Chapter 2 - More Advanced Applications

Abstract

Chapter 2 presents more advanced applications of the partition method for a power series expansion than those of the previous chapter. In presenting these applications, the method is both extended and adapted, thereby enabling a general theory to be developed in Chapter 4. The chapter begins by applying the method to the Bell polynomials of the first kind, denoted by Bk(x). From this study a general formula for the Stirling numbers of the second kind S(k,j), which represent the number of ways of partitioning k objects into j subgroups, is derived as a sum over the partitions summing to k. By setting j equal to several values from k down to k−3, explicit formulas are derived for the four highest order Stirling numbers of the second kind. In the following section the cosecant and secant numbers of Chapter 1 are generalized by introducing an arbitrary power of ρ to the trigonometric functions that generate them. As a consequence, the cosecant and secant numbers turn into polynomials of degree k in ρ that reduce to the results of Chapter 1 for ρ=1. In particular, the generalized cosecant numbers for integer values, viz. c2v,k where v is an integer, are shown to be related to the k-th symmetric polynomial s(v,k) obtained by summing the squares of the integers up to v−1. The next example deals with the generalization of the reciprocal logarithm numbers in Chapter 1 whereby a power s appears in the logarithm that generates them. Again, the numbers, which are denoted by Ak(s), are polynomials of degree k in s with many interesting properties. As in the case of the Bell polynomials, general expressions in k are obtained for the four highest order terms in the Ak(s). The section concludes by studying the power series expansion for the doubly-nested logarithm ln⁡ln⁡(1+z), whose coefficients are represented by Ak(2). These numbers are found to be related to the reciprocal logarithm numbers and thus converge in a similar manner to zero as k→∞. The final application in the chapter is concerned with the generalization of the integrands in the elliptic integrals F(x,κ) and E(x,κ). First the partition method for a power series expansion is applied to f(x)=(1+acos⁡x)−ρ in Theorem 2.4. Again, general formulas are derived for the three highest and lowest order terms in powers of a+1, while in a corollary the recurrence relation for the coefficients is derived. By putting a=−κ2/(κ2−2) and integrating the expansion over x, we obtain power series expansions for the elliptic integral of the first kind F(x,κ) and the second kind E(x,κ) when ρ equals 1/2 and −1/2, respectively. Then the power series expansion for the elliptic integral of the third kind Π(n,x,κ) is derived by combining these results. Finally, similar analyses are carried out when the cosine in f(x) is replaced by other trigonometric functions such as sin⁡x, tan⁡x and cot⁡x.