On the finite Mellin transform in quantum calculus and application

The aim of the present paper is to introduce and study a new type of q-Mellin transform 11], that will be called q-finite Mellin transform. In particular, we prove for this new transform an inversion formula and q-convolution product. The application of this transform is also earlier proposed in solving procedure for a new equation with a new fractional differential operator of a variational type.