In recent paper Dinesh Kumar et al. developed a generalized fractional kinetic equation involving generalized Bessel function of first kind. The object of this paper is to derive the solution of the fractional kinetic equation involving generalized Struve function of the first kind. The results obtained in terms of generalized Struve function of first kind are rather general in nature and can easily construct various known and new fractional kinetic equations

Two integral operators involving Appell′s functions, or Horn′s function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind is expressed in terms of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function, are also discussed.

CERTAIN UNIFIED INTEGRAL FORMULAS INVOLVING THE GENERALIZED MODIFIED k-BESSEL FUNCTION OF FIRST KIND k-Bessel function;Gamma function;hypergeometric function $_2F_1$;generalized hypergeometric function $_pF_q$;generalized (Wright) hypergeometric functions $_p{\Psi}_q$;Oberhettinger's integral formula; Generalized integral formulas involving the generalized modified k-Bessel function $J^{b,c,{\gamma},{\lambda}}_{k,{\upsilon}}(z)$ of first kind are expressed in terms generalized Wright functions. Some interesting special cases of the main results are also discussed.

In this paper, we introduce and study a generalization of the k-Bessel function of order ν given by $$ \mathtt{W}^{\mathtt{k}}_{\nu , c}(x):= \sum_{r=0}^{\infty } \frac{(-c)^{r}}{\Gamma_{\mathtt{k}} ( r \mathtt{k} +\nu +\mathtt{k} ) r!} \biggl( \frac{x}{2} \biggr) ^{2r+\frac{\nu }{\mathtt{k}}}. $$ We also indicate some representation formulae for the function introduced. Further, we show that the function \(\mathtt{W}^{ \mathtt{k}}_{\nu , c}\) is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function \(\mathtt{W}^{\mathtt{k}} _{\nu , c}\), particularly, in the case \(c=-1\). We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of \(\mathtt{W}^{ \mathtt{k}}_{\nu , c}\).

We aim to introduce a k-Struve function and investigate its various properties, including mainly certain inequalities associated with this function. One of the inequalities given here is pointed out to be related to the so-called classical Turán-type inequality. We also present a differential equation, several recurrence relations, and integral representations for this k-Struve function.