Series expansion of the Gamma function and its reciprocal

Ioana Petkova
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 104—115
DOI: 10.7546/nntdm.2021.27.4.104-115
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Authors and affiliations

Ioana Petkova
Faculty of Mathematics and Informatics, Sofia University
Sofia, Bulgaria


In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2):

    \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*}

Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.


  • Gamma function
  • Zeta function
  • Euler’s constant
  • Maclaurin series

2020 Mathematics Subject Classification

  • 33B15
  • 11M06
  • 30D10
  • 30D30
  • 30K05
  • 41A58


  1. Abramowitz, M. & Stegun, I. A. (1972). Handbook of Mathematical Functions, Tenth Printing, Washington.
  2. Choi, J., & Srivastava, H. M. (2000). Evaluation of higher-order derivatives of the Gamma function. University of Belgrade Publikacije Elektrotehnickog Fakulteta, Ser. Mat. 11 (2000), 9–18.
  3. Fekih-Ahmed, L. (2014). On the Power Series Expansion of the Reciprocal Gamma Function. Preprint. Retrieved from:, 9–18.
  4. Kono, K. (2017). Series Expansion of Gamma Function & the Reciprocal. Retrieved from:, Formula 12.1.0 (Masayuki Ui).
  5. Luke, Y. L. (1980). Special Mathematical Functions and Their Approximations, Mir Publishing House, Moscow (in Russian).
  6. Nielsen, N. (1906). Handbuch der theorie der gammafunktion, Teubner, Leipzig, 40–43.
  7. Olver, F. W. J. (1997). Asymptotics and Special Functions, A K Peters, Wellesley, Massachusetts.
  8. Sakata, M. (2017). Taylor series for the reciprocal Gamma function and multiple zeta values Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, 93, 47.
  9. Wikipedia contributors. (2021, November 30). Gamma function. Wikipedia, The Free Encyclopedia. Retrieved from:

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Cite this paper

Petkova, I. (2021). Series expansion of the Gamma function and its reciprocal. Notes on Number Theory and Discrete Mathematics, 27(4), 104-115, doi: 10.7546/nntdm.2021.27.4.104-115.

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