Números de Stirling do Primeiro Tipo e as Relações de Girard

ABSTRACT

The presented article discusses the Stirling numbers of the first kind, which is a special case of the study of polynomials on various indeterminates over the integers ring and has relations between the coefficients and the respective roots of a given algebraic equation. The idea consists of an expansion of a class of polynomials on the indeterminates x, x ₁ , x ₂ ,··· , x _n ∈ ℤ, defined by $p_n\left(x\right)=\prod _{j=1}^n(x-x_j)$, given a fixed positive integer n. The idea is even more special, because it comes from the Vieta’s formula of the study of homogeneous and symmetric polynomials, which consists of studying the polynomials in 𝔸(x), where the coefficients are on the ring 𝔸 = ℤ(x ₁ , x ₂ , · · · , x _n ), moreover the particular integer roots in Vieta’s formula, discussed here, are x ₁ = 0, x ₂ = −1,··· , x _n = −(n − 1) making interesting algebraic identities whose combinatorial nature is evident and the coefficients of the powers of x in p _n (x), in this case, can be the answer to various counting problems modeled after this generating function, more specifically, the sequence associated to p _n (x) generates the Stirling numbers of the first kind.

Keywords:
polynomials; Vieta’s formula; Stirling numbers