대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제37권2호
  • /
  • Pages.473-496
  • /
  • 2022
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

CERTAIN REDUCTION AND TRANSFORMATION FORMULAS FOR THE KAMPÉ DE FÉRIET FUNCTION

  • 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 : 2021.03.11
  • 심사 : 2021.05.06
  • 발행 : 2022.04.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In 2014, Liu and Wang established a large number of interesting reduction, transformation and summation formulas for the Kampé de Fériet function. Inspired by the work, we aim to find further several transformation and reduction formulas for the Kampé de Fériet function. Theses formulas are mainly based on the formulas given by Liu and Wang [33].

키워드

과제정보

The authors are grateful to the referee for carefully reading the manuscript and pointing out certain corrections.

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, Sur les fonctions hypergeometriques de plusieurs variables, Gauthier-Villars, Paris, 1925.
  4. P. Appell and J. Kampe de Feriet, Fonctions hypergeometriques et hyperspheriques: polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
  5. W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964.
  6. 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
  7. 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
  8. R. G. Buschman and H. M. Srivastava, Series identities and reducibility of Kampe de Feriet functions, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 3, 435-440. https://doi.org/10.1017/S0305004100059478
  9. L. Carlitz, Summation of a double hypergeometric series, Matematiche (Catania) 22 (1967), 138-142.
  10. W.-C. C. Chan, K. Chen, C. Chyan, and H. M. Srivastava, Some multiple hypergeometric transformations and associated reduction formulas, J. Math. Anal. Appl. 294 (2004), no. 2, 418-437. https://doi.org/10.1016/j.jmaa.2004.02.008
  11. K.-Y. Chen and H. M. Srivastava, Series identities and associated families of generating functions, J. Math. Anal. Appl. 311 (2005), no. 2, 582-599. https://doi.org/10.1016/j.jmaa.2005.03.030
  12. 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
  13. J. Choi and A. K. Rathie, General summation formulas for the Kampe de Feriet function, Montes Taures J. Pure Appl. Math. 1 (2019), no. 1, 107-128.
  14. J. Choi and A. K. Rathie, Further summation formulas for the Kampe de Feriet function, arxiv:1912.10441.
  15. J. Choi and A. K. Rathie, Reducibility of Kampe de Feriet function, Appl. Math. Sci. 9 (2015), no. 85, 4219-4232.
  16. 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
  17. J. Choi, A. K. Rathie, and H. M. Srivastava, Certain hypergeometric identities deducible by using the beta integral method, Bull. Korean Math. Soc. 50 (2013), no. 5, 1673-1681. https://doi.org/10.4134/BKMS.2013.50.5.1673
  18. D. Cvijovic and A. R. Miller, A reduction formula for the Kampe de Feriet function, Appl. Math. Lett. 23 (2010), no. 7, 769-771. https://doi.org/10.1016/j.aml.2010.03.006
  19. V. L. Deshpande, Some infinite summation formulae involving Kampe de Feriet function, Defence Sci. J. 21 (1971), 125-130.
  20. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. I, McGraw-Hill Book Company, Inc., New York, 1953.
  21. 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
  22. P. Humbert, The confluent hypergeometric functions of two variables, Proc. Royal Soc. of Edinburgh 41 (1922), 73-96. https://doi.org/10.1017/s0370164600009810
  23. R. N. Jain, Sum of a double hypergeometric series, Matematiche (Catania) 21 (1966), 300-301.
  24. J. Kampe de Feriet, Les fonctions hypergeometriques drdre superieura deux variables, CR Acad. Sci. Paris 173 (1921), 401-404.
  25. P. W. Karlsson, Some reduction formulae for double power series and Kampe de Feriet functions, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 1, 31-36. https://doi.org/10.1016/1385-7258(84)90053-2
  26. Y. S. Kim, On certain reducibility of Kampe de Feriet function, Honam Math. J. 31 (2009), no. 2, 167-176. https://doi.org/10.5831/HMJ.2009.31.2.167
  27. 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 (2000), Art. ID 309503, 26 pp. https://doi.org/10.1155/2010/309503
  28. 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
  29. E. D. Krupnikov, A register of computer-oriented reduction identities for the Kampe de Feriet function, Novosibirsk, Russia, 1996.
  30. J.-L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Watson's theorem on the sum of a 3F2, Indian J. Math. 34 (1992), no. 1, 23-32.
  31. J.-L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Whipple's theorem on the sum of a 3F2, J. Comput. Appl. Math. 72 (1996), no. 2, 293-300. https://doi.org/10.1016/0377-0427(95)00279-0
  32. J.-L. Lavoie, F. Grondin, A. K. Rathie, and K. Arora, Generalizations of Dixon's theorem on the sum of a 3F2, Math. Comp. 62 (1994), no. 205, 267-276. https://doi.org/10.2307/2153407
  33. H. Liu and W. Wang, Transformation and summation formulae for Kampe de Feriet series, J. Math. Anal. Appl. 409 (2014), no. 1, 100-110. https://doi.org/10.1016/j.jmaa.2013.06.068
  34. F. M. Ragab, Expansions of Kampe de Feriet's double hypergeometric function of higher order, J. Reine Angew. Math. 212 (1963), 113-119. https://doi.org/10.1515/crll.1963.212.113
  35. E. D. Rainville, Special Functions, The Macmillan Company, New York, (1960), Reprinted by Chelsea Publishing Company, New York, (1971).
  36. 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
  37. 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
  38. S. Saran, Reducibility of generalised Kampe de Feriet function, Gan. ita 31 (1980), no. 1-2, 89-98.
  39. O. Shanker and S. Saran, Reducibility of Kampe de Feriet function, Gan. ita 21 (1970), no. 1, 9-16.
  40. B. L. Sharma, Sum of a double series, Proc. Amer. Math. Soc. 52 (1975), 136-138. https://doi.org/10.2307/2040117
  41. R. P. Singal, Transformation formulae for the modified Kampe de Feriet function, Math. Student 40A (1972), 327-330.
  42. S. N. Singh and S. P. Singh, Transformation of Kampe de Feriet function, Indian J. Pure Appl. Math. 13 (1982), no. 10, 1163-1166.
  43. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
  44. 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
  45. 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
  46. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1985.
  47. 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
  48. R. Vidunas, Specialization of Appell's functions to univariate hypergeometric functions, J. Math. Anal. Appl. 355 (2009), no. 1, 145-163. https://doi.org/10.1016/j.jmaa.2009.01.047
  49. A. D. Wadhwa, A theorem on Kampe de Feriet function, Math. Education 4 (1970), A137-A139.
  50. A. D. Wadhwa, Fourier series for Kampe de Feriet function, Defence Sci. J. 21 (1971), 179-186.