The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions

D. K. Jain; Altaf Ahmad; Renu Jain; Farooq Ahmad

doi:doi:10.11648/j.dmath.20170201.11

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The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions

D. K. Jain, Altaf Ahmad, Renu Jain, Farooq Ahmad

Published in International Journal of Discrete Mathematics (Volume 2, Issue 1)

Received: 23 December 2016 Accepted: 16 January 2017 Published: 9 February 2017

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Abstract

The q-Calculus has served as a bridge between mathematics and physics, particularly in case of quantum physics. The q-generalizations of mathematical concepts like Laplace and Fourier transforms, Hypergeometric functions etc. can be advantageously used in solution of various problems arising in the field of physical and engineering sciences. The present paper deals with some of the important results of q-Laplace transform of Fox-Wright and Mittag-Leffler functions in terms of well-known Fox’s H-function. Some special cases have also been discussed.

Published in	International Journal of Discrete Mathematics (Volume 2, Issue 1)
DOI	10.11648/j.dmath.20170201.11
Page(s)	1-5
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Classical Laplace Transform, q-Image of Laplace Transform, ML-Function, Fox-Wright Function

References

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[2]	Zemanian, A. H. (1968). Generalized integral transform, Pure Appl. Math. 18, inter Science Publ. John Wiley and Sons, New York.
[3]	Rao, G. L. N. (1974). The Generalized Laplace transform of generalized function, Ranchi Univ, Math. J.5.
[4]	Saxena, R. K. (1960). Some theorems on generalized Laplace transform I, Proc. Nat. Inst. Sci. India, part A, 26,400-413.
[5]	Mathai, A. M. (2007). A Versatile integral, Preprint series, No.12, Centre for Mathematical Sciences, Pala, Kerala, India.
[6]	Hahn, W. (1949). Beitrage Zur Theorie der Heineshan Reihen. Die 24 integrale der hypergeometrischen q-Differenzengleichung. Das q-Analogon der Laplace-Transformation. Mathematische Nachrichten, 2(6), 340-379.
[7]	Rooney, P. G. (1982\83). On integral transformation with G-function kernels, proc. Royal. Soc. Edinburgh Sect. A. 93.
[8]	Saxena, R. K. (1961). Some theorems on generalized Laplace transform II, Riv. Mat. Univ. Parma(2)2, 287-299.
[9]	Saxena, R. K. (1966). Some theorems on generalize Laplace transform, Proc. Cambridge Philos. Soc; 62, 467-471.
[10]	Purohit, S. D., Yadav R. K. and Vyas, V. K. On applications of q-Laplace transforms to a basic analogue of the I-function. (Communicated).
[11]	Purohit S. D., and Yadav, R. K. On q-Laplace transforms of certain q-hypergeometric polynomials , Proc. Nat. Acad. Sci. India(2006) 235-242-III.
[12]	Heideman, M. T., Johnson, D. H., & Burrus, C. S. (1985). Gauss and the history of the fast Fourier transform. Archive for history of exact sciences, 34(3), 265-277.
[13]	Kac, V., & Cheung, P. (2001). Quantum calculus. Springer Science & Business Media.
[14]	Wright, E. M (1935), The asymptotic expansion of generalized hypergeometric function. J. London Math. So, 10, 286-293.
[15]	Mittag-Leffler function, M. G.: 1903 Sur la nouvelle function , Comptes Rendus Acad. Sci, Paris (Ser.II)137, 554-558.
[16]	Metzler, R and Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, phys. Rep. 339(2000), 1-77.
[17]	Prabhakar, T. R (1971), A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. Vol. 19, 7-15.
[18]	Sheikh, F. A., & Jain, D. K. Basic Hypergeometric Functions and q-Calculus. Corona publications (2015).
[19]	Sharma, S. K., & Jain, R. On Some Properties Of Generalized q-Mittag-Leffler Function. Mathematica Aeterna, Vol. 4, 2014, no.6, 613–619.

