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The Asymptotic Zero Distribution of Orthogonal Polynomials With Varying Recurrence Coefficients
"... this paper to ll this gap. To state our theorem we use the notation lim n=N!t X n;N = X 4 Kuijlaars and Van Assche to denote the property that in the doubly indexed sequence X n;N we have lim j!1 X n j ;N j = X whenever n j and N j are two sequences of natural numbers such that N j ! 1 and n j ..."
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Cited by 40 (9 self)
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this paper to ll this gap. To state our theorem we use the notation lim n=N!t X n;N = X 4 Kuijlaars and Van Assche to denote the property that in the doubly indexed sequence X n;N we have lim j!1 X n j ;N j = X whenever n j and N j are two sequences of natural numbers such that N j ! 1 and n j =N j ! t as j !1. For example, the convergence in Proposition 1.3 may be expressed by lim n=N!t (p n;N ) = w;t : We will use this notation throughout the rest of the paper. Our main result is the following. Theorem 1.4 Let for each N 2 N, two sequences fa n;N g 1 n=1 , a n;N > 0 and fb n;N g 1 n=0 of recurrence coecients be given, together with orthogonal polynomials p n;N generated by the recurrence xp n;N (x) = a n+1;N p n+1;N (x) + b n;N p n;N (x) + a n;N p n 1;N (x); n 0; (1.6) and the initial conditions p 0;N 1 and p 1;N 0. Suppose that there exist two continuous functions a : (0; 1) ! [0; 1), b : (0; 1) ! R, such that lim n=N!t a n;N = a(t); lim n=N!t b n;N = b(t) (1.7) whenever t > 0. Dene the functions (t) := b(t) 2a(t); (t) := b(t) + 2a(t); t > 0: (1.8) Then we have for every t > 0, lim n=N!t (p n;N ) = 1 t Z t 0 ! [(s);(s)] ds: (1.9) Here ! [;] is the measure given by (1.4) if < . If = , then ! [;] is the Dirac point mass at . Remark 1.5 The measure on the righthand side of (1.9) is the average of the equilibrium measures of the varying intervals [(s); (s)] for 0 < s < t. Its support is given by " inf 0<s<t (s); sup 0<s<t (s) # : (1.10) In particular, the support is always an interval. The support is unbounded if or are unbounded near 0. J. Approx. Theory 99 (1999), 167197. 5 Remark 1.6 Theorem 1.4 has an obvious extension to polynomials that are orthogonal with respect to a discrete measure supp...
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 15 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform
 J. Approx. Theory
"... The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) →−1asy→∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical p ..."
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Cited by 7 (0 self)
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The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) →−1asy→∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics. Key words and phrases: real Borel probability measure, convergence in distribution, Stieltjes transform, Lévy continuity theorem, AkhiezerKrein theorem, weak convergence of probability measures. Lévy’s classical continuity theorem says that if the pointwise limit of the characteristic functions of a sequence of real Borel probability measures (Pn) exists, then the limit function ϕ is itself the characteristic function for a probability measure P if and only if ϕ is continuous at zero, in which case Pn → P in distribution. The purpose of this note is to prove a direct analog of Lévy’s theorem for Stieltjes transforms, complementing those for other representing functions in [HS] and [HK], and to give several examples of applications. Throughout this note, R and C denote the real and complex numbers, respectively; p.m. and s.p.m. denote Borel probability measures, and subprobability (mass ≤ 1) measures, respectively, on R; and s.p.m.’s (µn) converge vaguely to a s.p.m. µ [C,p.80],ifthere exists a dense subset D of R such that for all a, b ∈ D, a<b,µn((a, b]) → µ((a, b]). (Thus if (µn), µ are p.m.’s, vague convergence is equivalent to convergence in distribution.)
Recent Problems from Uniform Asymptotic Analysis of Integrals In Particular in Connection with Tricomi's $Psi$function
, 1998
"... The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of TricomiCarlitz polynomials in terms of Hermite polynomials. We mention othe ..."
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The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of TricomiCarlitz polynomials in terms of Hermite polynomials. We mention other recent expansions for orthogonal polynomials that do not satisfy a differential equation, and for which methods based on integral representations produce powerful uniform asymptotic expansions. 1991 Mathematics Subject Classification: 41A60, 33B20, 33C10, 33C45, 11B73, 30E15. Keywords and Phrases: uniform asymptotic expansions, Tricomi's \Psi\Gammafunction, Kummer functions, confluent hypergeometric functions, Whittaker functions, Hermite polynomials, TricomiCarlitz polynomials. Note: Work carried out under project MAS2.8 Exploratory research. Extended version of a paper presented at the Conference Tricomi's Ideas and Contemporary Applied Mathematics to celebrate the 100th anniversary o...