On some generalized integral transforms and Parseval-Goldstein type relations

Abstract

In the present article, using the generalized Bessel-Maitland transform, the Laplace transform and the other known transforms, authors obtain new Parseval-Goldstein type relations. Using these relations, some generalized integrals involving Fox-Wright functions are evaluated. Illustrative examples are also given.

Keywords

• Generalized Bessel-Maitland integral transform
• Laplace transform
• Parseval-Goldstein type relations
• Special functions

References

1. [1] D. Albayrak, A. Dernek, F. Uçar and N. Dernek, Theorems on the Generalized Bessel- Maitland Integral Transform and its Applications, Math. Meth. Appl. Sci. 1–15, 2020.
2. [2] D. Brown, N. Dernek and O. Yürekli, Identities for the exponential integral and the complementary error transforms, Appl. Math. Comput. 182, 1377–1384, 2006.
3. [3] L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Chapman and Hall/CRC, 2006.
4. [4] N. Dernek, H.M. Srivastava and O. Yürekli, Parseval-Goldstein Type Identities In- volving the L4-Transform and the P4-Transform and Their Applications, Integral Transforms Spec. Funct. 18 (6), 397–408, 2007.
5. [5] A. Erdelyi, Table of Integral Transforms, Vol I, Mcgraw Hill, New York, 1954.
6. [6] A. Erdelyi, Table of Integral Transforms, Vol II, Mcgraw Hill, New York, 1954.
7. [7] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol.I,II, McGraw-Hill, New York Toronto and London, 1954. Reprinted: Krieger, Melbourne-Florida 1981.
8. [8] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience, 1993.

9. [9] F. Oberhettinger, Tables of Mellin Transforms, Springer-Verlag, New York, 1974.
10. [10] K.B. Oldham, J. Spanier and J. Myland, An atlas of functions, 2ed., Springer, 2010.
11. [11] I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differantial Equa- tions, to Methods of their Solution and Some of their Applications, Academic Press, San Diego and London, 1990.
12. [12] M. Singh, M.A. Khan and A.H. Khan, On Some Properties of a Generalization of Bessel-Maitland Function, International Journal of Mathematics Trend and Tecnol- ogy, 14 (1), 46–54, 2014.
13. [13] O. Yürekli, A Parseval-type theorem applied to certain integral transforms, IMA J. Appl. Math. 42, 241–249, 1989.
14. [14] O. Yürekli, A Theorem on the Generalized Stieltjes Transform and its Applications, J. Math. Anal. Appl. 168, 63–71, 1992.
15. [15] D.V. Widder, The Stieltjes Transform, Trans. Amer. Math. Soc. 43 (1), 7–60, 1938.
16. [16] E.M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. 10 (4), 286–293, 1935.