Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals

Abstract

This article is a short elementary review of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned. At the end of the paper one application is described in details -certain Fourier integrals involving and are evaluated in terms of Stirling numbers.

Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

“…The Pochhammer symbol can be rewritten using the Stirling numbers of the first kind (2) where we employed (17). Substituting this into the line above, we have proven (5). Hence all of our statements in Theorem 1.1 are proven.…”

“…A survey and some historical remarks related to these notions can be found in the paper [5] and references therein. The so-called -Stirling numbers of the first kind are determined as the coefficients of…”

On Multi-Order Logarithmic Polynomials and their Explicit Formulas, Recurrence Relations, and Inequalities

“…When m = 1 and x 1 = 1, the quantities Q 1,n (1) = B n were called the Bell numbers [1,5,8,17,18] or exponential numbers [2] and were generalized and applied [1]. When m = 1 and x 1 = x is a variable, the quantities Q 1,n (x) = B n (x) = T n (x) were called the Bell polynomials [20,21], the Touchard polynomials [19,22], or exponential polynomials [3,4,7] and were applied [9,10,11,12,19]. In the paper [22], explicit formulas, recurrence relations, determinantal inequalities, product inequalities, logarithmic convexity, logarithmic concavity, and applications of Q m,n (x) were investigated.…”

Lah numbers, Laguerre polynomials of order negative one, and the nth derivative of exp(1/x)

“…This will reveal interesting connections of Lah numbers to Laguerre polynomials and also to Stirling numbers. In Section 4 we present a new formula for D n e c x p in terms of the exponential polynomials ϕ n (x) considered in [4] and [5].…”