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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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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.
Uniform bounds for the complementary incomplete gamma function, Preprint at http://locutus.cs.dal.ca:8088/archive/00000335
"... Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : z > ca − 1} with a real and z complex. Our results show that within the hyperboloid, Γ ..."
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Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : z > ca − 1} with a real and z complex. Our results show that within the hyperboloid, Γ(a, z)  is of order z  a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z.
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...