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ON AN INTERESTING EXTENSION OF KUMMER'S SECOND THEOREM WITH APPLICATIONS

  • Awad, Mohammed M. (Department of Mathematics Faculty of Science Suez Canal University) ;
  • Mohammed, Asmaa O. (Department of Mathematics Faculty of Science Suez Canal University) ;
  • Rakha, Medhat A. (Department of Mathematics Faculty of Science Suez Canal University) ;
  • Rathie, Arjun K. (Department of Mathematics Vedant College of Engineering & Technology Rajasthan Technical University)
  • 투고 : 2020.04.29
  • 심사 : 2020.07.20
  • 발행 : 2021.01.31

초록

In this research paper, an attempt has been made to provide an interesting extension of the well-known and useful Kummer's second theorem. Several applications have also been given.

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참고문헌

  1. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC, 1964.
  2. G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999. https://doi. org/10.1017/CBO9781107325937
  3. P. Appell and J. Kampe de Feriet, Fonctions hypergeometriques et hyperspheriques: polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
  4. W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964.
  5. 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
  6. B. C. Berndt, Ramanujan's notebooks. Part II, Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4612-4530-8
  7. J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions, Quart. J. Math. Oxford Ser. 11 (1940), 249-270. https://doi.org/10.1093/qmath/os-11.1.249
  8. J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions. II, Quart. J. Math. Oxford Ser. 12 (1941), 112-128. https://doi.org/10.1093/qmath/os-12.1.112
  9. J. Choi and A. K. Rathie, On the reducibility of Kampe de Feriet function, Honam Math. J. 36 (2014), no. 2, 345-355. https://doi.org/10.5831/HMJ.2014.36.2.345
  10. J. Choi and A. K. Rathie, Reducibility of Kampe de Feriet Function, Appl. Math. Sci. 9(85) (2015), 4219-4232.
  11. J. Choi and A. K. Rathie, Reducibility of certain Kampe de Feriet function with an application to generating relations for products of two Laguerre polynomials, Filomat 30 (2016), no. 7, 2059-2066. https://doi.org/10.2298/FIL1607059C
  12. J. Choi, X. Wang, and A. K. Rathie, Reduction formulas for Srivastava's triple hypergeometric series F(3)[x, y, z], Kyungpook Math. J. 55 (2015), no. 2, 439-447. https://doi.org/10.5666/KMJ.2015.55.2.439
  13. H. Exton, On the reducibility of the Kampe de Feriet function, J. Comput. Appl. Math. 83 (1997), no. 1, 119-121. https://doi.org/10.1016/S0377-0427(97)86597-1
  14. C. F. Gauss, Disquisitiones generales circa seriem infinitam thesis, Gottingen, published in Ges Werke Gottingen (1866), Vol.II, 437-445; II 123-163, II 207-229; II 446-460.
  15. Y. S. Kim, J. Choi, and A. K. Rathie, An identity involving product of generalized hypergeometric series 2F2, Kyungpook Math. J. 59 (2019), no. 2, 293-299. https://doi.org/10.5666/KMJ.2019.59.2.293
  16. 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
  17. Y. S. Kim, A. Kilicman, and A. K. Rathie, On new reduction formulas for the Srivastava's triple hypergeometric series F3[x, y, z], New Trends in Math. Sci. 6 (2018), no.2, 1-5.
  18. Y. S. Kim, T. K. Pogany, and A. K. Rathie, On a reduction formula for the Kampe de Feriet function, Hacet. J. Math. Stat. 43 (2014), no. 1, 65-68.
  19. Y. S. Kim, M. A. Rakha, and A. K. Rathie, Generalization of Kummer's second theorem with applications, Comput. Math. Math. Phys. 50 (2010), no. 3, 387-402; translated from Zh. Vychisl. Mat. Mat. Fiz. 50 (2010), no. 3, 407-422. https://doi.org/10.1134/S0965542510030024
  20. Y. S. Kim and A. K. Rathie, Some results of terminating 2F1(2), J. Inequ. Appl. 2013 (2013), 365. https://doi.org/10.1186/1029-242X-2013-365
  21. Y. S. Kim, A. K. Rathie, and R. B. Paris, Generalization of two theorems due to Ramanujan, Integral Transforms Spec. Funct. 24 (2013), no. 4, 314-323. https://doi.org/10.1080/10652469.2012.689302
  22. Y. S. Kim, A. K. Rathie, and R. B. Paris, Evaluations of some terminating hypergeometric 2F1(2) series with applications, Turkish J. Math. 42 (2018), no. 5, 2563-2575. https://doi.org/10.3906/mat1804-67
