| MSC: | 33B15 | Gamma, beta and polygamma functions |
| 33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) | |
| 46F10 | Operations with distributions and generalized functions |
Abstract:
The incomplete Beta function $B(a,b;x)$ is
defined by
\[
B(a,b;x) = \int_0^x t^{a-1} (1-t)^{b-1}\,dt\,,
\]
for $a,b>0$ and $0<x<1$. This definition was extended to
negative integer values of $a$ and $b$ by \"Oz\c ca\=g et al.
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, it is proved that
\[
B_{0,1}(-1,1;x) = -\ln \frac{x}{ 1-x}- \frac{\ln (1-x)} {x}-1
\]
and
\[
n B_{0,1}(-n,1;x)=-\ln \frac{x}{1-x}- \frac{
\ln(1-x) }{ x^n}-n^{-1} +\sum _{i=1}^ {n-1} \frac{x^{-i} }{ i}\,,
\]
for $n=2,3,\ldots,$ where
\[
\frac{\partial^{{\kern.5pt}m+n}}{\partial a^m\partial
b^n}B(a,b;x) = B_{m,n}(a,b;x)\,.
\]
Further results are also given.