Volume 52, pp. 203-213, 2020.

The series expansions and Gauss-Legendre rule for computing arbitrary derivatives of the Beta-type functions

Junlin Li, Tongke Wang, and Yonghong Hao


The beta-type functions play an important role in many applied sciences. The partial derivatives of the beta function and the incomplete beta function are integrals involving algebraic and logarithmic endpoint singularities. In this paper, some series expansions for these beta-type functions are found, which are easily used to evaluate these functions with prescribed precision. On the other hand, an accurate Gauss-Legendre quadrature formula is designed to compute these beta-type functions and their partial derivatives based on the Puiseux series for the integrands at their singularities. Some numerical examples confirm the high accuracy and high efficiency of the two algorithms, and also show that the algorithms can be used to effectively evaluate the generalized beta-type functions.

Full Text (PDF) [230 KB], BibTeX

Key words

beta-type functions, higher-order partial derivatives, series expansion, modified Gauss-Legendre rule

AMS subject classifications

65D20, 33B15, 33B20

Links to the cited ETNA articles

[5]Vol. 48 (2018), pp. 450-461 Chelo Ferreira, José L. López, and Ester Pérez Sinusía: Uniform representations of the incomplete beta function in terms of elementary functions

< Back