Chapter 1 - Introduction

Abstract

Chapter 1 begins by discussing the origin of the partition method for a power series expansion. Then the reader is introduced to the method as it is applied to the basic transcendental functions of cosecant, secant and the reciprocal of the logarithmic function, ln⁡(1+z). The coefficients of the resulting power series expansions obtained from the two trigonometric functions are shown to be related to the famous Bernoulli and Euler numbers. All three cases yield resulting power series expansions, whose coefficients are not only rational but, unlike the Bernoulli and Eulers numbers, converge to zero for higher orders with those for the cosecant expansion converging the fastest. The chapter concludes by explaining how the method can be adapted to obtain the Bernoulli and Euler polynomials from their generating functions and relates them, respectively, to the cosecant and secant polynomials, which represent the polynomials in the generating functions of xcos⁡(xt)/sin⁡x and sin⁡(xt)/xcos⁡x. The latter polynomials are also found to possess numerous and interesting properties.