# A Generalized Fractional Calculus and Integral Transforms

• Virginia Kiryakova

## Abstract

In this paper a generalized fractional calculus and its applications to different topics in analysis, especially to some integral transforms, are discussed. The kernel-function of the generalized operators of integration of fractional multiorder considered here is a suitably chosen case of Meijer’s G-function:
$${\text{G}}_{{\text{pq}}}^{{\text{mn}}} \left[ {\sigma \left| {\begin{array}{*{20}c} {{\text{a}}_{\text{1}} , \ldots ,{\text{a}}_{\text{p}} } \\ {{\text{b}}_{\text{1}} , \ldots ,{\text{b}}_{\text{q}} } \\ \end{array} } \right.} \right] = \frac{1} {{2\pi {\text{i}}}}\int\limits_L {\frac{{\prod\limits_{{\text{k}} = {\text{1}}}^{\text{m}} {\Gamma \left( {{\text{b}}_{\text{k}} - {\text{s}}} \right)} \prod\limits_{{\text{j}} = {\text{1}}}^{\text{n}} {\Gamma \left( {{\text{1}} - {\text{a}}_{\text{j}} + {\text{s}}} \right)} }} {{\prod\limits_{{\text{k}} = {\text{m}} + {\text{1}}}^{\text{q}} {\Gamma \left( {1 - {\text{b}}_{\text{k}} + {\text{s}}} \right)} \prod\limits_{{\text{j}} = {\text{n}} + {\text{1}}}^{\text{p}} {\Gamma \left( {{\text{a}}_{\text{j}} - {\text{s}}} \right)} }}} \sigma ^{\text{s}} {\text{ds}}\quad \left( {\left[ {\text{1}} \right],\left[ 2 \right]} \right).$$
(1)

### Keywords

Convolution Meijer

## Preview

### References

1. 1.
H. Bateman, A. Erdely, “Higher transcedental functions”, Moscow, (1978), (in Russian).Google Scholar
2. 2.
A. M. Mathai, R. K. Saxena, “Generalized hypergeometric functions with applications ...”, Lecture Notes in Math. 348, Berlin, (1973).Google Scholar
3. 3.
V. Kiryakova, On operators of fractional integration involving Mejer’s G-function, C. R. Acad. Bulg. Sci., 39, 10, 25–28, (1986).
4. 4.
V. Kiryakova, Generalized operators of fractional integration and differentiation and applications, Author’s summary of Ph. D. Thesis, Sofia, (1986).Google Scholar
5. 5.
V. Kiryakova, On a class of generalized operators of fractional integration, Proc. Jubilee Sess. dev. to acad. Chakalov 86 (to appear).Google Scholar
6. 6.
S. L. Kalla, “Operators of fractional integration”, in Lecture Notes in Math., 798, Springer-Verlag, (1980).Google Scholar
7. 7.
I. Dimovski, Convolutional representation of the commutant of Gelfond-Leontiev integr. operator, C. R. Acad. Bulg. Sci., 34, 12, 1643, (1981).
8. 8.
I. Dimovski, V. Kiryakova, “Convolution and commutant of Gelfond-Leontiev integr. operator”, in Function Theory 81, Sofia, (1982).Google Scholar
9. 9.
I. Dimovski, V. Kiryakova, “Convolution and differential property of Borel-Džrbasjan transform”, in Complex Anal, and Appl. ’81, Sofia, (1984).Google Scholar
10. 10.
M. Saigo, “A generalization of fractional calculus”, in Fractional calculus, London, Pitman, (1985).Google Scholar
11. 11.
I. Dimovski, Operational calculus for a class of differential operators, C. R. Acad. Bulg. Sci., 19, 12, 1111–1114, (1966).
12. 12.
I. Dimovski, Foundations of operational calculi for the Bessel-type differential operators, Serdica, 1, 51–63, (1975).
13. 13.
I. Dimovski, A convolutional method in operational calculus, Author’s summary of Ph. D. Thesis, Sofia, (1977).Google Scholar
14. 14.
I. Dimovski, V. Kiryakova, “Transmutations, convolutions and fractional powers of Bessel-type operators via Meijer’s G- function”, in Complex Anal, and Appl. ‘83, Sofia, (1985).Google Scholar
15. 15.
I. Dimovski, V. Kiryakova, Complex inversion formulas for the Obreckhoff transform, Pliska, 4, 110–116.Google Scholar
16. 16.
I. Dimovski, On a Bessel-type integral transformation, due to N. Obrechkoff, C. R. Acad. Bulg. Sci., 27, 1, 23–26, (1974).
17. 17.
I. Dimovski, V. Kiryakova, “On an integral transformation, due to N. Obrechkoff”, in Lecture Notes in Math., 798, Springer-Verlag, (1980).Google Scholar