Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE INCOMPLETE GENERALIZED τ-HYPERGEOMETRIC AND SECOND τ-APPELL FUNCTIONS

  • Received : 2015.02.15
  • Published : 2016.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] and the second Appell function [Appl. Math. Comput. 219 (2013), 8332-8337] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we introduce here the family of the incomplete generalized ${\tau}$-hypergeometric functions $2{\gamma}_1^{\tau}(z)$ and $2{\Gamma}_1^{\tau}(z)$. The main object of this paper is to study these extensions and investigate their several properties including, for example, their integral representations, derivative formulas, Euler-Beta transform and associated with certain fractional calculus operators. Further, we introduce and investigate the family of incomplete second ${\tau}$-Appell hypergeometric functions ${\Gamma}_2^{{\tau}_1,{\tau}_2}$ and ${\gamma}_2^{{\tau}_1,{\tau}_2}$ of two variables. Relevant connections of certain special cases of the main results presented here with some known identities are also pointed out.

Keywords

References

  1. M. Abramowitz and I. A. Stegun (Editors), 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, D.C. 1964.
  2. 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.
  3. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Company, New York, 1984.
  4. W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935; Reprinted by Stechert-Hafner, New York, 1964.
  5. Yu. A. Brychkov and N. Saad, On some formulas for the Appell function $F_2$[a, b, b'; c, c';w, z], Integral Transforms Spec. Funct. 25 (2014), no. 2, 111-123.
  6. A. Cetinkaya, The incomplete second Appell hypergeometric functions, Appl. Math. Comput. 219 (2013), no. 15, 8332-8337. https://doi.org/10.1016/j.amc.2012.11.050
  7. M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall (CRC Press Company), Boca Raton, London, New York and Washington, D.C., 2001.
  8. J. Choi, R. K. Parmar, and P. Chopra, The incomplete Lauricella and first Appell functions and associated properties, Honam Math. J. 36 (2014), no. 3, 531-542. https://doi.org/10.5831/HMJ.2014.36.3.531
  9. J. Choi, R. K. Parmar, and P. Chopra, The incomplete Srivastava's triple hypergeometric functions ${\Gamma}\frac{H}{B}$ and ${\Gamma}\frac{H}{B}$, Filomat, In Press 2015.
  10. M. Dotsenko, On some applications of Wright's hypergeometric function, C. R. Acad. Bulgare Sci. 44 (1991), no. 6, 13-16.
  11. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
  12. L. Galue, A. Al-Zamel, and S. L. Kalla, Further results on generalized hypergeometric functions, Appl. Math. Comput. 136 (2003), no. 1, 17-25. https://doi.org/10.1016/S0096-3003(02)00014-0
  13. R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000.
  14. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.
  15. Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, San Francisco and London, 1975.
  16. M.-J. Luo and R. K. Raina, Extended generalized hypergeometric functions and their applications, Bull. Math. Anal. Appl. 5 (2013), no. 4, 65-77.
  17. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third Enlarged edition, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtingung der Anwendungsgebiete, Bd. 5, Springer-Verlag, Berlin, Heidelberg and New York, 1966.
  18. V. Malovichko, On a generalized hypergeometric function and some integral operators, Math. Phys. 19 (1976), 99-103.
  19. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Functions: Theory and Applications, Springer, New York, 2010.
  20. K. B. Oldham, J. Myland, and J. Spanier, An Atlas of Functions with Equator, The Atlas Function Calculator, Second edition [With 1 CD-ROM (Windows)], Springer-Verlag, Berlin, Heidelberg and New York, 2009.
  21. F.W. J. Olver, D.W. Lozier, R. F. Boisvert, and C.W. Clark (Editors), NIST Handbook of Mathematical Functions, [With 1 CD-ROM (Windows, Macintosh and UNIX)], US Department of Commerce, National Institute of Standards and Technology, Washington, D.C., 2010; Cambridge University Press, Cambridge, London and New York, 2010.
  22. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. II, Gordon and Breach Science Publishers, New York, 1990.
  23. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  24. S. B. Rao, J. C. Prajapati, A. K. D. Patel, and A. K. Shukla, Some properties of Wright-type generalized hypergeometric function via fractional calculus, Adv. Difference Equ. 2014 (2014), 119. https://doi.org/10.1186/1687-1847-2014-119
  25. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, 1993.
  26. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, London, and New York, 1966.
  27. H. M. Srivastava, A. Cetinkaya, and I. O. Kiymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 (2014), 484-491.
  28. H. M. Srivastava, M. A. Chaudhry, and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23 (2012), no. 9, 659-683. https://doi.org/10.1080/10652469.2011.623350
  29. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Acedemic Publishers, Dordrecht, Boston and London, 2001.
  30. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
  31. 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.
  32. 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.
  33. H. M. Srivastava, R. K. Parmar, and P. Chopra, A Class of extended fractional derivative operators and associated generating relations involving hypergeometric functions, Axioms 1 (2012), 238-258. https://doi.org/10.3390/axioms1030238
  34. H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernal, Appl. Math. Comput. 211 (2009), no. 1, 198-210. https://doi.org/10.1016/j.amc.2009.01.055
  35. R. Srivastava, Some properties of a family of incomplete hypergeometric functions, Russian J. Math. Phys. 20 (2013), no. 1, 121-128. https://doi.org/10.1134/S1061920813010111
  36. R. Srivastava, Some generalizations of Pochhammer's symbol and their associated families of hypergeometric functions and hypergeometric polynomials, Appl. Math. Inform. Sci. 7 (2013), no. 6, 2195-2206. https://doi.org/10.12785/amis/070609
  37. R. Srivastava, Some classes of generating functions associated with a certain family of extended and generalized hypergeometric functions, Appl. Math. Comput. 243 (2014), 132-137.
  38. R. Srivastava and N. E. Cho, Generating functions for a certain class of incomplete hypergeometric polynomials, Appl. Math. Comput. 219 (2012), no. 6, 3219-3225.
  39. R. Srivastava and N. E. Cho, Some extended Pochhammer symbols and their applications involving generalized hypergeometric polynomials, Appl. Math. Comput. 234 (2014), 277-285.
  40. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1996.
  41. N. Virchenko, On some generalizations of the functions of hypergeometric type, Fract. Calc. Appl. Anal. 2 (1999), no. 3, 233-244.
  42. 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. https://doi.org/10.1080/10652460108819336
  43. G. N.Watson, A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge, London and New York, 1944.

Cited by

  1. Incomplete Caputo fractional derivative operators vol.2018, pp.1, 2018, https://doi.org/10.1186/s13662-018-1656-1
  2. Some Families of the Incomplete H-Functions and the Incomplete $$\overline H $$H¯-Functions and Associated Integral Transforms and Operators of Fractional Calculus with Applications vol.25, pp.1, 2018, https://doi.org/10.1134/S1061920818010119