Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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CERTAIN DECOMPOSITION FORMULAS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS pFq AND SOME FORMULAS OF AN ANALYTIC CONTINUATION OF THE CLAUSEN FUNCTION 3F2

  • Received : 2010.07.27
  • Published : 2012.01.31
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Abstract

Here, by using the symbolical method introduced by Burchnall and Chaundy, we aim at constructing certain expansion formulas for the generalized hypergeometric function $_pF_q$. In addition, using our expansion formulas for $_pF_q$, we present formulas of an analytic continuation of the Clausen hypergeometric function $_3F_2$, which are much simpler than an earlier known result. We also give some integral representations for $_3F_2$.

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References

  1. P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite, Gauthier - Villars, Paris, 1926.
  2. 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
  3. 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
  4. T. W. Chaundy, Expansions of hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942), 159-171. https://doi.org/10.1093/qmath/os-13.1.159
  5. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions Vols. I, II, Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.
  6. A. Hasanov and H. M. Srivastava, Some decomposition formulas associated with the Lauricella function $F^{(\tau)}_A$ A and other multiple hypergeometric functions, Appl. Math. Lett. 19 (2006), no. 2, 113-121. https://doi.org/10.1016/j.aml.2005.03.009
  7. A. Hasanov and H. M. Srivastava, Decomposition formulas associated with the lauricella multivariable hypergeo-metric functions, Comput. Math. Appl. 53 (2007), no. 7, 1119-1128. https://doi.org/10.1016/j.camwa.2006.07.007
  8. O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane and Toronto, 1982.
  9. P. O. M. Olsson, Analytic continuations of higher-order hypergeometric functions, J. Math. Phys. 7 (1966), no. 4, 702-710. https://doi.org/10.1063/1.1704985
  10. R. C. Pandey, On the expansions of hypergeometric functions, Agra Univ. J. Res. Sci. 12 (1963), 159-169.
  11. E. G. Poole, Introduction to the Theory of Linear Differential Equations, Clarendon (Oxford University Press), Oxford, 1936.
  12. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  13. J. P. Singhal and S. S. Bhati, Certain expansions associated with hypergeometric functions n variables, Glasnik Mat. Ser. III 11 (31) (1976), no. 2, 239-245.
  14. H. M. Srivastava, Some integrals representing triple hypergeometric functions, Rend. Circ. Mat. Palermo (Ser. 2) 16 (1967), 99-115. https://doi.org/10.1007/BF02844089
  15. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane and Toronto, 1985.
  16. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane and Toronto, 1984.

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