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), 495505
MSC:
Primary 42C05; Secondary 05A15
MathSciNet review:
768722
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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 onedimensional lattice walk is solved and its relationship to a combinatorial identity of J. L. W. V. Jensen is explored.
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 [2]
 , On the greatest zero of an orthogonal polynomial, Acta Sci. Math. Szeged 24 (1973), 9197.
 [3]
 , On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A 76 (1976), 16. MR 0419895 (54:7913)
 [4]
 H. W. Gould, Generalization of a theorem of Jensen concerning convolutions, Duke Math. J. 27 (1960), 7176. MR 0114766 (22:5585)
 [5]
 H. W. Gould and J. Kauchy, Evaluation of a class of binomial coefficient summations, J. Combin. Theory 1 (1966), 233247. MR 0210607 (35:1493)
 [6]
 P. Henrici, Applied and computational complex analysis, Vol. 2, Wiley, New York and London, 1977. MR 0453984 (56:12235)
 [7]
 J. L. W. V. Jensen, Sur une identité d'Abel et sur d'autres formules analogues, Acta Math. 26 (1902), 307318. MR 1554966
 [8]
 C. Jordan, Calculus of finite differences, 3rd ed., Chelsea, New York, 1965. MR 0183987 (32:1463)
 [9]
 D. E. Knuth, The art of computer programming, Vol. 1, 2nd ed., AddisonWesley, Reading, Mass., 1973. MR 0378456 (51:14624)
 [10]
 J. S. Lew and D. A. Quarles, Jr., Nonnegative solutions of a nonlinear recurrence, J. Approx. Theory 38 (1983), 357379. MR 711463 (84m:39007)
 [11]
 A. Máté and P. Nevai, Asymptotics for solutions of smooth recurrence equations, Proc. Amer. Math. Soc. (to appear). MR 773995 (86d:39002)
 [12]
 H. N. Mhaskar and E. B. Saff, Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc. 285 (1984), 203234. MR 748838 (86b:41024)
 [13]
 P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc., Vol. 18 (1979), No. 213. MR 519926 (80k:42025)
 [14]
 , Orthogonal polynomials associated with , Canadian Math. Soc. Conference Proc. 3 (1983), 263285.
 [15]
 , Asymptotics for orthogonal polynomials associated with , SIAM J. Math. Anal. 15 (1984).
 [16]
 E. A. Rahmanov, On asymptotic properties of polynomials orthogonal on the real axis, Math. Sb. (N.S.) 119 (161) (1982), 163203. (In Russian) MR 675192 (84e:42025)
 [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 and II, 2nd ed., Cambridge Univ. Press, Cambridge and New York, 1977. MR 0236587 (38:4882)
 [19]
 Al. Magnus, A proof of Freud's conjecture about the orthogonal polynomials related to for integer (manuscript).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507687227
PII:
S 00029947(1985)07687227
Keywords:
Asymptotic expansion,
combinatorial enumerations,
combinatorial identities,
Freud's conjecture,
Jensen's identity,
orthogonal polynomials
Article copyright:
© Copyright 1985 American Mathematical Society
