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Binomial sums and properties of the Bernoulli transform

Laid Elkhiri, Miloud Mihoubi, Meriem Moulay

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.

Binomial sums and properties of the Bernoulli transform

Abstract

In this paper, we study the binomial sum for a given sequence of real or complex numbers. We express in function of the powers of and, we explicit it when the sequence is the sequence of Fibonacci numbers, Laguerre Polynomials, Meixner Polynomials, binomial coefficients and the sequence We establish later some properties, relations, probabilistic interpretations and generating functions between and Further identities related to Appell polynomials are also given in the last of the paper.
Paper Structure (4 sections, 16 theorems, 112 equations)

This paper contains 4 sections, 16 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. A principal elementary identity and its applications
  3. Some properties of the sequences $\left( S_{n}(q)\right)$
  4. Applications to Appell polynomials

Key Result

Proposition 1

For any real sequence $\left( a_{n};n\geq 0\right)$ there hold where

Theorems & Definitions (22)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • Remark 3
  • ...and 12 more