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P. Flajolet and R. Sedgewick, “Analytic Combinatorics, Chapter V: Application of Rational and Meromorphic Asymptotics, Chapter VIII: Saddle Point Asymptotics,” Cambridge University Press, Cambridge, 2009.

has been cited by the following article:

• TITLE: On Continuous Limiting Behaviour for the q(n)-Binomial Distribution with q(n)→1 as n→∞

  AUTHORS: Malvina Vamvakari

  KEYWORDS: Stirling Formula; q(n) -Factorial Number of Order n; Saddle Point Method; q(n)-Binomial Distribution; Pointwise Convergence; Gauss Distribution

  JOURNAL NAME: Applied Mathematics, Vol.3 No.12A, December 31, 2012

  ABSTRACT: Recently, Kyriakoussis and Vamvakari [1] have established a q-analogue of the Stirling type for q-constant which have lead them to the proof of the pointwise convergence of the q-binomial distribution to a Stieltjes-Wigert continuous distribution. In the present article, assuming a sequence q(n) of n with q(n)→1 as n→∞, the study of the affect of this assumption to the q(n)-analogue of the Stirling type and to the asymptotic behaviour of the q(n)-Binomial distribution is presented. Specifically, a q(n) analogue of the Stirling type is provided which leads to the proof of deformed Gaussian limiting behaviour for the q(n)-Binomial distribution. Further, figures using the program MAPLE are presented, indicating the accuracy of the established distribution convergence even for moderate values of n.