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

Abstract

Denote by σn 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 σn (x) = σj = 0n (2jj!)-1 qn - j (j) xj with polynomials qj of degree j. We deduce from this the polynomial identities σ/a + b + c + d = n (-1)d (x - 2 a - 2 b)3n-s-a-c/a!b!c!d! (3n - s - a - c)! = 0, for s ϵ ℤ≥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.