Brian Fisher;
Mongkolsery Lin;
Somsak Orankitjaroen

Results on partial Derivatives of the incomplete Beta Function

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 72, 3 - 10 (2019/2020)
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.

Keywords:   Beta function, incomplete Beta function, neutrix, neutrix limit
Seite generiert am 24.02.2021