magnus,*A proof of Freud's conjecture about the orthogonal polynomials related to**for integer*(manuscript).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
42C05,
05A15

Retrieve articles in all journals with MSC: 42C05, 05A15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768722-7

Keywords:
Asymptotic expansion,
combinatorial enumerations,
combinatorial identities,
Freud's conjecture,
Jensen's identity,
orthogonal polynomials

Article copyright:
© Copyright 1985
American Mathematical Society