Applications of extended Watson’s summation theorem

Applications of extended Watson’s summation theorem

n this research paper, several interesting applications of the extended classical summation theorem are given. As special cases, we recover several known results available in the literature.

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  • [1] Bailey WN. Products of generalized hypergeometric series. P Lond Math Soc 1928; 28: 242-254.
  • [2] Bailey WN. Generalized Hypergeometric Series. New York, NY, USA: Stechert-Hafner, 1964.
  • [3] Berndt BC. Ramanujan’s Notebooks, Parts II. New York, NY, USA: Springer-Verlag, 1989.
  • [4] Bhatt RC. Another proof of Watson’s theorem for summing 3F2(1). J London Math Soc 1965; 40: 47-48.
  • [5] Brychkov YA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct 2002; 13: 163-167.
  • [6] Choi J, Rathie AK. Evaluation of certain new classes of definite integrals. Integral Transforms Spec Funct 2015; 26: 282-294.
  • [7] Edwards J. A Treatise on the Integral Calculus with Applications, Examples and Problems, II. New York, NY, USA: Chelsea Publishing Company, 1954.
  • [8] Gaboury S, Rathie AK. Evaluation of a new class of double definite integrals. Comm Korean Math Soc 2017; 32: 979-990.
  • [9] Kim YS, Rakha MA, Rathie AK. Extensions of certain classical summation theorem for the series 2F1, 3F2 and 4F3 with applications in Ramanujan’s summations. Int J Math Math Sci 2010; 2010: 309503.
  • [10] Krattenthaler C, Rao KS. Automatic generation of hypergeometric identities by the beta integral method. J Comput Appl Math 2003; 160: 159-173.
  • [11] Lavoie JL, Grondin F, Rathie AK. Generalizations of Watson’s theorem on the sum of a 3F2. Indian J Math 1992; 34: 23-32.
  • [12] Lavoie JL, Grondin F, Rathie AK. Generalizations of Whipple’s theorem on the sum of a 3F2. J Comput Appl Math 1996; 72: 293-300.
  • [13] Lavoie JL, Grondin F, Rathie AK, Arora K. Generalizations of Dixon’s theorem on the sum of a 3F2. Math Comp 1994; 205: 267-276.
  • [14] Lewanowicz S. Generalized Watson’s summation formula for 3F2(1). J Comput Appl Math 1997; 86: 375-386.
  • [15] MacRobert TM. Functions of a Complex Variable. London, UK: Macmillan Company, 1962.
  • [16] Rathie AK, Pogany TK. New summation formula for 3F2(1/2) and a Kummer-type II transformation of 2F2(x). Math Commun 2008; 13: 63-66.
  • [17] Preece CT. The product of two generalized hypergeometric functions. P Lond Math Soc 1924; 22: 370-380.
  • [18] Prudnikov AP, Brychkov YA, Marichev OI. Intregrals and Series Vol. 3: More Special Functions. New York, NY, USA: Gordon and Breach Science Publishers, 1990.
  • [19] Rainville ED. Special Functions. New York, NY, USA: Macmillan Company, 1960.
  • [20] Rakha MA, Rathie AK. Generalizations of classical summation theorems for the series 2F1 and 3F2 with applica- tions. Integral Transforms Spec Funct 2011; 22: 823-840.
  • [21] Rathie AK. A short proof of Preece’s identity and other contiguous results. Rev Mat Estat 1997; 15: 207-210.
  • [22] Rathie AK, Choi J. A note on generalization of Preece’s identity and other contiguous results. B Korean Math Soc 1998; 35: 339-344.
  • [23] Rathie AK, Paris RB. A new proof of Watson’s theorem for the 3F2(1) series. Appl Math Sci 2009; 314: 161-164.
  • [24] Saad N, Hall RL. Integrals containing confluent hypergeometric functions with applications to perturbed singular potentials. J Phys A-Math Gen 2003; 36: 7771-7788.
  • [25] Srivastava HM, Choi J. Zeta and q-Zeta Functions and Associated Series and Integrals. New York, NY, USA: Elsevier Science Publishers, 2012.
  • [26] Vidunas R. A generalization of Kummer’s identity. Rocky Mount J Math 2002; 32: 919-936.
  • [27] Watson GN. A note on generalized hypergeometric series. P Lond Math Soc 1925; 2: 13-15.
  • [28] Whipple FJW. A group of generalized hypergeometric series: relations between 120 allied series of the type F(a; b; c; e; f). P Lond Math Soc 1925; 2: 104-114.