The basic analogue of Kummer’s theorem

About one hundred years ago, E. E. Kummer proved the formula (1) 2F1 1 + a – b a, b; -1 = [Γ(1 + a - b)Γ(1 + a/2) ÷ Γ(1+a)Γ(1+a/2 - b) which has since been known as Kummer's theorem. This appears to be the simplest relation involving a hypergeometric function with argument ( - 1). All the relations in the theory of hypergeometric series rF8 which have analogues in the theory of basic serie3 are those in which the argument is ( + 1 ) . Apparently, there has been no successful attempt to establish the basic analogue of any formula involving a function rF8 ( - 1). Since Kummer's theorem is fundamental in the proofs of numerous relations between hypergeometric functions of argument ( - 1), it seemed desirable that an attempt be made to prove the basic analogue of Kummer's theorem and to investigate the possibility of obtaining new relations in basic series with arguments corresponding to the argument ( - 1) in the classical case.