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 ${\left[n\right]}_p\mathrm{.}$ We establish later some properties, relations, probabilistic interpretations and generating functions between $S_n(q)$ and $S_n(x+q-xq)\mathrm{.}$ Further identities related to Appell polynomials are also given in the last of the paper.