General Properties of Generalized Gamma-Functions

Ніна Опанасівна Вірченко

Abstract


Background. The article is dedicated to studies of the main properties of new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions for their best applications in applied sciences, for calculations of integrals which are absent in scientific literature.

Objective. Introduction and study of the basic properties of the new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions and their applications.

Methods. We apply the following methods: the methods of the theory of functions of the real variable, the theory of the special functions, the theory of the mathematical physics, the methods of applied analysis.

Results. Some new forms of generalized gamma-functions, incomplete gamma-functions, digamma-functions are introduced. The main properties of these generalized special functions are explored. Examples of application of new generalized gamma-functions are given.

Conclusions. With the help of the r-generalized confluent hypergeometric functions the new generalization of gamma-functions, incomplete gamma-functions, digamma-functions are introduced. The main properties of the new generalized special functions are explored, examples of application of these functions are given.

Keywords


Generalized gamma-functions; Incomplete gamma-functions; Digamma-functions

References


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A.R. Miller, “Reduction of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals”, Rocky Mountain J. Math., vol. 30, no. 2, pp. 703–714, 2000.

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GOST Style Citations


  1. Alzer H. Some inequalities for the incomplete gamma function // Math. Comp. – 1997. – 66. – P. 771–778.

  2. Boudjelkha M., Chaudhry M.A. Asymptotic expansion of the generalized incomplete gamma functions with application to heat conduction problems // J. оf Math. Analysis and Applications. – 2000. – 248. – P. 509–519.

  3. Chaudhry M.A., Zubair S.M. Generalized incomplete gamma function with application // J. Comp. Appl. Math. – 1994. – 55. – P. 99–124.

  4. Miller A.R. Reduction of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals // Rocky Mountain J. Math. – 2000. – 30, № 2. – P. 703–714.

  5. Paris R.B., Wood A.D. Exponentially – improved asymptotic for the gamma function // J. Comp. Appl. Math. – 1992. – 41. – P. 135–143.

  6. Prym F.E. Zur Theorie der Gamma Function // J. Reine Angew. Math. – 1877. – 82. – P. 165–172.

  7. Tricomi F.G., Erdelyi A. The asymptotic expansion of a ratio of gamma functions // Pacific J. Math. – 1951. – 1. – P. 133–142.

  8. Zubair S.M., Chaudhry M.A. Temperature solutions due to gamma – type heat sources in a finite medium // ASME – HTD. – 1992. – 207. – P. 63–68.

  9. Kobayashi K. On generalized gamma functions occurring in diffraction theory // J. Phys. Soc. Japan. – 1991. – 60. – P. 1501–1512.

  10. Bateman H., Erdelyi A. Higher transcendental functions. Vol. 1. – New York: McGraw-Hill Book Company, 1953. – 296 р.

  11. Virchenko N.O. On some generalizations of gamma functions // Доп. АН України. – 1999. – № 10. – C. 39–44.

  12. Wright E.M. On the coefficient of power series having exponential singularities // J. London Math. Soc. – 1993. – 8. – P. 71–79.




DOI: https://doi.org/10.20535/1810-0546.2016.4.62211

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