On the Generalized Meijer Transformation

Following the method of Mikusiński [l], Ditkin [2] and later with Prudnikov [3] developed an operational calculus for the operator DtD. In the 60’s, Meller [4] generalized Ditkin’s calculus to the operator B_α = t-α Dt1+αD with α ∈ (-1,1). Generalizations to Bessel operators of a higher order were made by Botashev [5], Dimovski [6], Krätzel [7] and others. Koh [8] extended Meller’s results to α > 1 by using fractional calculus. Later [9], a direct extension was achieved in which the convolution of Ф(t) and ψ(t) is given by 1$$ \phi *\psi = \frac{1}{{\Gamma \left( {\alpha + 1} \right)}}{D^\alpha }DtD\int\limits_0^t {\int\limits_0^1 {{\eta ^\alpha }{{\left( {1 - x} \right)}^\alpha }\phi \left( {x\eta } \right)\left[ {\psi \left( {1 - x} \right)\left( {t - \eta } \right)} \right]dxd\eta ,} } $$ where α ≥ 0, Dα = DnIn-α and Iv is the Riemann-Liouville integral, 2 $${\text{I}}^v {\text{f}}\left( {\text{t}} \right) = \frac{1} {{\Gamma \left( v \right)}}\int\limits_0^{\text{t}} {\left( {{\text{t}} - \xi } \right)^{v - 1} {\text{f}}\left( \xi \right){\text{d}}\xi {\text{.}}}$$