A generalized beta integral on two intervals

Abstract

We give an explicit formula for the expansion coefficients of a generalized beta integral on the set [−1, −b] ∪ [b, 1] b ∈ (0, 1), in a power series in the parameter b, thus defining a generalized beta function of two complex variables.

Ultraspherical moments on a set of disjoint intervals

“…In this section we briefly review the construction of the moment integral for the case of one gap and use this opportunity to express the odd moments from [6] in a more convenient form for the purpose of this paper.…”

“…In [5] and [6] the following moment integrals over the set E 2 were computed where n = 0, 1, 2, ... :…”

“…The construction we use follows an analagous construction to that used in [5] and [6] where one gap was considered, and is founded on the relationship between a collection of disjoint intervals and a diophantine polynomial equation known as a generalized Pell equation. The generalized ultraspherical moments form a one parameter generalization of the classical case and we show that the moments can be explicitly evaluated in terms of the hypergoemteric function and this parameter.…”

“…This paper is organized as follows. In section 2 we give a brief summary of the construction of the 1-gap case, studied in [5] and [6], together with the explicit representation of the moments. Before moving to the two gap case we evaluate the odd moments again using a different method which is based on the polynomial mapping technique first outlined in [7].…”

A Fractional Newton Series for the Riemann Zeta Function

“…This theorem gives an asymptotic expansion for the coefficients of the Newton series in (4). The series is convergent in the whole plane.…”