ASYMPTOTIC PROPERTIES OF MEIJER`S G-FUNCTION WITH TWO IMAGINARY PARAMETERS

UDC 517.588

  • Yarotskaya Lyudmila Dmitrievna − PhD (Physics and Mathematics), Assistant Professor, the Department of Higher Mathematics. Belarusian State Technological University (13a, Sverdlova str., 220006, Minsk, Republic of Belarus). E-mail: yarockaya@belstu.by

DOI: https://doi.org/10.52065/2520-6141-2022-260-2-14-20

Key words: asymptotic expansion, Meijer`s G-function, index transform, Bessel functions, Stirling formula, Euler Gamma-function.

For citation: Yarotskaya L. D. Asymptotic properties of Meijer`s G-function with two imaginary parameters. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2022, no. 2 (260), pp. 14–20 (In Russian). DOI: https://doi.org/10.52065/2520-6141-2022-260-2-14-20.

Abstract

The problem of asymptotic expansions of special functions by their indices or parameters arises in connection with the investigation of some classes of integrals and index transforms. The Meyer G-function is the most common function of the hypergeometric type. The G-function is important in applied mathematics due to the ability to express through the G-symbol a large number of special functions and their combinations.

This paper deals with some asymptotic properties of the Meyer G-function of a special kind with two imaginary parameters, which are large enough by their absolute values. It is shown that particular cases of the considered function are the kernels of known integral transformations by index – the transformations of Kontorovich – Lebedev, Mehler – Fock, Olevsky, Lebedev and others. The representation of the G-function in the form of a linear combination of generalized hypergeometric series with power multipliers is based on the Slater's theorem.

The Mellin transforms of functions of the hypergeometric type are the ratio of the products of the Euler gamma functions whose asymptotics are known in accordance to the Stirling formula. We give the Stirling formula for the Euler gamma function of a complex argument, for which the imaginary part is unbounded and the real part is fixed. Asymptotic estimates for the Meijer`s G-function of a special form with respect to large values of the parameter are established. It is shown that such expansion includes, as special cases, earlier known representations Bessel functions and functions connected with them.

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27.04.2022