Research ArticleOpen AccessGoogle Scholar indexed
Modified Double Zeta Function and Its Properties
Department of Mathematics, Jodhpur Institute of Engineering & Technology, Jodhpur, India
- 1 Department of Mathematics, Jodhpur Institute of Engineering & Technology, Jodhpur, India
Advances in Pure Mathematics·Volume 06 (2016)·Pages 159–167·Published 26 February 2016·DOI10.4236/apm.2016.63013
Copy link · social · email
Abstract
The present paper aims at introducing and investigating a new class of generalized double zeta function i.e . modified double zeta function which involves the Riemann, Hurwitz, Hurwitz-Lerch, Barnes double zeta function and Bin-Saad generalized double zeta function as particular cases. The results are obtained by suitably applying Riemann-Liouville type and Tremblay fractional integral and differential operators. We derive the expansion formula for the proposed function with some of its properties via fractional operators and discuss the link with known results.
KeywordsModified Zeta FunctionRiemann-Liouville OperatorTremblay Fractional OperatorsHypergeometric Function
- Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1953) Higher Transcendental Functions, Vol. I. Mc-Graw-Hill, New York, Toronto and London.
- Goyal, S. and Laddha, R.K. (1997) On the Generalized Riemann Zeta Function and the Generalized Lambert Transform. Ganita Sandesh, 11, 99-108.
- Bin-Saad, M.G. and Al Gonah, A.A. (2006) On Hypergeometric Type Generating Functions Associated with Generalized Zeta Function. Acta Mathematica Universitatis Comenianae, 75, 253-266.
- Bin-Saad, M.G. (2007) Sums and Partial Sums of Double Power Series Associated with the Generalized Zeta Function and Their N Fractional Calculus. Mathematical Journal of Okayama University, 49, 37-52.
- Bin-Saad, M.G. (2009) Hypergeometric Series Associated with the Hurwitz-Lerch Zeta Function. Acta Mathematica Universitatis Comenianae, 78, 269-286.
- Renvillle, E.D. (1960) Special Functions. Macmillan Company, New York.
- Srivastava, H.M. and Karlsson, P.K. (1985) Multiple Gaussian Hypergeometric Series. Halsted Press Bristone, London and New York.
- Kilbas, A.A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Application of Fractional Differential Equations. North Holland Mathematical Studied, Vol. 204, Elsevier, Amsterdam.
- Hilfer, R., Ed. (2000) Application of Fractional Calculus in Physics. World (Sc.), Singapore.
- Prajapati, J.C., Saxena, R.K., Jana, R.K. and Shukla, A.K. (2013) Some Results on Mittag Leffler Function Operator. Journal of Inequalities and Applications, 2013, 33. http://dx.doi.org/10.1186/1029-242X-2013-33
- Rao, S.B., Salehbhai, I.A. and Shukla, A.K. (2013) On Sequence of Functions Containing Generalized Hypergeometric Function. Mathematical Sciences Research Journal, 17, 98-110.
- Rao, S.B., Prajapati, J.C., Patel, A.D. and Shukla A.K. (2014) Some Properties of Wright Type Hypergeometric Functionvia Fractional Calculus. Advances in Difference Equation, 2014, 119. http://dx.doi.org/10.1186/1687-1847-2014-119
- Shukla, A.K. and Prajapati, J.C. (2008) On Mittag Leffler Type Function and Generalized Integral Operator. Mathematical Sciences Research Journal, 12, 283-290.
- Srivastva, H.M. and Tomovski, Z. (2009) Fractional Calculus with an integral Operator Containing a Generalized Mittag Leffler in the Kernel. Applied Mathematics and Computation, 211, 198-210. http://dx.doi.org/10.1016/j.amc.2009.01.055