Some results on the beta function and the incomplete beta function

Abstract

The incomplete Beta function B(a, b; x) is defined for a, b > 0 and 0 < x < 1. This definition was extended to negative integer values of a and b byÖzçaḡ et al. using neutrix limits. Partial derivatives of the incomplete Beta function B(a, b; x) for negative integer values of a and b were then evaluated. In the following, a number of results are obtained for the incomplete Beta function B (a, b; x) and then the corresponding results for the Beta function B(a, b) are deduced.

The Numerical Evaluation Methods for Beta Function

“…The beta function for non-positive p or q integers is also available in the literature. Fisher, Orankitjaroe and Lin show that beta function has a finite value in case of p or q equal to zero [35]:…”

“…Table 2 contains the results obtained from Equation ( 35) using Algorithm 1, Algorithm 2 and Mathematica 10. If we taking account Equation (35), beta function on the right side of this equation consists of zero and positive integers while the function on the left side includes zero and negative integers. This provides a great advantage to calculate beta function with zero and negative integers.…”

“…Such an idea was devised by Hadamard and the finite integrals obtained with this method are called Hadamard type integrals [32]. Some other studies in the literature have shown that the beta function by taking the neutrix limit also has some values in cases in which that p and q are zero, negative or positive integers: [33][34][35][36]. First: one of p and q is positive integer and the other is negative integer.…”