DOI QR코드

DOI QR Code

On Extended Hurwitz-Lerch Zeta Function

  • Received : 2022.08.07
  • Accepted : 2023.02.22
  • Published : 2023.09.30

Abstract

This paper investigates an extended form Hurwitz-Lerch zeta function, as well as related integral images, ordinary and fractional derivatives, and series expansions, using the term extended beta function. We establish a connection between the extended Hurwitz-Lerch zeta function and the Laguerre polynomials. Furthermore, we present a probability distribution application of the extended Hurwitz-Lerch zeta function ζ𝛿,𝜇𝜈,λ. Several results, both known and new, are shown to follow as special cases of our findings.

Keywords

Acknowledgement

We appreciate the anonymous reviewers insightful comments, which improved the readability of the papers presentation.

References

  1. E. W. Barnes, The theory of the double gamma function, Philos. Trans. Roy. Soc.(A), 196(1901), 265-387. https://doi.org/10.1098/rsta.1901.0006
  2. E. W. Barnes, On the theory of the double gamma function, Trans. Cambridge Phil. Soc., 19(1904), 374-425.
  3. C. Berge, Principles of combinatorics, Academic Press(1971).
  4. Maged G. Bin-Saad, Sums and partial sums of double power series associated with the generalized zeta function and their N-fractional calculus, Math. J. Okayama University, 49(2007), 37-52.
  5. Maged G. Bin-Saad, Hypergeometric series associated with the Hurwitz-Lerch zeta function, Acta Math. Univ. Comenian, 78(2009), 269-286.
  6. M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma function with applications. J. Comput. Appl. Math., 55(1994), 99-124. https://doi.org/10.1016/0377-0427(94)90187-2
  7. M. A. Chaudhry, A. Qadir, M. Raque and S. M. Zubair, Extension of Euler's beta function, J. Comput. Appl. Math., 78(1997), 19-32. https://doi.org/10.1016/S0377-0427(96)00102-1
  8. M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159(2004), 589-602. https://doi.org/10.1016/j.amc.2003.09.017
  9. M. A. Chaudhry and S. M. Zubair, On a class of incomplete gamma functions with applications, CRC Press (Chapman and Hall), Boca Raton(2002).
  10. A. Erdelyi, W. Magnus and F. Oberhettinger, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York(1953).
  11. O. Espinosa and V. H. Moll, On some integrals involving the Hurwitz zeta function: Part 2, The Ramanujan J., 6(2002), 159-188. https://doi.org/10.1023/A:1015706300169
  12. O. Espinosa and V. H. Moll, A generalized poly gamma function, Integral Transforms Spec. Funct., 15(2)(2004), 101-115. https://doi.org/10.1080/10652460310001600573
  13. H. Exton, Multiple hypergeometric functions and applications, Halsted Press, London(1976).
  14. M. Garg, K. Jain and S. L. Kalla, A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom., 25(2008), 311-319.
  15. S. Goyal and R. K. Laddha, On the generalized Riemann zeta function and the generalized Lambert transform, Ganita Sandesh, 11(1997), 99-108.
  16. S. Kanemitsu, M. Katsurada and M. Yoshimoto, On the Hurwitz-Lerch zeta function. Aequationes Math., 59(2000), 1-19. https://doi.org/10.1007/PL00000117
  17. M. Katsurada, An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-function, Collect. Math., 48(1997), 137-153.
  18. D. Klusch, On the Taylor expansion of Lerch zeta-function, J. Math. Anal. Appl., 170(1992), 513-523. https://doi.org/10.1016/0022-247X(92)90034-B
  19. V. Kumar, On the generalized hurwitz-lerch zeta function and generalized lambert transform, J. Class. Anal., 17(1)(2021), 55-67. https://doi.org/10.7153/jca-2021-17-05
  20. C. A. Larry, Special functions of mathematics for engineers, SPIE Press and Oxford University Press, New York(1998).
  21. K. Matsumoto, On the analytic continuation of various multiple zeta-functions, Number theory for the millennium, II (Urbana, IL, 2000), 417-440. https://doi.org/10.1201/9780138747060-21
  22. K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, New York(1993).
  23. M. A. Ozaraslan, Some remarks on extended hypergeometric, extended confluent hypergeometric and extended Appell's functions, J. Comput. Anal. Appl., 14(6)(2012), 1148-1153.
  24. A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Volume 3, More special functions, Gordon and Breach Science Publishers, New York(1990).
  25. R. K. Parmar and R. K. Raina, On a certain extension of the Hurwitz-Lerch zeta functionm, An. Univ. Vest Timis. Ser. Mat.-Inform., 52(2)(2014), 157-170. https://doi.org/10.2478/awutm-2014-0017
  26. S. Ramanujan, A series for Euler's constant γ, Messenger Math., 46(1916-17), 73-80.
  27. R. K. Raina and P. K. Chhajed, Certain results involving a class of functions associated with the Hurwitz zeta function, Acta. Math. Univ. Comenianae, 1(2004), 89-100.
  28. L. J. Slater, Confluent hypergeometric functions, Cambridge Univ., Press(1960).
  29. H. M. Srivastava and P. K. Karlsson, Multiple Gaussian Hypergeometric Series. Halsted Press, Brimstone, London, New York(1985).
  30. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. Cambridge Univ. Press(1927).