# The Main Properties of q-Functions

## DOI:

https://doi.org/10.20535/1810-0546.2017.4.105300## Keywords:

Function of complex variable, Generalized q-function, Integral equation## 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.

## References

N. Virchenko, *Generalized Hypergeometric Functions*. Kyiv, Ukraine: NTUU KPI Publ., 2016 (in Ukrainian).

A.A. Kilbas and M. Saigo, *H-Transforms*. London, UK: Chapman and Hall/CRC, 2004.

S.* *Yakubovich, “Index transforms associated with generalized hypergeometric functions”, *Math. Methods** **Appl**. **Sci*., vol. 27, no. 1, pp. 35–46, 2004. doi: 10.1002/mma.436

E. Picard, *Sur la Representation Approchee des Fonctions*. Paris, France: C.R. Acad. Sci. Paris, 1891.

E. Beltrami, “Sulle funzioni potenziali di sistemi simmetrici intorno ad un asse”, *Opere mat. Milano*, vol. 3, pp. 115–128, 1911.

L. Bers and A. Gelbart, “*On a class of functions defined by partial differential equations**”*,* Trans. Amer. Math. Soc.*, vol. 56, pp. 67–93, 1944. doi: 10.1090/S0002-9947-1944-0010910-5

A. Weinstein, “*Generalized axially symmetric potential theory**”*, *Bull. Amer. Math. Soc.*, vol. 59, pp. 20–38, 1953. doi: 10.1090/S0002-9904-1953-09651-3

M.A. Lukomskaia, “*On cycles of systems of linear homogeneous differential equations**”*, *Mat. Sbornik N.S.*, no. 29 (71), pp. 551–558, 1951 (in Russian).

S. Agmon and* *L. Bers, “*The expansion theorem for pseudo-analytic functions**”*, *Proc. Amer. Math. Soc.*, vol. 3, pp. 757–764, 1952. doi: 10.1090/S0002-9939-1952-0057349-4

G.N. Polozii, *Theory and Application of p-Analytic and *(*p*, *q*)*-Analytic Functions*. Kyiv, Ukraine: Naukova Dumka, 1973 (in Russian).

B.V. Shabat, “*Cauchy’s theorem and formula for quasi-conformal mappings of linear classes**”*,* Doklady Akad. Nauk SSSR (N.S.)*, vol. 69, pp. 305–308, 1949 (in Russian).

Y.B. Lopatinskii, “On one generalization of analytic function”, *UMJ*, no. 2, pp. 56–73, 1950 (in Russian).

L. Bers, “*Remark on an application of pseudo-analytic functions**”**, **Bull**. **Amer**. **Math**. **Soc*., vol. 62, pp. 291–331, 1956.

I.N. Vekua,* **Generalized Analytic Functions*. Moscow, SU: Fizmatgiz, 1959 (in Russian).

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2017 Igor Sikorsky Kyiv Polytechnic Institute

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work