The Main Properties of q-Functions

Nina O. Virchenko, Olena V. Ovcharenko


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 = xk-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.


Function of complex variable; Generalized q-function; Integral equation


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).

GOST Style Citations



  • There are currently no refbacks.

Copyright (c) 2017 Igor Sikorsky Kyiv Polytechnic Institute

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