  23. C. Krattenthaler and K. Srinivasa Rao, Automatic generation of hypergeometric identities by the beta integral method, J. Comput. Appl. Math. 160 (2003), no. 1-2, 159-173. https://doi.org/10.1016/S0377-0427(03)00629-0
  24. E. E. Kummer, Uber die hypergeometrische Reihe, J. Reine Angew. Math. 15 (1836), 127-172. https://doi.org/10.1515/crll.1836.15.127
  25. E. E. Kummer, Collected papers, Springer-Verlag, Berlin, 1975.
  26. N. N. Lebedev, Special functions and their applications, Second, revised and augmented edition, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963.
  27. R. B. Paris, A Kummer-type transformation for a 2F2 hypergeometric function, J. Comput. Appl. Math. 173 (2005), no. 2, 379-382. https://doi.org/10.1016/j.cam.2004.05.005
  28. T. K. Pogany and A. K. Rathie, Reduction of Srivastava-Daoust S function of two variables, Math. Pannon. 24 (2013), no. 2, 253-260.
  29. 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
  30. 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.
  31. E. D. Rainville, Special Functions, The Macmillan Co., New York, 1960.
  32. M. A. Rakha, M. M. Awad, and A. K. Rathie, On an extension of Kummer's second theorem, Abstr. Appl. Anal. 2013 (2013), Art. ID 128458, 6 pp. https://doi.org/10.1155/2013/128458
  33. M. A. Rakha, M. M. Awad, and A. K. Rathie, Extension of identities due to Preece and Bailey involving product of generalized hypergeometric function 2F2, Wulfenia 21 (2014), no. 7, 56-103.
  34. M. A. Rakha, M. M. Awad, and A. K. Rathie, Extension of a theorem due to Ramanujan, J. Interpolat. Approx. Sci. Comput. 2014 (2014), Art. ID 00072, 9 pp. https://doi.org/10.5899/2014/jiasc-00072
  35. M. A. Rakha, M. M. Awad, and A. K. Rathie, On a reducibility of the Kampe de Feriet function, Math. Methods Appl. Sci. 38 (2015), no. 12, 2600-2605. https://doi.org/10.1002/mma.3245
  36. M. A. Rakha and A. K. Rathie, On an extension of Kummer-type II transformation, TWMS J. Appl. Eng. Math. 4 (2014), no. 1, 80-85.
  37. A. K. Rathie, A short proof of Preece's identities and other contiguous results, Rev. Mat. Estatist. 15 (1997), 207-210.
  38. A. K. Rathie and J. Choi, A note on an identity due to Preece, Far East J. Math. Sci. 6 (1998), no. 2, 205-209.
  39. A. K. Rathie and J. Choi, Another proof of Kummer's second theorem, Commun. Korean Math. Soc. 13 (1998), no. 4, 933-936.
  40. 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.
  41. A. K. Rathie and J. Choi, Some more results contiguous to a Bailey's formula, Far East J. Math. Sci. 6(1998), no. 5, 743-750.
  42. 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.
  43. A. K. Rathie and T. K. Pogany, New summation formula for $_3F_2(\frac{1}{2})$ and a Kummer-type II transformation of 2F2(x), Math. Commun. 13 (2008), no. 1, 63-66.
  44. L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, New York, 1960.
  45. H. M. Srivastava and J. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier, Inc., Amsterdam, 2012. https://doi.org/10.1016/B978-0-12-385218-2.00001-3
  46. H. M. Srivastava and M. C. Daoust, Certain generalized Neumann expansions associated with the Kampe de Feriet function, Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31 (1969), 449-457.
  47. H. M. Srivastava and M. C. Daoust, On Eulerian integrals associated with Kampe de Feriet's function, Publ. Inst. Math. (Beograd) (N.S.) 9(23) (1969), 199-202.
  48. H. M. Srivastava and M. C. Daoust, A note on the convergence of Kampe de Feriet's double hypergeometric series, Math. Nachr. 53 (1972), 151-159. https://doi.org/10.1002/mana.19720530114
  49. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1985.
  50. H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1984.
  51. H. M. Srivastava and R. Panda, An integral representation for the product of two Jacobi polynomials, J. London Math. Soc. (2) 12 (1975/76), no. 4, 419-425. https://doi.org/10.1112/jlms/s2-12.4.419
  52. F. G. Tricomi, Fonctions hypergeometriques confluentes, Memorial des Sciences Mathematiques, Fasc. CXL, Gauthier-Villars, Paris, 1960.
  53. M. I. Zurina and L. N. Osipova, Tables of the confluent hypergeometric function, Vycisl. Centr Akad. Nauk SSSR, Moscow, 1964.