Partition Method for a Power Series Expansion

This book explores how the method known as the partition method for a power series expansion, which has been developed by the author, can be applied to a host of previously intractable problems in mathematics and physics. In particular, this book describes how the method can be used to determine the Bernoulli, cosecant and reciprocal logarithm numbers, which appear as the coefficients of the resulting power series expansions and then extends the method to more complicated situations where the coefficients become polynomials or mathematical functions. From these examples a general theory for the method is presented, which enables a programming methodology to be established. Because the coefficients in the power series expansions require the compositions of all the partitions that sum to their order, this book also presents the bivariate recursive central partition (BRCP) algorithm, which is able to process the partitions more efficiently via a tree approach as opposed to standard lexicographic methods. Another advantage of the algorithm is its ability to solve diverse problems in the theory of partitions with minor modification. Finally, the programming techniques of previous chapters are used to derive power series expansions for complex generating functions arising in the theory of partitions and in lattice models of statistical mechanics. Explains the partition method by presenting elementary applications involving the Bernoulli, cosecant and reciprocal logarithm numbersCompares generating partitions via the BRCP algorithm with the standard lexicographic approachesDescribes how to programme the partition method for a power series expansion and BRCP algorithm