Binomial sums and properties of the Bernoulli transform

In this paper, we study the binomial sum $S_{n}(q):=$ 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.