On the link between Binomial Theorem and Discrete Convolution of Power Function

Petro Kolosov (1)

1 Odessa National I.I.Mechnikov University

In this manuscript we introduce and discuss the $2m+1$-degree integer valued polynomials $\mathbf{P}^{m}_{b}(n)$. These polynomials are in strong relation with discrete convolution of power function. It is also shown that odd binomial expansion is partial case of $\mathbf{P}^{m}_{b}(n)$. Basis on above, we show the relation between Binomial theorem and discrete convolution of power function.

Mots clés: Binomial theorem; Discrete convolution; Power function; Polynomials; Convolution; Multinomial theorem; Binomial coefficients; Bernoulli numbers; Pascal's triangle; Faulhaber's formula

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]; Mathématiques générales [math.GM]

Petro Kolosov. On the link between Binomial Theorem and Discrete Convolution of Power Function. 2018. ⟨hal-01283042v14⟩