Certain Applications of Generalized Kummer’s Summation Formulas for ₂<i>F</i>₁

Abstract

We present generalizations of three classical summation formulas 2F1 due to Kummer, which are able to be derived from six known summation formulas of those types. As certain simple particular cases of the summation formulas provided here, we give a number of interesting formulas for double-finite series involving quotients of Gamma functions. We also consider several other applications of these formulas. Certain symmetries occur often in mathematical formulae and identities, both explicitly and implicitly. As an example, as mentioned in Remark 1, evident symmetries are naturally implicated in the treatment of generalized hypergeometric series.

Keywords

• Gamma function
• Beta function
• Psi function
• generalized hypergeometric function <i>_pF_q</i>
• summation formulas for ₂<i>F</i>₁ and <i>_pF_q</i>
• transformation formulas for ₂<i>F</i>₁ and <i>_pF_q</i>