Nonnegative solutions of a nonlinear recurrence

Lew, John S.; Quarles, Donald A

doi:10.1016/0021-9045(83)90154-5

Cited by 62 publications

(35 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for a discussion of asymptotic expansions). A result analogous to (27) was obtained by J. S. Lew and D. A. Quarles, Jr., [3] for the weight function exp(-;c4/4), and the polynomials orthonormal with respect to this weight function are studied by Nevai [5 and 6].…”

mentioning

confidence: 57%

“…..,yn + k,\/n) belongs to a convex neighborhood of o in which H is m times continuously differentiable). The left-hand side here is zero according to (3). In view of the continuity of the rath derivatives of H at o, (2) implies that the right-hand side will change only slightly if we replace the argument of H with o in the last term; estimating the magnitude of this change we obtain the following (note that the modified last term of the preceding formula being incorporated into the sum below, / now goes to m rather than m -1): (the denominator here is not zero according to (1)) and…”

mentioning

confidence: 99%

“…Thus, by the above Theorem, >>" has an asymptotic expansion as described in (4), i.e., we have /in\i/6 m (27) a if =£<VT'+0(n-', as n -> oo for all integers ra with some constants c0,cx,...,;cQ = 1 by (23). (The Theorem gives the expansion of yn_2 in terms of «"', sinceyn_2 in (25) corresponds to yn in (3); that is, the expansion of yn is obtained in terms of (n + 2)"' = n~l(\ + 2/n)'1. Expansion (27) can then be derived by using the binomial expansion for (1 + 2/n)'1.)…”

mentioning

confidence: 99%

See 2 more Smart Citations

Asymptotics for solutions of smooth recurrence equations

Mate¹,

Nevai²

1985

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract.It is shown that convergent solutions of a smooth recurrence equation whose gradient satisfies a certain " nonunimodularity" condition can be approximated by an asymptotic expansion. The lemma used to show this has some features in common with Poincaré's theorem on homogeneous linear recurrence equations. An application to the study of polynomials orthogonal with respect to the weight function exp(-Ar6/6) is given.

show abstract

mentioning

confidence: 57%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Asymptotics for solutions of smooth recurrence equations

Mate¹,

Nevai²

1985

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…(8) below). For ß = 4, Theorem 1 was proved by J. S. Lew and D. A. Quarks, Jr. [10], and for ß = 6, it was established in Máté-Nevai [11]. The polynomials pn for ß -4 were discussed by Nevai in [14 and 15].…”

mentioning

confidence: 91%

“…In hght of the description of the polynomial P above, it is easy to give a description of the value of the partial derivative on the left-hand side of (10). It is the number of all passages between floors / and / -1 (up or down) summed over all trips of length 2k + 1 from floor 0 to floor -1.…”

Section: ] Frommentioning

confidence: 99%

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

Mate¹,

Nevai²,

Zaslavsky³

1985

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract.Letp"(x) = y"x" + •■ ■ denote the nth polynomial orthonormal with respect to the weight exp (-x^/ß) where ß > 0 is an even integer. G. Freud conjectured and Al. Magnus proved that, writing a" = t"-\/in, the expression a"n~1/P has a limit as n -» oo. It is shown that this expression has an asymptotic expansion in terms of negative even powers oí 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.Consider the polynomials pn that are orthonormal with respect to the weight function exp(-\x\ß/ß) on the real line, where ß is a positive real number. Denoting by y" the leading coefficient of pn (n > 0) and writing an = y" i/y" for n > 1 and an = 0 for n < 0, G. Freud conjectured thatholds for every positive even ß (see [3, Conjecture, p. 5]; his conjecture has a slightly different form, as he considered the weight function |jc|pexp(-|jc|^) rather than the one above). He also entertained the possibility that this conjecture is valid for all positive real ß. In case ß > 0 is even, he proved that if the limit on the left exists then it must have the value on the right-hand side (see [3, Theorem 1 on p. 4]), and he established the conjecture for ß = 2, 4, and 6 (see [3, pp. 5-6]). He accomplished these by extracting information from the formula (2) -=\ Pn(x)p"-X(x)x-l\x\ exp -|x| /ß) dx,

show abstract

A Uniform Asymptotic Formula for Orthogonal Polynomials Associated with exp(−x4)

Rui

Wong

1999

Journal of Approximation Theory

View full text Add to dashboard Cite

Nonnegative solutions of a nonlinear recurrence

Cited by 62 publications

References 1 publication

Asymptotics for solutions of smooth recurrence equations

Asymptotics for solutions of smooth recurrence equations

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

A Uniform Asymptotic Formula for Orthogonal Polynomials Associated with exp(−x4)

Contact Info

Product

Resources

About