The Gauss product and Raabe’s integral for k-gamma functions

József Sándor
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310-5132, Online ISSN 2367-8275
Volume 26, 2020, Number 4, Pages 122—127
DOI: 10.7546/nntdm.2020.26.4.122-127
Download full paper: PDF, 156 Kb

Details

Authors and affiliations

József Sándor
Department of Mathematics, Babes-Bolyai University
Cluj-Napoca, Romania

Abstract

We obtain an extension of the famous Gauss product formula to the case of k-gamma functions. The Sándor–Tóth short product formula [16] is also attended to these functions. An asymptotic formula and Raabe’s integral analogue are also considered.

Keywords

  • Arithmetic function
  • Gamma function
  • k-gamma function

2010 Mathematics Subject Classification

  • 11A25
  • 33B15

References

  1. Allouche, J.-P. (2015). Paperfolding infinite products and the gamma function. J. Number Theory, 148, 95-111.
  2. Bachraoui, M. E., & Sándor, J. (2019). On a theta product of Jacobi and its applications to q-gamma products. J. Math. Anal. Appl., 472(1), 814-826.
  3. Ben-Ari, I., Hay, D., & Roitershtein, A. (2014). On Wallis-type products and Pólya’s urn schemes. Amer. Math. Monthly, 121(5), 422-432.
  4. Chamberland, M., & Straub, A. (2013). On gamma quotients and infinite products. Adv. Appl. Math, 51(5), 546-562.
  5. Diaz, R., & Teruel, C. (2005). q, h-generalized gamma and beta functions. J. Nonlinear Math, Phys., 12(1), 118-134.
  6. Diaz, R., & Pariguan, E. (2007). On hypergeormetric functions and k-Pochhamer symbol. Div. Mat., 15(2), 179-192.
  7. Hardy, G. H., & Wright, E. M. (1968). An Introduction to the Theory of Numbers, Clarendon Press, Oxford.
  8. Kokologiannaki, C. G. (2010). Properties and inequalities of generalized k-gamma, beta and zeta functions. Int. J. Contemp. Math. Sci., 5(14), 653-660
  9. Kokologiannaki, C. G., & Krasniqi, V. (2013). Some properties of the k-gamma functions. Le Matematiche (Catania), LXVIII(I), 13-22.
  10. Krasniqi, V. (2010). Inequalities and monotonicity for the ration of k-gamma function. Scientia Magna, 6(1), 40-45.
  11. Mansour, M. (2009). Determining the k-generalized gamma function Γk (x) by functional equations. Int. J Contemp. Math. Sci., 4(21), 1037-1042.
  12. Martin, G. (2009). A product of gamma function values at fractions with the same denominator. Preprint. Available online: http://arxiv.org/abs/0907.4383.
  13. Nijenhuis, A. (2010). Short gamma products with simple values. Amer. Math. Monthly, 117(8), 733-737.
  14. Nimbran, A. S. (2016). Interesting infinite products of rational functions motivated by Euler. Math. Student, 85, 117-133.
  15. Nishizawa, K. (2012). Finite gamma products. In: Akashi, S., et al. (Eds.) Nonlinear Analysis and Optimization, 239-249. Available online: http://www.ybook.co.jp/online2/nao2012matsue.html
  16. Sándor, J., & Tóth, L. (1989). A remark on the gamma function. Elem. Math. (Basel), 44(3), 73-76.

Related papers

Cite this paper

Sándor, J. (2020). The Gauss product and Raabe’s integral for k-gamma functions. Notes on Number Theory and Discrete Mathematics, 26 (4), 122-127, doi: 10.7546/nntdm.2020.26.4.122-127.

Comments are closed.