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.

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.

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.

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 Laguerre-Polya 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

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 Laguerre-P'olya class, /(x) = P 1 n=0 fl nx n =n!. 1.

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.

this paper we present a concise survey of the research in Computational

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.

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 Riemann-Hilbert approach. A comparison with classical steepest descent analysis shows the advantages of the Riemann-Hilbert analysis in particular for points z that are close to the critical points of φ. 1.

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