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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
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Abstract: Let $ {p_n}(x) = {\gamma _n}{x^n} + \cdots $ denote the $ n$th polynomial orthonormal with respect to the weight $ \exp ( - {x^\beta }/\beta )$ where $ \beta > 0$ is an even integer. G. Freud conjectured and Al. Magnus proved that, writing $ {a_n} = {\gamma _{n - 1}}/{\gamma _n}$, the expression $ {a_n}{n^{ - 1/\beta }}$ has a limit as $ n \to \infty $. It is shown that this expression has an asymptotic expansion in terms of negative even powers of $ n$. 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.


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  • [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] -, On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A 76 (1976), 1-6. MR 0419895 (54:7913)
  • [4] H. W. Gould, Generalization of a theorem of Jensen concerning convolutions, Duke Math. J. 27 (1960), 71-76. MR 0114766 (22:5585)
  • [5] H. W. Gould and J. Kauchy, Evaluation of a class of binomial coefficient summations, J. Combin. Theory 1 (1966), 233-247. 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), 307-318. 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., Addison-Wesley, 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), 357-379. 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), 203-234. 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 $ \exp ({x^{ - 4}})$, Canadian Math. Soc. Conference Proc. 3 (1983), 263-285.
  • [15] -, Asymptotics for orthogonal polynomials associated with $ \exp ({x^{ - 4}})$, 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), 163-203. (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 $ \vert x{\vert^\rho }\exp ( - {x^{2m}})$ for integer $ m$ (manuscript).

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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

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