Notes on the Binomial Transform

Theory and Table with Appendix on Stirling Transform

Khristo N Boyadzhiev

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813234994
• Topic: Mathematics
• Subtopic: Discrete Mathematics
• Index: Mathematics

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The binomial transform is a discrete transformation of one sequence into another with many interesting applications in combinatorics and analysis. This volume is helpful to researchers interested in enumerative combinatorics, special numbers, and classical analysis. A valuable reference, it can also be used as lecture notes for a course in binomial identities, binomial transforms and Euler series transformations. The binomial transform leads to various combinatorial and analytical identities involving binomial coefficients. In particular, we present here new binomial identities for Bernoulli, Fibonacci, and harmonic numbers. Many interesting identities can be written as binomial transforms and vice versa.

The volume consists of two parts. In the first part, we present the theory of the binomial transform for sequences with a sufficient prerequisite of classical numbers and polynomials. The first part provides theorems and tools which help to compute binomial transforms of different sequences and also to generate new binomial identities from the old. These theoretical tools (formulas and theorems) can also be used for summation of series and various numerical computations.

In the second part, we have compiled a list of binomial transform formulas for easy reference. In the Appendix, we present the definition of the Stirling sequence transform and a short table of transformation formulas.

--> Contents:

• Theory of the Binomial Transform:
  • Introduction
  • Prerequisite: Special Numbers and Polynomials
  • Euler's Transformation for Series
  • Melzak's Formula and Related Formulas
  • Special Properties. Creating New Identities
  • Binomial Transforms of Products
  • Special Formulas and Power Series with Binomial Sums
• Table of Binomial Transforms:
  • Assorted Binomial Formulas
  • Identities Involving Harmonic Numbers
  • Transforms of Binomial Coefficients
  • Transforms of Special Numbers and Polynomials
  • Transforms of Trigonometric and Hyperbolic Functions and Applications to Some Trigonometric Integrals
  • Transforms of Some Special Functions
• Appendix:
  • The Stirling Transform of Sequences

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--> Readership: Graduate and researchers in the areas of number theory, discrete mathematics, combinatorics, statistics working with applications using the binomial transform. -->
Keywords:Binomial Coefficients;Binomial Identities;Binomial Sums;Binomial Transform;Euler's Series Transformation;Discrete Mathematics;Finite Differences;Stirling Numbers of the First Kind;Stirling Numbers of the Second Kind;Stirling Transform;Special Numbers and Polynomials;Harmonic Numbers;Bernoulli Numbers;Fibonacci Numbers;Melzak's Formula;Exponential Polynomials;Geometric Polynomials;Laguerre Polynomials;Trigonometric IntegralsReview: Key Features:

• This is the first, long-overdue book on the subject. (At present, there are no competing books)
• The book provides interesting new material for researchers in discrete mathematics and will serve as a valuable reference for binomial identities, binomial transform formulas, and Euler series transformations

Part 1

Theory of the Binomial Transform

Chapter 1

Introduction

We assume basic knowledge about the binomial coefficients

and throughout we use the agreement that

if k < 0. Here p

does not need to be an integer.

Now we define the binomial transform. Given a sequence {a_k}, k = 0,1,2…, its binomial transform is the new sequence {b_n}, n = 0,1,2… generated by the formula

with inversion

which can be written also as

The symmetric version of the binomial transform is used very often. In this case the sequence {b_k} is defined by

and the inversion formula is

Here the factor (−1)^k can be replaced by (−1)^k−1.

A simple computation shows that (1.3) implies the formula

If the sequence {a_k} is indexed from k = 1,2,… we can assume that a₀ = 0 in order to use the same formulas. In this case we have also b₀ = 0.

Here is a proof of the inversion formula (1.2).

Here the last equality follows from the convolution identity

where δ_nj is the Kronecker symbol. The inversion formula (1.4) follows from (1.2) as we can write

The binomial transform naturally appears in the theory of finite differences. Let

be a sequence and consider the forward difference operator Δ defined by Δa_k = a_k+1 − a_k. It is easy to compute that

and because of the well-known property

this can be written also as

or, equivalently, as

Another elementary property involves the transform of the shifted sequence. If (1.1) is true then

for n ≥ 1 (see [64]). Here is the simple proof

where for the last sum we have set j = k − 1. Notice that the summation ...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Preface
• Contents
• Part 1. Theory of the Binomial Transform
• Part 2. Table of Binomial Transforms
• Appendix A The Stirling Transform of Sequences
• References
• Index