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Review - Computational Techniques for the Summation of Series
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Review of the book

Computational Techniques for the Summation of Series


Anthony Sofo

Kluwer Academic/Plenum Publishers, New York, 2003

ISBN 0-306-47805-6.

At about the time when one of the greatest adventurers of all time, Captain James Cook (1728-1779), finished his exploits on the east coast of Australia (New Holland), one of the greatest mathematician, Carl Friedrich Gauss (1777-1855) was born in Braunschweig, Germany.

Legend has it that at a very early age Gauss, and his fellow pupils, were given the task of adding the first one hundred natural numbers. Gauss's ingenuity was evident even at this tender age, he noticed there were fifty groups of numbers each summing to one hundred and one, giving a grand total of five thousand and fifty. In a very elementary way the author's book is concerned with exactly this problem of adding together a finite or infinite number of terms and being able to express the result in closed form. It is an old an important problem, one that has been tackled by many mathematicians including Euler, Ramanujan, Jensen and more recently Borwein and Petkovsek, Wilf and Zeilberger.

The author uses function theoretic methods in his investigations and the generation of the series is achieved through the consideration of differential difference equations and generalized difference delay equations. The book is well planned and written in a style that is pleasing to read. It has some deep technical results as well as a smattering of historical connections. The conjunction of technical results and historical connections is indeed appreciated.

An outline of the chapters is now given. The first chapter is an introductory one in which some results and methods for closed form summation are given. This chapter prepares the reader with a few types of techniques, in particular residue theory, hypergeometric summation and contour integration, that will be useful in the following chapters. Chapters two to five deal with non-hypergeometric summation, or the summation of Abel type series. The author develops techniques for the summation of

$\displaystyle S(a,b,k,m,R,t)=\sum_{n\geq 0}\left( \begin{array}{c} n+R-1 \ n \end{array} \right) \frac{(b^{k}e^{ab})^{n}(t-an)^{nk+Rk+m-1}}{(nk+Rk+m-1)!}$    

and gives convergence criteria for $ S(a,b,k,m,R,t)$ to be expressed in closed form.

I particularly like the connection of $ S(a,b,1,0,1,t)$ with a result given independently, and using entirely different methods than those developed by the author, by Euler in 1779, Jensen in 1902, and questioned by Ramanujan in about 1914. Moreover the author demonstrates a practical connection of $ S(a,b,1,0,1,t)$ with Neutron behaviour, Grazing systems and Renewal problems.

Chapters six to eight deal with hypergeometric summation, that is, series in which subsequent terms can be put in the form

$\displaystyle \frac{\left( a_{1}+n\right) \cdots \left( a_{p}+n\right) z}{\left( n+1\right) \left( b_{1}+n\right) \cdots \left( b_{q}+n\right) }.$    
Beginning with arbitrary order difference-delay systems, and using residue theory, the author develops both finite and infinite sums of the form
$\displaystyle T(a,b,c,k,m,R,t)=\sum_{r\geq 0}\left( \QATOP{r+R-1}{r}\right) \left( \QATOP{ t-akr}{rk+Rk+m-1}\right) \left( \frac{b}{c}\right) ^{t-akr-Rk-m+1}.$    
In the finite case the author obtains closed form expressions for
$\displaystyle \sum_{r=0}^{n}\left( \QATOP{r+R-1}{r}\right) \left( \QATOP{na+n+R-ar-1}{R+r-1}\right) b^{na+n-ar}$    
which in some cases can be expressed in trigonometric form, such as
$\displaystyle \sum_{r=1}^{n}r\left( \QATOP{2n-r}{r}\right) \left( -\frac{1}{4}\right) ^{r}$ $\displaystyle =-\dprod\limits_{j=1}^{n}\sin ^{2}\left( \frac{\pi j}{2n+1}\right) \sum_{k=1}^{n}\cot ^{2}\left( \frac{\pi k}{2n+1}\right)$    
  $\displaystyle =\frac{n\left( 4n^{2}-1\right) }{3.4^{n}}.$    
The methods used for the evaluation of the infinite binomial type series mirror the methods that the author developed in the first five chapters. In this sense the author has indeed achieved the main aim of his book as stated in the preface "to present a unified treatment of summation of sums and series using function theoretic methods. The author develops a technique, based on residue theory, that is useful for the summation of both non-hypergeometric and hypergeometric form".

There are a remarkable few, in number, of typographical errors and certainly none that detract from the meticulous presentation of the theorems and proofs. Table 8.3, on page 164 should read "The constant lambda of (8.30)". The Lambda, $ \lambda _{R,j}$ values in Table 8.3, which have been developed from a Gauss hypergeometric function, maybe represented in the double binomial form

$\displaystyle \lambda _{R,j}=\left( \begin{array}{c} R-1 \ R-2j-1 \end{array} \right) \left( \begin{array}{c} 2j \ j \end{array} \right) .$    
It is also pleasing to see that the author has spent a considerable amount of space and time to the question of convergence criteria for the generic Abel type and binomial type series.

It was a pleasure reviewing this book and it should be of interest to mathematicians, engineers and to researchers in related fields.

I strongly recommend this book for libraries and to all JIPAM readers.

S.S. Dragomir

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