Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

INCOMPLETE EXTENDED HURWITZ-LERCH ZETA FUNCTIONS AND ASSOCIATED PROPERTIES

  • Parmar, Rakesh K. (Department of Mathematics Government College of Engineering and Technology) ;
  • Saxena, Ram K. (Department of Mathematics and Statistics Jai Narain Vyas University)
  • Received : 2015.11.30
  • Published : 2017.04.30
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] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we first introduce incomplete Fox-Wright function. We then define the families of incomplete extended Hurwitz-Lerch Zeta function. We then systematically investigate several interesting properties of these incomplete extended Hurwitz-Lerch Zeta function which include various integral representations, summation formula, fractional derivative formula. We also consider an application to probability distributions and some special cases of our main results.

Keywords

References

  1. E. W. Barnes, The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), 249-297. https://doi.org/10.1098/rsta.1906.0019
  2. 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
  3. M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994), no. 1, 99-124. https://doi.org/10.1016/0377-0427(94)90187-2
  4. 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.
  5. J. Choi, D. S. Jang, and H. M. Srivastava, A generalization of the Hurwitz-Lerch zeta function, Integral Transforms Spec. Funct. 19 (2008), no. 1-2, 65-79. https://doi.org/10.1080/10652460701528909
  6. J. Choi and R. K. Parmar, The incomplete Lauricella and fourth Appell functions, Far East J. Math. Sci. 96 (2015), 315-328.
  7. 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
  8. J. Choi, R. K. Parmar, and P. Chopra, The incomplete Srivastava's triple hypergeometric functions ${\gamma}_{B}^{H}$ and ${\Gamma}_{B}^{H}$, Filo-mat, In Press 2015.
  9. 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.
  10. M. Garg, K. Jain, and S. L. Kalla, A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), 311-319.
  11. S. P. Goyal and R. K. Laddha, On the generalized zeta function and the generalized Lambert function, Ganita Sandesh 11 (1997), 99-108.
  12. D. Jankov, T. K. Pogany, and R. K. Saxena, An extended general HurwitzLerch zeta function as a Mathieu (a, $\lambda$)-series, Appl. Math. Lett. 24 (2011), 1473-1476. https://doi.org/10.1016/j.aml.2011.03.040
  13. 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.
  14. S. D. Lin and H. M. Srivastava, Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), no. 3, 725-733. https://doi.org/10.1016/S0096-3003(03)00746-X
  15. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Functions: Theory and Appli-cations, Springer, New York, 2010.
  16. R. K. Parmar and R. K. Saxena, The incomplete generalized ${\tau}$-hypergeometric and second ${\tau}$-Appell functions, J. Korean Math. Soc. 53 (2016), no. 2, 363-379. https://doi.org/10.4134/JKMS.2016.53.2.363
  17. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives:Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications ("Nauka i Tekhnika", Minsk, 1987); Gordon and Breach Science Publishers: Reading, UK, 1993.
  18. R. K. Saxena, A remark on a paper on M-series, Fract. Calc. Appl. Anal. 12 (2009), no. 1, 109-110.
  19. M. Sharma, Fractional integration and fractional differentiation of the M-series, Fract. Calc. Appl. Anal. 11 (2008), no. 2, 187-191.
  20. M. Sharma and R. Jain, A note on a generalized M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), no. 4, 449-452.
  21. H. M. Srivastava, A new family of the ${\lambda}$-generalized Hurwitz-Lerch zeta functions with applications, Appl. Math. Inf. Sci. 8 (2014), no. 4, 1485-1500. https://doi.org/10.12785/amis/080402
  22. H. M. Srivastava, M. A. Chaudhry, and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Trans-forms Spec. Funct. 23 (2012), no. 9, 659-683. https://doi.org/10.1080/10652469.2011.623350
  23. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer, Acedemic Publishers, Dordrecht, Boston and London, 2001.
  24. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Sci-ence, Publishers, Amsterdam, London and New York, 2012.
  25. H. M. Srivastava, D. Jankov, T. K. Pogany, and R. K. Saxena, Two-sided inequalities for the extended Hurwitz-Lerch zeta function, Comput. Math. Appl. 62 (2011), no. 1, 516-522. https://doi.org/10.1016/j.camwa.2011.05.035
  26. 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.
  27. H. M. Srivastava, M.-J. Luo, and R. K. Raina, New results involving a class of gen-eralized Hurwitz-Lerch zeta functions and their applications, Turkish J. Anal. Number Theory 1 (2013), no. 1, 26-35.
  28. 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.
  29. H. M. Srivastava, R. K. Saxena, T. K. Pogany, and R. Saxena, Integral and computational representations of the extended Hurwitz-Lerch zeta function, Integral Transforms Spec. Funct. 22 (2011), no. 7, 487-506. https://doi.org/10.1080/10652469.2010.530128
  30. 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
  31. 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
  32. 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.
  33. 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. https://doi.org/10.1016/j.amc.2012.09.059
  34. R. Srivastava and N. E. Cho, Some extended Pochhammer symbols and their applications involving generalized hypergeometric polynomials, Appl. Math. Comput. 234 (2014), 277-285.

Cited by

  1. 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