On Some Properties of a Generalization of Bessel-Maitland Function

Manoj Singh; Mumtaz Ahmad Khan; Abdul Hakim Khan

doi:https://doi.org/10.14445/22315373/IJMTT-V14P507

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 14 | Number 1 | Year 2014 | Article Id. IJMTT-V14P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V14P507

On Some Properties of a Generalization of Bessel-Maitland Function

Manoj Singh , Mumtaz Ahmad Khan , Abdul Hakim Khan

Abstract

The present paper is the investigation of certain properties of generalized Bessel-Maitland function, written in the form J(ν,q)(μ,γ) (z)=∑(n=0)∞‍ ((γ)_qn (-z)n)/n!Γ(μn+ν+1) , where μ,ν,γ∈C;Re(μ)≥0, Re(ν)≥-1,Re(γ)≥0 and q∈(0,1)∪N. For the function J(ν,q)(μ,γ) (z), a number of results including differentiation and integration formulas, Mellin-Barnes integral representation, Laplace transform, Euler transform, k-transform, Varma transform, Mellin transform. Various relationship with other functions including Fox's H-function and Wright hypergeometric function were also established. In the end certain relations have been obtained by using the Riemann-Liouville fractional integrals and derivatives.

Keywords

Generalized Bessel-Maitland function, integral transform, Wright hypergeometric function, Fox's ࡴ -function, generalized hypergeometric function, fractional calculus.

References

1.A. Erde'lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol.I and II. McGraw-Hill, New York Toronto and London, 1954. Reprinted: Krieger, Melbourne-Florida (1981).
2.A. Wiman, U ̈ber den fundamental Satz in der Theorie der Funktionen E_α (x), Acta Math., 29 (1905), 191-201.
3.A.K. Shukla and J.C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 797-811.
4.A.M. Mathai, and R.K. Saxena and H.J. Haubold, The H-Function Theory and Applications, Springer New York Dordrecht Heidelberg London, 2010.
5.A.M. Mathai, and R.K. Saxena, On linear combinations of stochastic variables, Metrika, 20(3) (1973), 160-169.
6.C. Fox, The G and H-functions and symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98 (1961), 395-429.
7.C.S. Meijer, U ̈ber eine Erweiterung der Laplace-Transform, Neder Wetensch Proc. 3:599-608, 702-711=Indag Math 2:229-238, 269-278, 1940.
8.E. D . Rainville, Special Functions, Macmillan, New York, 1960.
9.E.M. Wright, On the coefficients of power series having exponential singularities., Proc. London Math. Soc., 8 (1933), 71-79.
10.E.M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. London Math. Soc., 10 (1935), 286-293.
11.G. M. Mittag-Leffler, Sur la nouvelle fonction E_α (x), C. R. Acad. Sci. Paris, 137 (1903), 554-558.
12.G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1962.
13.I.N. Sneddon, The use of Integral Transform, Tata Mc Graw-Hill, New Delhi, 1979.
14.L. Carlitz, Some expansion and convolution formulas related to Mac Mohan's master theorem, SIAM J. Math. Anal., 8(2) (1977), 320-336.
-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
15.O.I. Marichev, Handbook of Integral Transform and Higher Transcendent -al functions. Theory and algorithm tables, Ellis Horwood, Chichester[John Wiley and Sons], New York, 1983.
16.S. G. Samko, A.A. Kilbas, and O.I.Marichev, Fractional Integrals and derivatives. Theory and Applications, Gordan and Breach, Yverdon et al., 1993.
17.T. R. Prabhakar, A singular integral equation with a generalized Mittag
18.Y. N. Robotnov, Elements of Hereditary Solid Mechanics (in English), MIR Publishers, Moscow, 1993.

Citation :

Manoj Singh , Mumtaz Ahmad Khan , Abdul Hakim Khan, "On Some Properties of a Generalization of Bessel-Maitland Function," International Journal of Mathematics Trends and Technology (IJMTT), vol. 14, no. 1, pp. 46-54, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V14P507