Asymptotics for a Determinant with a Confluent Hypergeometric Kernel

A Connection Formula for the q-Confluent Hypergeometric Function

We show a connection formula for the q-confluent hypergeometric functions 2φ1(a, b; 0; q, x). Combining our connection formula with Zhang’s connection formula for 2φ0(a, b;−; q, x), we obtain the connection formula for the q-confluent hypergeometric equation in the matrix form. Also we obtain the connection formula of Kummer’s confluent hypergeometric functions by taking the limit q → 1− of our...

Asymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals

The sine kernel K(x, y) := sin(x− y) π(x− y) arises in many areas of mathematics and mathematical physics. There is an extensive literature on the asymptotics of the eigenvalues of Ks, the operator with this kernel on an interval of length s, as s→ ∞, for example [4, 6, 8, 12], and asymptotic fomulas of various kinds were obtained. Some of these derivations were rigorous, others were more heuri...

Gauss Quadrature Approximations to Hypergeometric and Confluent Hypergeometric Functions

Confluent A-hypergeometric functions and rapid decay homology cycles

We study confluent A-hypergeometric functions introduced by Adolphson [1]. In particular, we give their integral representations by using rapid decay homology cycles of Hien [17] and [18]. The method of toric compactifications introduced in [27] and [31] will be used to prove our main theorem. Moreover we apply it to obtain a formula for the asymptotic expansions at infinity of confluent A-hype...

Asymptotic Representations of Confluent Hypergeometric Functions.

I A more general theory will result if in the place of R we employ an abstract normed ring. s We use the symbols =, ... ... in more than one sense. No confusion need arise as tie context makes clear the meaning of each such symbol. It is worth while to mention here that the relation of equality = for E1 as well as for E2 is not an independent primitive idea; for, an equivalent set of postulates...