# Pathway fractional integral operators involving $$\mathtt {k}$$-Struve function

• Kottakkaran Sooppy Nisar
• Saiful Rahman Mondal
• Guotao Wang
Article

## Abstract

Fractional calculus has gained more attention during the last decade due to their effectiveness and potential applicability in various problems of mathematics and statistics. Several authors have studied the pathway fractional operator representations of various special functions such as Bessel function, generalized Bessel functions, Struve function and generalized Struve function. Many researchers have established the significance and great consideration of Struve function in the theory of special functions for exploring the generalization and some applications. A new generalization called $$\mathtt {k}$$-Struve function $$\mathtt {S}_{\nu ,c}^{\mathtt {k}}\left( x\right)$$ defined by
\begin{aligned} \mathtt {S}_{\nu ,c}^{\mathtt {k}}(x):=\sum _{r=0}^{\infty }\frac{(-c)^r}{\varGamma _{\mathtt {k}}\left( r\mathtt {k}+\nu +\frac{3\mathtt {k}}{2}\right) \varGamma \left( r+\frac{3}{2}\right) } \left( \frac{x}{2}\right) ^{2r+\frac{\nu }{\mathtt {k}}+1}. \end{aligned}
where $$c,\nu \in {C}, \nu >\frac{3}{2}\mathtt {k}$$ is given by Nisar and Saiful very recently. In this paper, we establish the pathway fractional integral representation of $$\mathtt {k}$$-Struve function. Also, we give the relationship between trigonometric function and $$\mathtt {k}$$-Struve function and establish the pathway fractional integration of cosine, hyperbolic cosine, sine and hyperbolic sine functions. Some special cases also established to obtain the pathway integral representation of classical Struve function. It is pointed out that the main results presented here are general enough to be able to be specialized to yield many known and (presumably) new results.

## Keywords

Struve function $$\mathtt {k}$$-Struve functions Fractional calculus Pathway integral

26A33 33E20

## References

1. 1.
Agarwal, P., Purohit, S.D.: The unified pathway fractional integral formulae. Fract. Calc. Appl. 4(9), 1–8 (2013)Google Scholar
2. 2.
Baleanu, D., Agarwal, P.: A Composition Formula of the Pathway Integral Transform Operator. Note di Matematica, Note Mat. 34(2), 145–155 (2014)
3. 3.
Das, S.: Introduction to Fractional Calculus. In: Functional Fractional Calculus. Springer, Berlin (2011)Google Scholar
4. 4.
Fox, C.: The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. 27(4), 389–400 (1928)
5. 5.
Kilbas, A.A., Saigo, M., Trujillo, J.J.: On the generalized Wright function. Fract. Calc. Appl. Anal. 5(4), 437–460 (2002)
6. 6.
Kilbas, A.A., Sebastian, N.: Fractional integration of the product of Bessel function of the first kind. Fract. Calc. Appl. Anal. 13(2), 159–175 (2010)
7. 7.
Kilbas, A.A., Sebastian, N.: Generalized fractional integration of Bessel function of the first kind. Integral Transforms Spec. Funct. 19(11–12), 869–883 (2008)
8. 8.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006)Google Scholar
9. 9.
Mathai, A.M., Haubold, H.J.: On generalized distributions and path-ways. Phys. Lett. A 372, 2109–2113 (2008)
10. 10.
Mathai, A.M., Haubold, H.J.: Pathway model, superstatistics, Tsallis statistics and a generalize measure of entropy. Phys. A 375, 110–122 (2007)
11. 11.
Mathai, A.M.: A pathway to matrix-variate gamma and normal densities. Linear Algebra Appl. 396, 317–328 (2005)
12. 12.
Misra, V.N., Suthar, D.L., Purohit, S.D.: Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function. Cogent Math. 4, 1320830 (2017).
13. 13.
Mondal, S.R., Nisar, K.S.: Marichev-Saigo-Maeda fractional integration operators involving generalized Bessel functions. Math. Probl. Eng. 2014, 274093 (2014)
14. 14.
Nair, S.S.: Pathway fractional integration operator. Fract. Calc. Appl. Anal. 12(3), 237–252 (2009)
15. 15.
Nisar, K.S., Mondal, S.R., Choi, J.: Certain inequalities involving the $$\mathtt k$$-Struve function. J. Inequal. Appl. 2017, 17 (2017)Google Scholar
16. 16.
Nisar, K.S., Purohit, S.D., Abouzaid, M.S., Al-Qurashi, M., Baleanu, D.: Generalized k-Mittag-Leffler function and its composition with pathway integral operators. J. Nonlinear Sci. Appl. 9, 3519–3526 (2016)
17. 17.
Nisar, K.S., Eata, A.F., Dhaifallah, M.D., Choi, J.: Fractional calculus of generalized $$k$$-Mittag-Leffler function and its applications to statistical distribution. Adv. Differ. Equ. 2016, 304 (2016).
18. 18.
Purohit, S.D., Suthar, D.L., Kalla, S.L.: Marichev-Saigo-Maeda fractional integration operators of the Bessel function. Le Matematiche 67, 21–32 (2012)
19. 19.
Rahman, G., Nisar, K.S., Choi, J., Mubeen, S., Arshad, M.: Pathway Fractional Integral Formulas Involving Extended Mittag-Leffler Functions in the Kernel. Kyungpook Math. J. 59, 125–34 (2019)
20. 20.
Rainville, E.D.: Special functions. Macmillan, New York (1960)
21. 21.
Saigo, M., Maeda, N.: More generalization of fractional calculus. In: Transform Methods & Special Functions; Bulgarian Academy of Sciences: Sofia, Bulgaria, 96, 386–400 (1998)Google Scholar
22. 22.
Wright, E.M.: The asymptotic expansion of integral functions defined by Taylor series. Philos. Trans. R. Soc. London, Ser. A 238, 423–451 (1940)Google Scholar
23. 23.
Wright, E.M.: The asymptotic expansion of the generalized hypergeometric function. Proc. Lond. Math. Soc. 2(46), 389–408 (1940)

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

## Authors and Affiliations

• Kottakkaran Sooppy Nisar
• 1
• Saiful Rahman Mondal
• 2
• Guotao Wang
• 3
1. 1.Department of Mathematics, College of Arts and SciencesPrince Sattam bin Abdulaziz UniversityWadi AldawaserKingdom of Saudi Arabia
2. 2.Department of MathematicsKing Faisal UniversityAl AhsaSaudi Arabia
3. 3.School of Mathematics and computer ScienceShanxi Normal UniversityLinfenChina