A SUMMATION FORMULA FOR THE SERIES 3F2 DUE TO FOX AND ITS GENERALIZATIONS
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Fox \cite{Fox} presented an interesting identity for ${}_pF_q$ which is expressed in terms of a finite summation of ${}_pF_q$'s whose involved numerator and denominator parameters are different from those in the starting one. Moreover Fox \cite{Fox} found a very interesting and general summation formula for ${}_3F_2(1/2)$ as a special case of his above-mentioned general identity with the help of Kummer's second summation theorem for ${}_2F_1(1/2)$. Here, in this paper, we show how two general summation formulas for $$ {}_{3}F_{2} \left[ \aligned \alpha,\, \beta,\,\gamma &;\\ \alpha-m,\, \frac{1}{2}(\beta+\gamma+i+1) &; \endaligned \,\, \frac{1}{2} \right],$$ $m$ being a nonnegative integer and $i$ any integer, can be easily established by suitably specializing the above-mentioned Fox's general identity with, here, the aid of generalizations of Kummer's second summation theorem for ${}_2F_1(1/2)$ obtained recently by Rakha and Rathie \cite{Ra-Ra}. Several known results are also seen to be certain special cases of our main identities.
Abstract
Fox \cite{Fox} presented an interesting identity for ${}_pF_q$ which is expressed in terms of a finite summation of ${}_pF_q$'s whose involved numerator and denominator parameters are different from those in the starting one. Moreover Fox \cite{Fox} found a very interesting and general summation formula for ${}_3F_2(1/2)$ as a special case of his above-mentioned general identity with the help of Kummer's second summation theorem for ${}_2F_1(1/2)$. Here, in this paper, we show how two general summation formulas for $$ {}_{3}F_{2} \left[ \aligned \alpha,\, \beta,\,\gamma &;\\ \alpha-m,\, \frac{1}{2}(\beta+\gamma+i+1) &; \endaligned \,\, \frac{1}{2} \right],$$ $m$ being a nonnegative integer and $i$ any integer, can be easily established by suitably specializing the above-mentioned Fox's general identity with, here, the aid of generalizations of Kummer's second summation theorem for ${}_2F_1(1/2)$ obtained recently by Rakha and Rathie \cite{Ra-Ra}. Several known results are also seen to be certain special cases of our main identities.
참고문헌 (9)
- W. N. Bailey, "Generalized Hypergeometric Series" (1935), Cambridge University Press
- C. Fox, "The expression of hypergeometric series in terms of a similar series", Proc. London Math. Soc, 26, pp.201~210 (1927)
- Y. S. Kim, "Extensions of classical summation theorems for the series 2F1, 3F2 and 4F3 with applications in Ramanujan’s summations", Int. J. Math. Math. Sci, 2010, pp.26~ (2010)
- Y. S. Kim, "Applications of generalized Gauss’s second summation theorem for the series 2F1", Math. Commun, 16, pp.481~489 (2011)
- E. D. Rainville, "Special Functions" (1960), Macmillan Company
- A. K. Rathie, "New summation formula for 3F2(1", 2) and Kummer-type II transformation, 2008, pp.13(1), 63~66
- M. A. Rakha, "Generalizations of classical summation theorems for the series 2F1 and 3F2 with applications", Integral Transforms Spec. Funct, 22(11), pp.823~840 (2011)
- L. J. Slater, "Generalized Hypergeometric Functions" (1966), Cambridge University Press
- H. M. Srivastava, "Zeta and q-Zeta Functions and Associated Series and Integrals" (2012), Elsevier Science Publishers