Rybachuk L. Generalized confluent hypergeometric functions and their applications

New generalizations of the confluent hypergeometric functions, gamma-, beta-, psi-functions, Hurwitz zeta-functions, Laguerre and Volterra functions are introduced. Integral, differential and functional relations for the introduced functions are proved. A number of integrals which are absent in available scientific publications are calculated. Volterra and Fredholm integral equations of the first kind and Cauchy problem for the fractional differential equation with the partial derivatives are solved and investigated. The theorems about the existence and uniqueness of the solutions of these equations are given. New generalizations of Lambert, Hurwitz integral transforms are constructed. Their applications are given.