Binomial sums and properties of the Bernoulli transform

TL;DR

This paper investigates binomial sums involving various sequences, expressing them in terms of powers of q, and explores their properties, relations, and interpretations, including connections to special polynomials and identities.

Contribution

It introduces explicit formulas and properties for binomial sums with different sequences, extending understanding of their structure and relations to special functions and polynomials.

Findings

Explicit expressions for binomial sums with Fibonacci, Laguerre, Meixner polynomials, and binomial coefficients.

Properties, relations, and probabilistic interpretations of these sums.

Connections to Appell polynomials and generating functions.

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.