Binomial sums and properties of the Bernoulli transform

Abstract

In this paper, we study the binomial sum $S_{n}(q):=% \overset{n}{\underset{k=0}{\sum }}a_{k}\binom{n}{k}\left( 1-q\right) ^{k}q^{n-k}$ for a given sequence $\left( a_{n}\right) $ of real or complex numbers. We express $S_{n}(q)$ in function of the powers of $q,$ and, we explicit it when the sequence $\left( a_{n}\right) $ is the sequence of Fibonacci numbers, Laguerre polynomials, Meixner polynomials, binomial coefficients and the sequence $\left[ n\right] _{p}.$ We establish later some properties, relations, probabilistic interpretations and generating functions between $S_{n}(q)$ and $S_{n}(x+q-xq).$ Further identities related to Appell polynomials are also given in the last of the paper.