Communications of the
Korean Mathematical Society CKMS

Our aim is to establish certain image formulas of the $(p,q)$--extended modified Bessel function of the second kind $M_{\nu,p,q} (z)$ by employing the Marichev-Saigo-Maeda fractional calculus (integral and differential) operators including their composition formulas and using certain integral transforms involving $(p,q)$--extended modified Bessel function of the second kind $M_{\nu,p,q} (z)$. Corresponding assertions for the Saigo's, Riemann-Liouville (R-L) and Erd\'elyi-Kober (E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the $(p,q)$--extended modified Bessel function of the second kind $M_{\nu,p,q} (z)$ and Fox-Wright function $_{r}\Psi_{s}(z)$.

Keywords: $(p,q)$--extended Gauss hypergeometric function, $F_{\,p,q}(a,b,c,z)$, extended beta function, $(p,q)$--extended modified Bessel function of the second kind $M_{\nu,p,q} (z)$, fractional calculus operators

MSC numbers: Primary 26A33, 33B20, 33C20; Secondary 26A09, 33B15, 33C05