INVESTIGATION OF TIME SCALING FOR THE INVERTED BETA FUNCTIONS
Ivanna M. Dronyuk, Zoreslava Shpak, B. A. Demyda · Ukrainian Journal of Information Technology · 2019
The use of Ateb-functions is determined by those areas where ordinary trigonometric functions are used. Modern advances in physics have led to the development of new mathematical areas that require the relativity or variability of time. The current researches in this field and main results of studies of the ordinary Ateb functions are briefly described. To take into account compression/slow-down as a property of time parameter, the q-analogs of Ateb-sine (q-Ateb-sine) and Ateb-cosine (q-Ateb-cosine) are constructed by inverting the incomplete q-Beta functions. The change in parameter q corresponds to the time scaling in the studies. q-analogs of Ateb-tangent (q-Ateb-tangent), Ateb-cotangent (q-Ateb-cotangent), Ateb-secant (q-Ateb-secant) and Ateb-cosecant (q-Ateb-cosecant) are introduced. Theorems characterizing the basic properties of the constructed functions are proved. In particular, it is shown that when q→1, taking the limit we obtain ordinary Ateb-functions. The introduced functions are periodic with the period corresponding to q-analogue periods of the ordinary Ateb-functions. The representation of the period using the q-analogue of the Gamma-function is constructed. The generalized Pythagorean identity for the q-analogues of trigonometric Ateb-functions is proved. Also the properties of the parity and oddity of these functions are considered and proved. The intervals of increasing/decreasing for all functions are found. The q-analogues of the identities formulas for the trigonometric Ateb-functions are presented. Formulas for calculating q-derivatives for the q-analogue of trigonometric Ateb-functions are constructed. It is proved that constructed functions satisfy the system of q-derivative differential equations. Results of the presented studies can be used in the time series theory and signal processing.