This book deals with the theory and some applications of integral transforms that involve integration with respect to an index or parameter of a special function of hypergeometric type as the kernel (index transforms). The basic index transforms are considered, such as the Kontorovich-Lebedev transform, the Mehler-Fock transform, the Olevskii Transform and the Lebedev-Skalskaya transforms. The Lp theory of index transforms is discussed, and new index transforms and convolution constructions are demonstrated. For the first time, the essentially multidimensional Kontorovich-Lebedev transform is announced. General index transform formulae are obtained. The connection between the multidimensional index kernels and G and H functions of several variables is presented. The book is self-contained, and includes a list of symbols with definitions, author and subject indices, and an up-to-date bibliography.This work will be of interest to researchers and graudate students in the mathematical and physical sciences whose work involves integral transforms and special functions.
Hypergeometric type special functions; Fourier, Laplace and Mellin transforms; Mellin convolution type integral transforms; Kontorovich-Lebedev transform; Mehler-Fock index transform; the Kontorovich-Lebedev transform of arbitrary index; asymptotic expansions of the index transforms; index-convolution Kontorovich-Lebedev transform; generalized Mehler-Fock transform; the Olevskii index transform; the index transforms with hypergeometric functions as the kernels; index transforms with the product of legendre functions of the first and second kind; the Lebedev-Skalskaya index transforms; index transforms involving Meijer G and H functions as the kernels; index transforms with the Whittaker functions; index transforms with cylindrical functions; convolutions of the index transforms; index transforms of distributions; multidimensional Kontorovich-Lebedev transform; multidimensional index transforms.