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Urbanik type subclasses of free-infinitely~divisible~transforms

Jurek Zbigniew J.

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 1

23--39

For the class of free-infinitely divisible transforms we introduce three families of increasing Urbanik type subclasses. They begin with the class of free-normal transforms and end up with the whole class of free- infinitely divisible transforms. Those subclasses are derived from the ones of classical infinitely divisible measures for which random integral repre- sentations are known. Special functions like Hurwitz–Lerch, polygamma and hypergeometric functions appear in kernels of the corresponding integral representations.

Niejednoznaczności wyznaczania ułamkowej potęgi wektora wirującego

Zawadzki A., Włodarczyk M.

Przegląd Elektrotechniczny

2017

R. 93, nr 11

115--118

W pracy przedstawiono problem niejednoznaczności wyznaczania pochodnych ułamkowego rzędu wektora wirującego. Jest to związane z określeniem ułamkowej potęgi liczby zespolonej.

The paper presents problems of ambiguity of calculation of fractional derivatives for the rotating vector. This is related to the determining of fractional power of the imaginary number.

The application of hypergeometric functions to computing fractional order derivatives of sinusoidal functions

Włodarczyk M., Zawadzki A.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2016

Vol. 64, nr 1

243--248

In the paper, the analytical forms of fractional order derivatives of sinusoidal function according to definitions of Riemann - Liouville and Caputo are presented. To determine the analytical form of the integrals appearing in definitions of derivatives of fractional order the Lommel functions from hypergeometrical functions family were applied. With the use of properties of the derivatives of fractional order - differential-integral there were presented the conception of generalized element of a single equation, which depending on the value of the derivative order, can be inductor, resistor, capacitor, or a hypothetical element of a fractional order differential equation.

Applications of convolution properties

Trojnar-Spelina L.

Demonstratio Mathematica

2013

Vol. 46, nr 3

513--523

K. I. Noor (2007 Appl. Math. Comput. 188, 814–823) has defined the classes Qk(a, b, λ, γ) and Tk(a, b, λ, γ) of analytic functions by means of linear operator connected with incomplete beta function. In this paper, we have extended some of the results and have given other properties concerning these classes.

On a certain class of analytic functions involving hypergeometric functions

Khosravianarab Sh., Kulkarni S. R., Raina R. K.

Journal of Mathematics and Applications

2010

Vol. 32

53-61

A certain class of functions analytic in the open unit disk and defined in terms of hypergeometric functions is introduced and investigated. We establish starlikeness, convexity and spirallikeness properties for this class of functions. Special cases and some useful consequences of our main results are aslo mentioned.

Properties of a class of analytic functions involving a linear operator

Patel J., Sahoo P.

Demonstratio Mathematica

2002

Vol. 35, nr 3

495-507

Generating relations involving hypergeometric functions by means of integral operators

Pathan M., Khan S.

Demonstratio Mathematica

2002

Vol. 35, nr 1

35-39

In this paper, the focus is on the results which involve exponential functions. The results of Pathan and Yasmeen [6] and Exton [3] are used with a view to obtaining generating functions which are partly unilateral and partly bilateral.