Asymptotic expansions for solutions of smooth recurrence equations

Authors:
Shing-Whu Jha, Attila Máté and Paul Nevai

Journal:
Proc. Amer. Math. Soc. **110** (1990), 365-370

MSC:
Primary 33C45; Secondary 39A10, 41A60

MathSciNet review:
1014646

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a convergent sequence of reals, where for each the tuple satisfies one of equations, depending on the residue class of , for some given and . Assume these equations are smooth, they have the same gradient in the first variables, and this gradient satisfies a certain nonmodularity condition. We then show that has asymptotic expansions, depending on the residue class of , in terms of powers of . This result enables us to discuss the asymptotic behavior of the recurrence coefficients associated with certain orthogonal polynomials. A key ingredient in the proof of the main result is a lemma involving exponential sums.

**[1]**William C. Bauldry,*Estimates of asymmetric Freud polynomials on the real line*, J. Approx. Theory**63**(1990), no. 2, 225–237. MR**1079852**, 10.1016/0021-9045(90)90105-Y**[2]**William C. Bauldry, Attila Máté, and Paul Nevai,*Asymptotics for solutions of systems of smooth recurrence equations*, Pacific J. Math.**133**(1988), no. 2, 209–227. MR**941919****[3]**Stanford S. Bonan and Dean S. Clark,*Estimates of the Hermite and the Freud polynomials*, J. Approx. Theory**63**(1990), no. 2, 210–224. MR**1079851**, 10.1016/0021-9045(90)90104-X**[4]**Géza Freud,*On the greatest zero of an orthogonal polynomial. I*, Acta Sci. Math. (Szeged)**34**(1973), 91–97. MR**0318761****[5]**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****[6]**D. S. Lubinsky, H. N. Mhaskar, and E. B. Saff,*Freud’s conjecture for exponential weights*, Bull. Amer. Math. Soc. (N.S.)**15**(1986), no. 2, 217–221. MR**854558**, 10.1090/S0273-0979-1986-15480-7**[7]**D. S. Lubinsky, H. N. Mhaskar, and E. B. Saff,*A proof of Freud’s conjecture for exponential weights*, Constr. Approx.**4**(1988), no. 1, 65–83. MR**916090**, 10.1007/BF02075448**[8]**Attila Máté and Paul Nevai,*Asymptotics for solutions of smooth recurrence equations*, Proc. Amer. Math. Soc.**93**(1985), no. 3, 423–429. MR**773995**, 10.1090/S0002-9939-1985-0773995-6**[9]**Attila Máté, Paul Nevai, and Vilmos Totik,*Asymptotics for the greatest zeros of orthogonal polynomials*, SIAM J. Math. Anal.**17**(1986), no. 3, 745–751. MR**838252**, 10.1137/0517053**[10]**A. Máté, P. Nevai, and V. Totik,*Asymptotics for the zeros of orthogonal polynomials associated with infinite intervals*, J. London Math. Soc. (2)**33**(1986), no. 2, 303–310. MR**838641**, 10.1112/jlms/s2-33.2.303**[H]**Attila Máté, Paul Nevai, and Thomas Zaslavsky,*Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights*, Trans. Amer. Math. Soc.**287**(1985), no. 2, 495–505. MR**768722**, 10.1090/S0002-9947-1985-0768722-7**[12]**Paul Nevai,*Orthogonal polynomials associated with 𝑒𝑥𝑝(-𝑥⁴)*, Second Edmonton conference on approximation theory (Edmonton, Alta., 1982), CMS Conf. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 1983, pp. 263–285. MR**729336****[13]**Paul Nevai,*Asymptotics for orthogonal polynomials associated with 𝑒𝑥𝑝(-𝑥⁴)*, SIAM J. Math. Anal.**15**(1984), no. 6, 1177–1187. MR**762973**, 10.1137/0515092

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DOI:
https://doi.org/10.1090/S0002-9939-1990-1014646-7

Keywords:
Asymptotic expansions,
exponential sums,
orthogonal polynomials,
recurrence equations

Article copyright:
© Copyright 1990
American Mathematical Society