SOME REPRESENTATIONS OF THE GAMMA FUNCTION

Pozhidaev Aleksandr Vasil'evich, Doctor of physical and mathematical sciences, professor, head of sub-department of higher mathematics, Siberia State University of Railway Transportation (191 D. Kovalchuk street, Novosibirsk, Russia), math@stu.ru
Pekel'nik Natal'ya Mikhaylovna, Candidate of pedagogical sciences, associate professor, sub-department of higher mathematics, Siberia State University of Railway Transportation (191 D. Kovalchuk street, Novosibirsk, Russia), pekelniknm@mail.ru
Khaustova Olesya Igorevna, Candidate of pedagogical sciences, associate professor, sub-department of higher mathematics, Siberia State University of Railway Transportation (191 D. Kovalchuk street, Novosibirsk, Russia), lex711@yandex.ru
Trefilova Irina Aleksandrovna, Lecturer, sub-department of higher mathematics, Siberia State University of Railway Transportation (191 D. Kovalchuk street, Novosibirsk, Russia), koja@mail.ru

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Background. One of the most important functions, expressed by an improper integral containing a parameter, is the gamma function. It occurs naturally in many areas of modern mathematics and applications. The special role of this function in mathematical analysis is that some important definite integrals, infinite series and infinite products are expressed through it. In recent years, the efforts of many authors have been aimed at getting different estimates of this function. The purpose of this paper is to make one of the possible decompositions of the gamma function into an infinite product and the analysis of this representation.
Materials and methods. The authors used suitable integral representations of functions, various properties of convergent improper integrals with a parameter and their behavior in the limit. Herewith, the mathedo of mathematical induction was applied.
Results and conclusions. The researchers have obtained some representation of the gamma function as an infinite product at some point. The analysis of the obtained results allowed to establish a connection between the gamma function and the Poisson distribution.

gamma function, Euler's constant, infinite product, differentiation of an improper integral by a parameter, two-sided estimates of the gamma function, behavior of an integral in the limit.

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