Introduction to the theory of $(\nu_1,\dots,\nu_{r-1})$-transforms

The transforms
\begin{gather*} \varphi_\nu(x)=\int_0^\infty\dotsi\int_0^\infty f\Bigl(x\prod t_i\Bigr)e^{-\sum{t_i^r}}\prod t _i^{r{\nu_i}+r-1}\,dt_i, \\ f(x)=\biggl(\frac r{2\pi i}\biggr)^{r-1}\int_{-\infty}^{(0+)}\dotsi\int_{-\infty}^{(0+)}\varphi_\nu\Bigl(x\prod t_i^{-\frac1r}\Bigr)e^{\sum{t_i}}\prod t_i^{{-\nu_i}-1}\,dt_i \end{gather*}
are introduced for an integer $r\geqslant2$ and a given vector $\nu=(\nu_1,\dots,\nu_{r-1})$. Their duality is substantiated, applications of the differentiation operations are studied, and other properties of $\nu$-transforms are established. A number of examples are given to illustrate the method of $\nu$-transforms for solving some classes of differential equations and boundary value problems for partial differential equations.
Bibliography: 9 titles.