On an extension of a quadratic transformation formula due to Kummer

The aim of this research note is to prove the following new transformation formula (1 - x)^-2a ₃F₂ [ _{c + 3/2, d} ^{a, a + 1/2, d + 1}; x²/(1 - x)₂ ] = ₄F₃ [ _{2c + 2, 2d + 1/2A - 1/2, 2d - 1/2A - 1/2} ^{2a, c, 2d + 1/2A +1/2, 2d - 1/2A + 1/2}; 2x ] valid for |x| < 1/2 and if |x| = 1/2, then Re(c - 2a) > 0, where A = √16d² - 16cd - 8d + 1. For d = c + 1/2, we get quadratic transformations due to Kummer. The result is derived with the help of the generalized Gauss's summation theorem available in the literature.