Kummer’s Theorems, Popular Solutions and Connecting Formulas on Hypergeometric Function

Abstract

The hypergeometric series is an extension of the geometric series. The confluent hypergeometric function is the solution of the hypergeometric differential equation [θ(θ +b−1)−z(θ +a)]w = 0. Kummer’s first formula and Kummer’s second formula are of significant importance in solving the hypergeometric differential equations. Kummer has developed six solutions for the differential equation and twenty connecting formulas during the period of 1865-1866. Each connecting formula consist of a solution expressed as the combination of two other solutions. Recently in 2021, these solutions were extensively used by Schweizer [13] in practical problems specially in Physics. Here we extend the connecting formulas obtained by Kummer to obtain the other six solutions w1(z), w2(z), w3(z), w4(z), w5(z) and w6(z) as the combination of three solutions.