Representing the Stirling polynomials σn(x) in dependence of n and an application to polynomial zero identities

Abstract

Denote by σn{\sigma }_{n} the n-th Stirling polynomial in the sense of the well-known book Concrete Mathematics by Graham, Knuth and Patashnik. We show that there exist developments xσn(x)=∑j=0n(2jj!)−1qn−j(j)xjx{\sigma }_{n}\left(x)={\sum }_{j=0}^{n}{\left({2}^{j}j\!)}^{-1}{q}_{n-j}\left(j){x}^{j} with polynomials qj{q}_{j} of degree j.j. We deduce from this the polynomial identities ∑a+b+c+d=n(−1)d(x−2a−2b)3n−s−a−ca!b!c!d!(3n−s−a−c)!=0,fors∈Z≥1,\sum _{a+b+c+d=n}{\left(-1)}^{d}\frac{{\left(x-2a-2b)}^{3n-s-a-c}}{a\!b\!c\!d\!\left(3n-s-a-c)\!}=0,\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}s\in {{\mathbb{Z}}}_{\ge 1}, found in an attempt to find a simpler formula for the density function in a five-dimensional random flight problem. We point out a probable connection to Riordan arrays.

Keywords

• stirling polynomials
• polynomial identities
• difference equations
• random flights
• riordan arrays
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• 11b37
• 11b73
• 39a06