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APA Style

D. K. Jain, Altaf Ahmad, Renu Jain, Farooq Ahmad. (2017). The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions. International Journal of Discrete Mathematics, 2(1), 1-5. https://doi.org/10.11648/j.dmath.20170201.11

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ACS Style

D. K. Jain; Altaf Ahmad; Renu Jain; Farooq Ahmad. The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions. Int. J. Discrete Math. 2017, 2(1), 1-5. doi: 10.11648/j.dmath.20170201.11

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AMA Style

D. K. Jain, Altaf Ahmad, Renu Jain, Farooq Ahmad. The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions. Int J Discrete Math. 2017;2(1):1-5. doi: 10.11648/j.dmath.20170201.11

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@article{10.11648/j.dmath.20170201.11,
  author = {D. K. Jain and Altaf Ahmad and Renu Jain and Farooq Ahmad},
  title = {The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions},
  journal = {International Journal of Discrete Mathematics},
  volume = {2},
  number = {1},
  pages = {1-5},
  doi = {10.11648/j.dmath.20170201.11},
  url = {https://doi.org/10.11648/j.dmath.20170201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170201.11},
  abstract = {The q-Calculus has served as a bridge between mathematics and physics, particularly in case of quantum physics. The q-generalizations of mathematical concepts like Laplace and Fourier transforms, Hypergeometric functions etc. can be advantageously used in solution of various problems arising in the field of physical and engineering sciences. The present paper deals with some of the important results of q-Laplace transform of Fox-Wright and Mittag-Leffler functions in terms of well-known Fox’s H-function. Some special cases have also been discussed.},
 year = {2017}
}

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TY  - JOUR
T1  - The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions
AU  - D. K. Jain
AU  - Altaf Ahmad
AU  - Renu Jain
AU  - Farooq Ahmad
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T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
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AB  - The q-Calculus has served as a bridge between mathematics and physics, particularly in case of quantum physics. The q-generalizations of mathematical concepts like Laplace and Fourier transforms, Hypergeometric functions etc. can be advantageously used in solution of various problems arising in the field of physical and engineering sciences. The present paper deals with some of the important results of q-Laplace transform of Fox-Wright and Mittag-Leffler functions in terms of well-known Fox’s H-function. Some special cases have also been discussed.
VL  - 2
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Author Information

D. K. Jain

Madhav Institute of Technology and Science, Gwalior (M.P.), India
Altaf Ahmad

School of Mathematics and Allied Sciences Jiwaji University, Gwalior (M.P.), India
Renu Jain

School of Mathematics and Allied Sciences Jiwaji University, Gwalior (M.P.), India
Farooq Ahmad

Department of Mathematics, Govt. Degree College, Kupwara (J&K), India

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APA Style

D. K. Jain, Altaf Ahmad, Renu Jain, Farooq Ahmad. (2017). The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions. International Journal of Discrete Mathematics, 2(1), 1-5. https://doi.org/10.11648/j.dmath.20170201.11

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ACS Style

D. K. Jain; Altaf Ahmad; Renu Jain; Farooq Ahmad. The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions. Int. J. Discrete Math. 2017, 2(1), 1-5. doi: 10.11648/j.dmath.20170201.11

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AMA Style

D. K. Jain, Altaf Ahmad, Renu Jain, Farooq Ahmad. The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions. Int J Discrete Math. 2017;2(1):1-5. doi: 10.11648/j.dmath.20170201.11

Copy | Download

@article{10.11648/j.dmath.20170201.11,
  author = {D. K. Jain and Altaf Ahmad and Renu Jain and Farooq Ahmad},
  title = {The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions},
  journal = {International Journal of Discrete Mathematics},
  volume = {2},
  number = {1},
  pages = {1-5},
  doi = {10.11648/j.dmath.20170201.11},
  url = {https://doi.org/10.11648/j.dmath.20170201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170201.11},
  abstract = {The q-Calculus has served as a bridge between mathematics and physics, particularly in case of quantum physics. The q-generalizations of mathematical concepts like Laplace and Fourier transforms, Hypergeometric functions etc. can be advantageously used in solution of various problems arising in the field of physical and engineering sciences. The present paper deals with some of the important results of q-Laplace transform of Fox-Wright and Mittag-Leffler functions in terms of well-known Fox’s H-function. Some special cases have also been discussed.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - The Classical Laplace Transform and Its q-Image of the Most Generalized Hypergeometric and Mittag-Leffler Functions
AU  - D. K. Jain
AU  - Altaf Ahmad
AU  - Renu Jain
AU  - Farooq Ahmad
Y1  - 2017/02/09
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170201.11
DO  - 10.11648/j.dmath.20170201.11
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 1
EP  - 5
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20170201.11
AB  - The q-Calculus has served as a bridge between mathematics and physics, particularly in case of quantum physics. The q-generalizations of mathematical concepts like Laplace and Fourier transforms, Hypergeometric functions etc. can be advantageously used in solution of various problems arising in the field of physical and engineering sciences. The present paper deals with some of the important results of q-Laplace transform of Fox-Wright and Mittag-Leffler functions in terms of well-known Fox’s H-function. Some special cases have also been discussed.
VL  - 2
IS  - 1
ER  -

Copy | Download