Recursive formulas for $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series

TL;DR

This paper develops recursive formulas for $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series using contiguous relations, facilitating symbolic and numerical evaluation, and potentially extending to other hypergeometric sums.

Contribution

The paper introduces new recursive formulas for hypergeometric series that enhance computational methods and can be applied to extend various known summation formulas.

Findings

Recursive formulas for $_{2}F_{1}$ and $_{3}F_{2}$ series derived.

Formulas are suitable for computer algebra systems.

Method can be applied to extend other hypergeometric sums.

Abstract

Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by means of computer algebra. On the other hand, sometimes closed-forms of such extensions can be derived by induction. It is expected that the method used to obtain the different recursive equations can be applied to extend other hypergeometric summation formulas given in the literature.