23-05E

We study the implications of a hypergeometric transformation formula by Mingari Scarpello and Ritelli [Open J. Math. Sci. 2 84–92 (2018)] derived from results given by Legendre in 1825 and essentially extend its validity range. Applications include closed-form evaluations of hypergeometric functions ₂F₁ with different sets of parameters and arguments. Relations to special values of elliptic integrals of first kind in the singular value theory are established. Connections with a hypergeometric transformation due to Ramanujan and his explicit determinations of certain ₂F₁ functions are discussed.