Aleph Function

In this book we present some more general properties and their applications of Aleph function which is more general extension of Saxena I-function. The usefulness and the importance of the Marichev-Saigo-Maeda fractional integral operators, many authors have presented a number of interesting integral formulas involving special functions by using the Marichev-Saigo-Maeda fractional integral operator. By virtue of the unified nature of Marichev-Saigo-Maeda fractional integral operators, a large number of new and known results involving Saigo, Riemann-Liouville and Erdélyi-Kober fractional integral operators follow as special cases of our main formulas. All the results derived here are of general character and can yield a number of results in the theory of fractional calculus. A variant of such operators (integral transforms) was introduced by Marichev as Mellin type convolution operators with a special function \(F_{3}(\cdot)\) in the kernel. These operators were rediscovered and studied by Saigo as a generalization of the so-called Saigo fractional integral operators.