Abstract
We aim at establishing two identities contiguous to Kummer’s transformation :(1 − z)−a2F1" 12a, 12a +12;b +12;z1 − z2#= 2F1a, b ;2b ;2z
by using two different methods. They are further applied to prove two summation formulas for the series 3F2(1), closely related to the classical Watson’s theorem due to Lavoie.
Keywords
Gamma function Hypergeometric function Generalized hypergeometric function Kamp´e de F´eriet function Kummer’s second theorem Dixon and Whipple’s summation theorems 2000 AMS Classification: Primary 33 B 20
References
- [1] Bailey, W. N. Product of generalized hypergeometric series, Proc. London Math. Soc. (ser. 2) 28, 242–254, 1928.
- [2] Mortici, C. New improvements of the Stirling formula, Appl. Math. Comput. 217, 699–704, 2010.
- [3] Mortici, C. A quicker convergence toward the gamma constant with the logarithm term involving the constant e, Carpathian J. Math. 26 (1), 86–91, 2010.
- [4] Erd´elyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. Tables of Integral Transforms (Vol. 2) (McGraw-Hill Book Company, New York, Toronto and London, 1954).
- [5] Kim, Y. S., Rakha, M. A. and Rathie, A. K. Generalization of Kummer’s second theorem with applications, Comput. Math. Math. Phys. 50 (3), 387–402, 2010.
- [6] Kim, Y. S., Rakha, M. A. and Rathie, A. K. Extensions of certain classical summation theorems for the series 2F1 and 3F2 with applications in Ramanujan’s summations, Int. J. Math. Math. Sci., 2011, to appear.
- [7] Lavoie, J. L. Some summation formulas for the series 3F2(1), Math. Comput. 49 (179), 269–274, 1987.
- [8] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Watson’s theorem on the sum of a 3F2, Indian J. Math. 34, 23–32, 1992.
- [9] Lavoie, J. L., Grondin, F., Rathie, A. K. and Arora, K. Generalizations of Dixon’s theorem on the sum of a 3F2, Math. Comput. 62, 267–276, 1994.
- [10] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Whipple’s theorem on the sum of a 3F2, J. Comput. Appl. Math. 72, 293–300, 1996.
- [11] Lewanowicz, S. Generalized Watson’s summation formula for 3F2(1), J. Comput. Appl. Math. 86, 375–386, 1997.
- [12] Milgram, M. On hypergeometric 3F2(1), Arxiv: math. CA/ 0603096, 2006. [13] Rainville, E. D. Special Functions (Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971).
- [14] Rakha, M. A. and Rathie, A. K. Generalizations of classical summation theorems for the series 2F1 and 3F2 with applications, Integral Transformations and Special Functions, 2011, to appear.
- [15] Rathie, A. K. and Nagar, V. On Kummer’s second theorem involving product of generalized hypergeometric series, Le Math. (Catania) 50, 35–38, 1995.
- [16] Srivastava, H. M and Choi, J. Series Associated with the Zeta and Related Functions (Kluwer Academic Publishers, Dordrecht, Boston and London, 2001).
- [17] Vidunas, R. A generalization of Kummer’s identity, Rocky Mount. J. Math. 32, 919–935, 2002; also available at http://arxiv.org/abs/math CA/ 005095
Abstract
References
- [1] Bailey, W. N. Product of generalized hypergeometric series, Proc. London Math. Soc. (ser. 2) 28, 242–254, 1928.
- [2] Mortici, C. New improvements of the Stirling formula, Appl. Math. Comput. 217, 699–704, 2010.
- [3] Mortici, C. A quicker convergence toward the gamma constant with the logarithm term involving the constant e, Carpathian J. Math. 26 (1), 86–91, 2010.
- [4] Erd´elyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. Tables of Integral Transforms (Vol. 2) (McGraw-Hill Book Company, New York, Toronto and London, 1954).
- [5] Kim, Y. S., Rakha, M. A. and Rathie, A. K. Generalization of Kummer’s second theorem with applications, Comput. Math. Math. Phys. 50 (3), 387–402, 2010.
- [6] Kim, Y. S., Rakha, M. A. and Rathie, A. K. Extensions of certain classical summation theorems for the series 2F1 and 3F2 with applications in Ramanujan’s summations, Int. J. Math. Math. Sci., 2011, to appear.
- [7] Lavoie, J. L. Some summation formulas for the series 3F2(1), Math. Comput. 49 (179), 269–274, 1987.
- [8] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Watson’s theorem on the sum of a 3F2, Indian J. Math. 34, 23–32, 1992.
- [9] Lavoie, J. L., Grondin, F., Rathie, A. K. and Arora, K. Generalizations of Dixon’s theorem on the sum of a 3F2, Math. Comput. 62, 267–276, 1994.
- [10] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Whipple’s theorem on the sum of a 3F2, J. Comput. Appl. Math. 72, 293–300, 1996.
- [11] Lewanowicz, S. Generalized Watson’s summation formula for 3F2(1), J. Comput. Appl. Math. 86, 375–386, 1997.
- [12] Milgram, M. On hypergeometric 3F2(1), Arxiv: math. CA/ 0603096, 2006. [13] Rainville, E. D. Special Functions (Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971).
- [14] Rakha, M. A. and Rathie, A. K. Generalizations of classical summation theorems for the series 2F1 and 3F2 with applications, Integral Transformations and Special Functions, 2011, to appear.
- [15] Rathie, A. K. and Nagar, V. On Kummer’s second theorem involving product of generalized hypergeometric series, Le Math. (Catania) 50, 35–38, 1995.
- [16] Srivastava, H. M and Choi, J. Series Associated with the Zeta and Related Functions (Kluwer Academic Publishers, Dordrecht, Boston and London, 2001).
- [17] Vidunas, R. A generalization of Kummer’s identity, Rocky Mount. J. Math. 32, 919–935, 2002; also available at http://arxiv.org/abs/math CA/ 005095
Details
| Primary Language | English |
|---|---|
| Subjects | Statistics |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | June 1, 2011 |
| Published in Issue | Year 2011 Volume: 40 Issue: 6 |