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On the functional equations of the q-Gamma function

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Abstract

In this paper, we present some characterizations of the q-gamma function by some functional equations using E. Artin’s technique.

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References

  1. Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press Inc., New York (1966)

  2. Aczél, J., Dhombres, J.: Functional equations in several variables, vol. 31 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1989)

  3. Alzer H., Grinsphan A.Z.: Inequalities for the gamma and q-gamma functions, J. Approx. Theory 144, 67–83 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Artin, E.: The Gamma function, translated by M. Butler, Holt, Rinehart and Winston, New York (1964)

  5. Brillouët-Belluot, N.: Problem 15, Report of meeting. In: The Thirty-eighth International Symposium on Functionl Equations (Noszvaj, 2000). Aequationes Math. 61, p. 304 (2001)

  6. Castillo, E., Ruiz Cobo, M.R.: Functional Equations and Modelling in Science and Engineering. Marcel Dekker, New York (1992)

  7. Chung J.Y.: Stability of a conditional Cauchy equation. Aequationes Math. 83(3), 313–320 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge (2002)

  9. Davis, P.J.: The Schwarz function and its applications. The Carus Mathematical Monographs, no. 17, The Mathematical Association of America, Washington, DC (1974)

  10. Eichhorn, W.: Functional Equations in Economics, Applied Mathematics and Computation, vol. 11. Addison-Wesley Publishing Co., Reading (1978)

  11. Gao P.: Some completely monotonic functions involving the q-gamma function. Math. Inequal. Appl. 17(2), 451–460 (2014)

    MATH  MathSciNet  Google Scholar 

  12. Gao, P.: Some monotonicity properties of gamma and q-gamma functions. ISRN Math. Anal. 2011, 15 (2011). Art. ID:375715

  13. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)

  14. Gong, X.: Convexity of solutions for an iterative equation in Banach spaces. Abst. Appl. Anal. 2013, 8 (2013). Art. ID:164851

  15. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of functional equations in several variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1998)

  16. Ismail M.E.H., Muldoon M.E.: Higher monotonicity properties of q-gamma and q−psi functions. Adv. Dyn. Syst. Appl. 8(2), 247–259 (2013)

    MathSciNet  Google Scholar 

  17. Jackson F.H.: On q−definite integrals. Quart. J. Pure Appl. Math. 41, 193–203 (1910)

    MATH  Google Scholar 

  18. Járai, A.: Regularity properties of functional equations in several variables. In: Advances in Mathematics (Springer), vol. 8. Springer, New York (2005)

  19. Jun K.W., Kim G.H., Lee Y.W.: Stability of generalized gamma and beta functional equations. Aequat. Math. 60, 15–24 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jun K.W., Kim H.M., Chang I.S.: On the Hyers–Ulam stability of an Euler-Lagrange type cubic functional equation. J. Comput. Anal. Appl. 7, 21–33 (2005)

    MATH  MathSciNet  Google Scholar 

  21. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

  22. Kannappan, P.L.: Functional Equations and Inequalities with Applications. Springer Monographs in Mathematics. Springer, Dordrecht, Heidelberg, London, New York (2009)

  23. Kim G.H.: On the stability of generalized Gamma functional equation. Int. J. Math. Math. Sci. 23, 513–520 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q−analogue. Technical University Delft, Report 17 (1998)

  25. Kuczma, M., Choczewski, B., Ger, R.: Iterative fnctional equations, vol. 32 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1990)

  26. Moak D.S.: The q-gamma function for q > 1. Aequat. Math. 20, 278–285 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  27. Qi F.: A completely monotonic function related to the q−trigamma function. Politehn. Univ. Bucharest Sci. Bull. Ser. A—Appl. Math. Phys. 76(1), 107–114 (2014)

    MathSciNet  Google Scholar 

  28. Rahbarnia F., Rassias Th.M., Saadati R., Sadeghi Gh.: Forti’s approach in fixed point theory and the stability of a functional equation on metric and ultra metric spaces. J. Comput. Anal. Appl. 13(3), 458–462 (2011)

    MATH  MathSciNet  Google Scholar 

  29. Rassias, Th.M.: Functional equations and inequalities, vol. 518 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (2000)

  30. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1978)

  31. Xu T.Z., Rassias J.M., Xu W.X.: A generalized mixed additive-cubic functional equation. J. Comput. Anal. Appl. 13(7), 1273–1282 (2011)

    MATH  MathSciNet  Google Scholar 

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Correspondence to Mansour Mahmoud.

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Mahmoud, M. On the functional equations of the q-Gamma function. Aequat. Math. 89, 1041–1050 (2015). https://doi.org/10.1007/s00010-014-0291-5

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  • DOI: https://doi.org/10.1007/s00010-014-0291-5

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