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

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.