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Upper Bounds and Asymptotics for the q ‐Binomial Coefficients
Author(s) -
Kirousis Lefteris M.,
Stamatiou Yannis C.,
Vamvakari Malvina
Publication year - 2001
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.1071177
Subject(s) - mathematics , upper and lower bounds , binomial coefficient , saddle point , expression (computer science) , binomial theorem , binomial (polynomial) , gaussian binomial coefficient , generalization , exponential function , saddle , gaussian , combinatorics , mathematical analysis , negative binomial distribution , poisson distribution , statistics , geometry , mathematical optimization , physics , quantum mechanics , computer science , programming language
In this article, we a derive an upper bound and an asymptotic formula for the q ‐binomial, or Gaussian, coefficients. The q ‐binomial coefficients, that are defined by the expression are a generalization of the binomial coefficients, to which they reduce as q tends toward 1. In this article, we give an expression that captures the asymptotic behavior of these coefficients using the saddle point method and compare it with an upper bound for them that we derive using elementary means. We then consider as a case study the case q =1+ z / m , z <0, that was actually encountered by the authors before in an application stemming from probability and complexity theory. We show that, in this case, the asymptotic expression and the expression for the upper bound differ only in a polynomial factor; whereas, the exponential factors are the same for both expressions. In addition, we present some numerical calculations using MAPLE (a computer program for performing symbolic and numerical computations), that show that both expressions are close to the actual value of the coefficients, even for moderate values of m .