A limiting case of a generalized beta integral on two intervals

Abstract

Akhiezer Polynomials orthogonal on several intervals are used to define a generalization of the beta integral where the integral is over two disjoint intervals of the real line, [−1, −β] ∪ [β, 1]. An explicit evaluation of the integral is given in the limiting case as β → 1.

A generalized beta integral on two intervals

“…In this situation we will hope to evaluate the moment integrals in terms of the parameters α and β. Owing to the nature of the set, it seems that the combination of the α and β that will replace the parameter b will be related to the gamma points for the set E. For a definition and discussion of the gamma points associated with a union of disjoint intervals see [2]. Finally, it is interesting to note that if we substitute b = 1 into the series (2.4) we should obtain a series expansion for the limiting case as b → 1 − that was calculated in [4]. The resultant identity in this case is just a special case of a hypergeometric summation but the relationship between different limiting values of integrals of these kinds over a general set E as the lengths of the disjoint intervals tend to zero is something that warrants further investigation.…”

“…In this paper we will give a series expansion of the generalized beta integral on two intervals that was defined in [4]. We will proceed directly with the definition of the integral since the motivation for the definition can be found in [4].…”

“…We will proceed directly with the definition of the integral since the motivation for the definition can be found in [4].…”

“…For a definition and discussion of the Akhiezer polynomials see [1,2] and [4]. It is clear that as b → 0 we obtain the classical beta integral.…”

“…It is clear that as b → 0 we obtain the classical beta integral. In [4] this integral and others related to it were evaluated in the limit as b → 1 − . In this paper we expand (1.1) in a power series in terms of the parameter b and evaluate the coefficients in terms of hypergeometric functions.…”