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

Abstract

Abstract Denote by σ n {\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 x σ n ( x ) = ∑ j = 0 n ( 2 j j ! ) − 1 q n − j ( j ) x j x{\sigma }_{n}\left(x)={\sum }_{j=0}^{n}{\left({2}^{j}j\!)}^{-1}{q}_{n-j}\left(j){x}^{j} with polynomials q j {q}_{j} of degree j . j. We deduce from this the polynomial identities ∑ a + b + c + d = n ( − 1 ) d ( x − 2 a − 2 b ) 3 n − s − a − c a ! b ! c ! d ! ( 3 n − s − a − c ) ! = 0 , for s ∈ 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.