Mathematica Eterna
Open Access

Abstract

The asymptotics of a generalised Beta function

R. B. Paris

We consider the generalised Beta function introduced by Chaudhry et al. [J. Comp. Appl. Math. 78 (1997) 19–32] defined by B(x, y; p) = Z 1 0 t x−1 (1 − t) y−1 exp  −p 4t(1 − t)  dt, where ℜ(p) > 0 and the parameters x and y are arbitrary complex numbers. The asymptotic behaviour of B(x, y; p) is obtained when (i) p large, with x and y fixed, (ii) x and p large, (iii) x, y and p large and (iv) either x or y large, with p finite. Numerical results are given to illustrate the accuracy of the formulas obtained.