Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE q-DEFORMED GAMMA FUNCTION AND q-DEFORMED POLYGAMMA FUNCTION

  • Received : 2013.09.19
  • Published : 2014.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we rederive the identity ${\Gamma}_q(x){\Gamma}_q(1-x)={\frac{{\pi}_q}{sin_q({\pi}_qx)}$. Then, we give q-analogue of Gauss' multiplication formula and study representation of q-oscillator algebra in terms of the q-factorial polynomials.

Keywords

References

  1. M. Arik and D. Coon, Hilbert spaces of analytic functions and generalized coherent states, J. Mathematical Phys. 17 (1976), no. 4, 524-527. https://doi.org/10.1063/1.522937
  2. R. Askey, The q-gamma and q-beta functions, Appl. Anal. 8 (1978), no. 2, 125-141. https://doi.org/10.1080/00036817808839221
  3. N. M. Atakishiyev, A. Frank, and K. B. Wolf, A simple difference realization of the Heisenberg q-algebra, J. Math. Phys. 35 (1994), no. 7, 3253-3260. https://doi.org/10.1063/1.530464
  4. R. W. Gosper, Experiments and Discoveries in q-Trigonometry, In Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, Proceedings of the Conference Held at the University of Florida, Gainesville, FL, 1999.
  5. F. H. Jackson, A generalization of the functions ${\Gamma}(n)\;and\;x^n$, Proc. Roy. Soc. London. 74 (1904), 64-72. https://doi.org/10.1098/rspl.1904.0082
  6. F. H. Jackson, The basic gamma function and the elliptic functions, Proc. Roy. Soc. London. A 76 (1905), 127-144. https://doi.org/10.1098/rspa.1905.0011
  7. T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288-299.
  8. T. Kim and C. Adiga, On the q-analogue of gamma functions and related inequalities, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 118, 4 pp.
  9. A. Macfarlane, On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q, J. Phys. A 22 (1989), no. 21, 4581-4588. https://doi.org/10.1088/0305-4470/22/21/020
  10. M. Mansour, An asymptotic expansion of the q-gamma function ${\Gamma}_q(x)$, J. Nonlinear Math. Phys. 13 (2006), no. 4, 479-483. https://doi.org/10.2991/jnmp.2006.13.4.2
  11. T. Mansour and A. Shabani, Some inequalities for the q-digamma function, J. Inequal. Pure Appl. Math. 10 (2009), no. 1, Article 12, 8 pp.
  12. S.-H. Rim and T. Kim, A note on the q-analogue of p-adic log-gamma function, Adv. Stud. Contemp. Math. 18 (2009), no. 2, 245-248.
  13. A. Salem, An infinite class of completely monotonic functions involving the q-gamma function, J. Math. Anal. Appl. 406 (2013), no. 2, 392-399. https://doi.org/10.1016/j.jmaa.2013.04.059
  14. A. Salem, A completely monotonic function involving q-gamma and q-digamma functions, J. Approx. Theory 164 (2012), no. 7, 971-980. https://doi.org/10.1016/j.jat.2012.03.014
  15. W. T. Sulaiman, Some inequalities for the q-digamma functions, J. Concr. Appl. Math. 10 (2012), no. 3-4, 301-308.
  16. N. J. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2010.
  17. J. Thomae, Beitrage zur Theorie der durch die Heinesche Reihe, J. Reine Angew. Math. 70 (1869), 258-281.

Cited by

  1. Inequalities for q-gamma function ratios pp.1664-235X, 2019, https://doi.org/10.1007/s13324-017-0198-0
  2. Further extended Caputo fractional derivative operator and its applications vol.24, pp.4, 2017, https://doi.org/10.1134/S106192081704001X