On Asymptotics for the Signless Noncentral q‐Stirling Numbers of the First Kind

Abstract

In this paper, we derive asymptotic formulas for the signless noncentral q‐Stirling numbers of the first kind and for the corresponding series. The signless noncentral q‐Stirling numbers of the first kind appear as coefficients of a polynomial of q‐number [t]q, expressing the noncentral ascending q‐factorial of t of order m and noncentrality parameter k. In this paper, we have two main purposes. The first is to give an expression by which we obtain the asymptotic behavior of these coefficients, using the saddle point method. The second main purpose is to derive an asymptotic expression for the signless noncentral q‐Stirling of the first kind series by using the singularity analysis method. We then apply our first formula to provide asymptotic expressions for probability functions of the number of successes in m trials and of the number of trials until the occurrence of the nth success in sequences of Bernoulli trials with varying success probability which are both written in terms of the signless noncentral q‐Stirling numbers of the first kind. In addition, we present some numerical calculations using the computer program MAPLE indicating that our expressions are close to the actual values of the signless noncentral q‐Stirling numbers of the first kind and of the corresponding series even for moderate values of m.