A NOTE ON CLASSICAL DIXON’S THEOREM FOR THE SERIES 3F2

R Sheeba, Ujjawal Pandey

Abstract


Classical summation theorems such as those of Gauss, Gauss second, Kummer and Bailey for the series 2F1; Watson, Dixon , Whipple and Saalschutz for the series 3F2 and others play an important role in the theory of hypergeometric and generalized hypergeometric series. Out of the above mentioned classical summation theorem, Dixon’s summation theorem has wide applications. For applications of Dixon’s summation theorem, we refer standard research books by Andrews,et.al. and Bailey.Also, in a very popular, interesting and useful research paper, Bailey has given a large number of applications of products of generalized hypergeometric series. Also, Berndt has pointed out that interesting summations due to Ramanujan can be obtained quickly by employing Dixon’s summation theorem.

Moreover, as mentioned in almost all the books on Special Functions that the Watson’s theorem can be obtained with the help of Thomae transformation together with Dixon’s theorem.

In this research note, we observe for the first time that the Dixon’s theorem can also be obtained from the Thomae transformation together with Watson’s theorem and thus this is our aim to prove Dixon’s theorem in this research note.


Keywords


Generalized hypergeometric series, Dixon’s and Watson’s summation theorems, Thomae transformation

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