Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A NOTE ON GENERALIZATIONS OF BAILEY'S IDENTITY INVOLVING PRODUCTS OF GENERALIZED HYPERGEOMETRIC SERIES

  • Kilicman, Adem (Department of Mathematics Institute for Mathematical Research University Putra Malaysia (UPM)) ;
  • Kurumujji, Shantha Kumari (Department of Mathematics A J Institute of Engineering and Technology(Affiliated to Visvesvaraya Technological University - Belagavi)) ;
  • Rathie, Arjun K. (Department of Mathematics Vedant College of Engineering & Technology (Rajasthan Technical University))
  • Received : 2021.05.13
  • Accepted : 2021.09.13
  • Published : 2022.04.30
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Abstract

In the theory of hypergeometric and generalized hypergeometric series, the well-known and very useful identity due to Bailey (which is a generalization of the Preece's identity) plays an important role. The aim of this research paper is to provide generalizations of Bailey's identity involving products of generalized hypergeometric series in the most general form. A few known, as well as new results, have also been obtained as special cases of our main findings.

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References

  1. W. N. Bailey, Products of generalized hypergeometric series, Proc. London Math. Soc. (2) 28 (1928), no. 4, 242-254. https://doi.org/10.1112/plms/s2-28.1.242
  2. W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964.
  3. Y. S. Kim, S. Gaboury, and A. K. Rathie, Applications of extended Watson's summation theorem, Turkish J. Math. 42 (2018), no. 2, 418-443. https://doi.org/10.3906/mat1701-31
  4. Y. S. Kim, M. A. Rakha, and A. K. Rathie, Extensions of certain classical summation theorems for the series 2F1, 3F2, and 4F3 with applications in Ramanujan's summations, Int. J. Math. Math. Sci. 2010 (2010), Art. ID 309503, 26 pp. https://doi.org/10.1155/2010/309503
  5. I. Kim, V. Rohira, and A. K. Rathie, Note on the generalization of Bailey's identity and two contiguous results, Bull. Kerala Math. Assoc. 16 (2019), no. 2, 1-11.
  6. C. T. Preece, The Product of two Generalised Hypergeometric Functions, Proc. London Math. Soc. (2) 22 (1924), 370-380. https://doi.org/10.1112/plms/s2-22.1.370
  7. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 3, translated from the Russian by G. G. Gould, Gordon and Breach Science Publishers, New York, 1990.
  8. E. D. Rainville, Special Functions, The Macmillan Co., New York, 1960.
  9. M. A. Rakha and A. K. Rathie, Generalizations of classical summation theorems for the series 2F1 and 3F2 with applications, Integral Transforms Spec. Funct. 22 (2011), no. 11, 823-840. https://doi.org/10.1080/10652469.2010.549487
  10. A. K. Rathie, A short proof of Preece's identities and other contiguous results, Rev. Mat. Estatist. 15 (1997), 207-210.
  11. A. K. Rathie and J. Choi, Another proof of Kummer's second theorem, Commun. Korean Math. Soc. 13 (1998), no. 4, 933-936.
  12. A. K. Rathie and J. Choi, A note on generalizations of Preece's identity and other contiguous results, Bull. Korean Math. Soc. 35 (1998), no. 2, 339-344.
  13. A. K. Rathie and V. Nagar, On Kummer's second theorem involving product of generalized hypergeometric series, Matematiche (Catania) 50 (1995), no. 1, 35-38.
  14. A. K. Rathie and T. K. Pogany, New summation formula for $_2F_1({\frac{1}{2}})$ and a Kummertype II transformation of 2F2(x), Math. Commun. 13 (2008), no. 1, 63-66.