Results 1  10
of
10
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 ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
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.
V.: A sequence of unimodal polynomials
 Jour. Math. Anal. Appl
, 1999
"... Abstract. The purpose of this paper is to establish the unimodality and to determine the mode of a class of Jacobi polynomials which arises in the exact integration of certain rational functions as well as in the Taylor expansion of the double square root. 1. ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Abstract. The purpose of this paper is to establish the unimodality and to determine the mode of a class of Jacobi polynomials which arises in the exact integration of certain rational functions as well as in the Taylor expansion of the double square root. 1.
Asymptotics of Zeros of Incomplete Gamma Functions
 Annals of Numerical Mathematics
, 1994
"... We consider the complex zeros with respect to z of the incomplete gamma functions fl(a; z) and \Gamma(a; z), with a real and positive. In particular we are interested in the case that a is large. The zeros are obtained from approximations that are computed by using uniform asymptotic expansions of ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We consider the complex zeros with respect to z of the incomplete gamma functions fl(a; z) and \Gamma(a; z), with a real and positive. In particular we are interested in the case that a is large. The zeros are obtained from approximations that are computed by using uniform asymptotic expansions of the incomplete gamma functions. The complex zeros of the complementary error function are used as a first approximations. Applications are discussed for the zeros of the partial sums s n (z) = P n j=0 z j =j! of exp(z). 1991 Mathematics Subject Classification: 33B15, 33B20, 41A60, 65U05. Keywords & Phrases: incomplete gamma functions, zeros of incomplete gamma functions, error function, uniform asymptotic expansion. 1.
Higher Order Turán Inequalities
"... The celebrated Tur'an inequalities P 2 n (x) \Gamma Pn\Gamma1 (x)Pn+1 (x) 0; x 2 [\Gamma1; 1]; n 1, where Pn (x) denotes the Legendre polynomial of degree n, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the ine ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
The celebrated Tur'an inequalities P 2 n (x) \Gamma Pn\Gamma1 (x)Pn+1 (x) 0; x 2 [\Gamma1; 1]; n 1, where Pn (x) denotes the Legendre polynomial of degree n, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities fl 2 n \Gamma fl n\Gamma1 fl n+1 0; n 1, which hold for the Maclaurin coefficients of the real entire function / in the LaguerreP'olya class, /(x) = P 1 n=0 fl nx n =n!. 1.
A Problem of Pólya Concerning Polynomials Which Obey Descartes' Rule of Signs
 East J. Approx
, 1997
"... We formulate and duscuss two open problems. The first one is due to P'olya. It states that the sequence of polynomials formed by a polynomial p with only real zeros and all its derivatives, obeys Descartes' rule of signs for any x, greater than the largest zero of p. The other problem is due to Karl ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
We formulate and duscuss two open problems. The first one is due to P'olya. It states that the sequence of polynomials formed by a polynomial p with only real zeros and all its derivatives, obeys Descartes' rule of signs for any x, greater than the largest zero of p. The other problem is due to Karlin and states that certain Hankel determinants associated with an entire function in the LaguerrePolya class do not change their signs. We prove that the statement of Karlin's problem is a consequence of that of P'olya's problem. The interest in these problems is motivated by the fact that once Karlin's problem is solved, it would yield necessary conditions that the Riemann hypothesis holds. 1
Best Uniform Rational Approximation of x
, 1993
"... For the uniform approximation of x on [0, 1] by rational functions the following strong error estimate is proved: Let Enn (x , [0, 1]) denote the minimal approximation error in the uniform norm on [0, 1] of rational approximants to x with numerator and denominator degree at most n, then ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
For the uniform approximation of x on [0, 1] by rational functions the following strong error estimate is proved: Let Enn (x , [0, 1]) denote the minimal approximation error in the uniform norm on [0, 1] of rational approximants to x with numerator and denominator degree at most n, then the the limit Enn (x holds for all # > 0.
LOCATING THE ZEROS OF PARTIAL SUMS OF e z WITH RIEMANNHILBERT METHODS
, 709
"... Dedicated to Percy Deift with gratitude and admiration. Abstract. In this paper we derive uniform asymptotic expansions for the partial sums of the exponential series. We indicate how this information will be used in a later publication to obtain full and explicitly computable asymptotic expansions ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Dedicated to Percy Deift with gratitude and admiration. Abstract. In this paper we derive uniform asymptotic expansions for the partial sums of the exponential series. We indicate how this information will be used in a later publication to obtain full and explicitly computable asymptotic expansions with error bounds for all zeros of the Taylor polynomials pn−1(z) = Pn−1 k=0 zk / k!. Our proof is based on a representation of pn−1(nz) in terms of an integral of the form R e γ nφ(s) ds. We demonstrate how to derive uniform s−z expansions for such integrals using a RiemannHilbert approach. A comparison with classical steepest descent analysis shows the advantages of the RiemannHilbert analysis in particular for points z that are close to the critical points of φ. 1.
Computational Number Theory at CWI in 19701994
, 1994
"... this paper we present a concise survey of the research in Computational ..."
ASYMPTOTIC ESTIMATION OF ξ (2n) (1/2): ON A CONJECTURE OF FARMER AND RHOADES
"... Abstract. We verify a very recent conjecture of Farmer and Rhoades on the asymptotic rate of growth of the derivatives of the Riemann xi function at s =1/2. We give two separate proofs of this result, with the more general method not restricted to s =1/2. We briefly describe other approaches to our ..."
Abstract
 Add to MetaCart
Abstract. We verify a very recent conjecture of Farmer and Rhoades on the asymptotic rate of growth of the derivatives of the Riemann xi function at s =1/2. We give two separate proofs of this result, with the more general method not restricted to s =1/2. We briefly describe other approaches to our results, give a heuristic argument, and mention supporting numerical evidence.
Best Uniform Rational Approximation of x^α on [0, 1]
, 1993
"... A strong error estimate for the uniform rational approximation of x α on [0,1] is given, and its proof is sketched. Let Enn(x α, [0,1]) denote the minimal approximation error in the uniform norm. Then it is shown that holds true for each α> 0. lim n→ ∞ e2π √ αn Enn(x α ..."
Abstract
 Add to MetaCart
A strong error estimate for the uniform rational approximation of x α on [0,1] is given, and its proof is sketched. Let Enn(x α, [0,1]) denote the minimal approximation error in the uniform norm. Then it is shown that holds true for each α> 0. lim n→ ∞ e2π √ αn Enn(x α