Comparative study between exponential Boltzmann and Lambert Boltzmann distributions for heat capacity calculation

The Stirling’s estimation to [Formula: see text](N!) is typically introduced to students as a step in the derivation of the statistical expression for the heat capacity. However, naïve application of this estimation leads to wrong conclusions. In this paper, firstly, the heat capacity of some semiconductor compounds was calculated using exponential Boltzmann distribution and compared with experimental data. It has shown a disagreement between experimental results and those calculated. Secondly, by applying the more exact Stirling formula, an analytical formulation of Boltzmann statistics using Lambert W function is shown to be a very good model and proves an excellent agreement between calculated and experimental data for heat capacity over the entire temperature range.

Polychronakos statistics and α-deformed Bose condensation of α-bosons

In this paper, we consider the Polychronakos statistics for [Formula: see text]. We use the Stirling formula for the [Formula: see text]-Gamma function to find the distribution function for the [Formula: see text]-bosons. As application, we discuss the [Formula: see text]-deformed Bose condensation for [Formula: see text]-boson gas.

A generalization of Mortici lemma

The aim of this note is to obtain a generalization of a very simple, elegant but powerful convergence lemma introduced by Mortici [Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comp., 215 (2010), No. 11, 4044–4048; Mortici, C., Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), No. 5, 434–441; Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93 (2009), No. 1, 37–45; Mortici, C., Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 25 (2009), No. 2, 186–191] and exploited by him and other authors in an impressive number of recent and very recent papers devoted to constructing asymptotic expansions, accelerating famous sequences in mathematics, developing approximation formulas for factorials that improve various classical results etc. We illustrate the new result by some important particular cases and also indicate a way for using it in similar contexts.