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

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.