Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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AN EXTENSION OF THE EXTENDED HURWITZ-LERCH ZETA FUNCTIONS OF TWO VARIABLES

  • Choi, Junesang (Department of Mathematics Dongguk University) ;
  • 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 : 2016.07.05
  • Accepted : 2017.02.20
  • Published : 2017.11.30
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Abstract

We aim to introduce a further extension of a family of the extended Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate several interesting properties of the extended function such as its integral representations which provide extensions of various earlier corresponding results of two and one variables, its summation formula, its Mellin-Barnes type contour integral representations, its computational representation and fractional derivative formulas. A multi-parameter extension of the extended Hurwitz-Lerch Zeta function of two variables is also introduced. Relevant connections of certain special cases of the main results presented here with some known identities are pointed out.

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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. 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.
  3. 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
  4. O. Daman and M. A. Pathan, A further generalization of the Hurwitz Zeta function, Math. Sci. Res. J. 16 (2012), no. 10, 251-259.
  5. 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.
  6. M. Garg, K. Jain, and S. L. Kalla, A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), 311-319.
  7. S. P. Goyal and R. K. Laddha, On the generalized Zeta function and the generalized Lambert function, Ganita Sandesh 11 (1997), no. 2, 99-108.
  8. N. T. Hai and S. B. Yakubovich, The Double Mellin-Barnes Type Integrals and Their Applications to Convolution Theory, World Scientific, Singapore, 1992.
  9. D. Jankov, T. K. Pogany, and R. K. Saxena, An extended general Hurwitz-Lerch Zeta function as a Mathieu (${\alpha}$, $\lambda$ )-series, Appl. Math. Lett. 24 (2011), no. 8, 1473-1476. https://doi.org/10.1016/j.aml.2011.03.040
  10. 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.
  11. 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
  12. A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applica- tions in Statistics and Physical Sciences, Lecture Notes Series No. 348, Springer-Verlag, Berlin, New York. Heidelberg, Germany, 1973.
  13. A. M. Mathai, The H-function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd. New Delhi and John Wiley and Sons, Inc. New York, 1978.
  14. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Functions: Theory and Applications, Springer, New York, 2010.
  15. R. B. Paris and D. Kaminski, Asymptotic and Mellin-Barnes Integrals, Cambridge University Press, Cambridge, 2001.
  16. M. A. Pathan and O. Daman, On generalization of Hurwitz zeta function, Submitted.
  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. 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
  19. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer, Acedemic Publishers, Dordrecht, Boston and London, 2001.
  20. H. M. Srivastava, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science, Publishers, Amsterdam, London and New York, 2012.
  21. 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
  22. 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.
  23. H. M. Srivastava, M.-J. Luo and R. K. Raina, New results involving a class of generalized Hurwitz-Lerch Zeta functions and their applications, Turkish J. Anal. Number Theory 1 (2013), no. 1, 26-35.
  24. 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.
  25. 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