• KALLA, SHYAM LAL (Department of Mathematics Vyas Institute of Higher Education) ;
  • PARMAR, RAKESH KUMAR (Department of Mathematics Government College of Engineering and Technology) ;
  • PUROHIT, SUNIL DUTT (Department of Mathematics Rajasthan Technical University)
  • Received : 2015.04.18
  • Published : 2015.06.30


Motivated mainly by certain interesting extensions of the ${\tau}$-hypergeometric function defined by Virchenko et al. [11] and some ${\tau}$-Appell's function introduced by Al-Shammery and Kalla [1], we introduce here the ${\tau}$-Lauricella functions $F_A^{(n),{\tau}_1,{\cdots},{\tau}_n}$, $F_B^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and $F_D^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and the confluent forms ${\Phi}_2^{(n),{\tau}_1,{\cdots},{\tau}_n}$ and ${\Phi}_D^{(n),{\tau}_1,{\cdots},{\tau}_n}$ of n variables. We then systematically investigate their various integral representations of each of these ${\tau}$-Lauricella functions including their generating functions. Various (known or new) special cases and consequences of the results presented here are also considered.


generalized hypergeometric function;generlalized ${\tau}$-hypergeometric function;Appell's and Lauricella functions;${\tau}$-Appell's function;${\tau}$-Lauricella functions of several variables;generating function


  1. A. H. Al-Shammery and S. L. Kalla, An extension of some hypergeometric functions of two variables, Rev. Acad. Canaria Cienc. 12 (2000), no. 1-2, 189-196.
  2. W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935; Reprinted by Stechert-Hafner, New York, 1964.
  3. M. Dotsenko, On some applications of Wright's hypergeometric function, C. R. Acad. Bulgare Sci. 44 (1991), no. 6, 13-16.
  4. L. Galue, An Extension of some Humbert's functions, Int. J. Appl. Math. 17 (2005), no. 1, 91-106.
  5. L. Galue, A. Al-Zamel, and S. L. Kalla, Further results on generalized hypergeometric functions, Appl. Math. Comput. 136 (2003), no. 1, 17-25.
  6. V. Malovicko, A generalized hypergeometric function and some integral operators that contains it, Mat. Fiz. 19 (1976), 99-103.
  7. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  8. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
  9. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
  10. N. Virchenko, On some generalizations of the functions of hypergeometric type, Frac. Calc. Appl. Anal. 2 (1999), no. 3, 233-244.
  11. N. Virchenko, S. L. Kalla, and A. Al-Zamel, Some results on a generalized hypergeometric function, Integral Transform. Spec. Funct. 12 (2001), no. 1, 89-100.
  12. E. M. Wright, On the coffecient of power series having exponential singularities, J. London Math. Soc. 8 (1933), 71-79.
  13. E. M. Wright, The asymptotic expansion of the generalized hypergeometric functions, J. London Math. Soc. 19 (1935), 286-293.

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