### The Main Properties of q-Functions

#### Abstract

**Background.** The new generalization of the function of complex variable (*q*-function) is considered, its main properties are investigated. Such distributions have a special place among the special functions due to their widespread use in many areas of applied mathematics.

**Objective**. The aim of the paper is to study the new generalization of the function of complex variable for application in applied sciences.

**Methods.** To obtain scientific results the general methods of the mathematical analysis, and the theory of special functions have been used.

**Results.** The article deals with new generalization of the function of complex variable – *q*-functions, its main properties are investigated. The theorem on integral representation of *q* = *x ^{k}*-analytical functions is proved, its inverse formula is constructed.

**Conclusions.** Considered in the article new generalization of the function of complex variable opens up opportunities for the use of *q*-functions in the theory of special functions, and in the applications of mathematical and physical problems. In the future we plan to use the results to solve the boundary value problems of mathematical physics, in the theory of elasticity, for solving of, the theory of integral equations, etc.

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DOI: https://doi.org/10.20535/1810-0546.2017.4.105300

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