Boyadzhiev Khristo N - Notes On The Binomial Transform: Theory And Table With Appendix On Stirling Transform

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office : 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. NOTES ON THE BINOMIAL TRANSFORM
Theory and Table with Appendix on Stirling Transform Copyright 2018 by World Scientific Publishing Co. Ltd. Ltd.

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http://www.worldscientifc.com/worldscibooks/10.1142/10848#t=suppl Printed in Singapore This book is dedicated to my wife Irina and daughters Marinella and Alexandra. Preface The binomial transform is a discrete transformation of one sequence into another with many interesting applications in combinatorics and analysis.

The binomial transform theory belongs to the expanding field of discrete mathematics. The material of this book will be helpful to researchers interested in enumerative combinatorics, special numbers, and classical analysis. The book can also serve as a valuable reference. It can be used as lecture notes for a course on binomial identities, binomial transforms and Euler series transformations. The binomial transform leads to various combinatorial and analytical identities involving binomial coefficients. Many interesting identities can be written as binomial transforms and vice versa.

The book 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 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. Contents 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. Contents 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 {ak}, k = 0,1,2, its binomial transform is the new sequence {bn}, 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 {bk} is defined by and the inversion formula is Here the factor (1)k can be replaced by (1)k1. A simple computation shows that (1.3) implies the formula If the sequence {ak} 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 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 defined by ak = ak+1ak. 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 []).

Here is the simple proof