Programming the Partition Method for a Power Series Expansion

Kowalenko, Victor

doi:10.1016/b978-0-12-804466-7.00005-x

Cited by 2 publications

(10 citation statements)

References 18 publications

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“…Then one equates like powers of x. For the special case of n = 1, the numbers are related to the cosecant-squared numbers discussed in [25] and [28]. As a consequence, we find that…”

Section: Generalized Cosecant Numbersmentioning

confidence: 78%

“…This method first appeared in [31]. Then, Table 2.1: Multiplicities of the partitions summing to 5 and the sums of the parts it was developed further in a series of articles [26,[28][29][30], culminating in [25]. In particular, [25] not only provides the most comprehensive account of the method by including most of the existing material from the other articles, but also introduces more intricate applications and derivations.…”

Section: Generalized Cosecant Numbersmentioning

confidence: 99%

“…Then, Table 2.1: Multiplicities of the partitions summing to 5 and the sums of the parts it was developed further in a series of articles [26,[28][29][30], culminating in [25]. In particular, [25] not only provides the most comprehensive account of the method by including most of the existing material from the other articles, but also introduces more intricate applications and derivations. For example, general formulas for the three highest and three lowest order coefficients of the generalized cosecant numbers are derived for the first time in Chapter 2 of this book in addition to derivations of the generating functions for generalizations of the elliptic integrals.…”

Section: Generalized Cosecant Numbersmentioning

confidence: 99%

“…To circumvent this problem, a general computing methodology has been developed to determine higher order coefficients via the partition method for a power series expansion. This methodology, which is based on representing all the partitions summing to a specific order as a tree diagram and invoking the bivariate recursive central partition (BRCP) algorithm, is described in great detail in [25] and [26]. The expressions for the coefficients arising from this general theory can then be imported into a mathematical software package such as Mathematica [46] whereupon its symbolic routines can be used to generate the final values displayed in Table 2.2. A particularly interesting property of the generalized cosecant numbers is that the power of the cosecant or ρ appears as the variable in the polynomials displayed in Table 2.2.…”

Section: Generalized Cosecant Numbersmentioning

confidence: 99%

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Generalized cosecant numbers and trigonometric inverse power sums

Carlos

Glasser

Kowalenko

2018

Appl Anal Discrete Math

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The generalized cosecant numbers denoted here by c ρ,k represent the coefficients of the power series expansion or generating function of the fundamental function x ρ / sin ρ x. In actual fact, these interesting numbers are polynomials in ρ of degree k, whose coefficients are only dependent upon k. In this paper we show how they emerge in the calculation of trigonometric inverse power sums. After introducing the generalized cosecant numbers we present a novel and elegant integral approach for computing the Gardner-Fisher trigonometric inverse power sum, which is given by Sv,2(m) = π 2m 2v m−1 k=1 cos −2v kπ 2m , where m and v are positive integers. This method not only confirms the solutions obtained earlier by an empirical method, but it is also much more expedient from a computational point of view. By comparing the formulas from both methods, we derive several new and interesting number-theoretical results involving symmetric polynomials over the set of quadratic powers up to (v − 1) 2 and the generalized cosecant numbers.

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“…Then one equates like powers of x. For the special case of n = 1, the numbers are related to the cosecant-squared numbers discussed in [25] and [28]. As a consequence, we find that…”

Section: Generalized Cosecant Numbersmentioning

confidence: 78%

Section: Generalized Cosecant Numbersmentioning

confidence: 99%

Section: Generalized Cosecant Numbersmentioning

confidence: 99%

Section: Generalized Cosecant Numbersmentioning

confidence: 99%

See 2 more Smart Citations

Generalized cosecant numbers and trigonometric inverse power sums

Carlos

Glasser

Kowalenko

2018

Appl Anal Discrete Math

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“…Generally, when k is large, most of the multiplicities for a partition vanish as we shall see shortly. Using (2) one finds that c 0 = 1, c 1 = 1/6, c 2 = 7/360, c 4 = 31/3•7! and so on.…”

Section: Introductionmentioning

confidence: 99%

Generalized Cosecant Numbers and the Hurwitz Zeta Function

Kowalenko

2017

Preprint

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This paper presents recent developments concerning the generalized cosecant numbers c ρ,k , which emerge as the coefficients of the power series expansion for the important fundamental function z ρ / sin ρ z. These coefficients can be computed for all, including complex, values of ρ by using the relatively novel graphical method known as the partition method for a power series expansion or by using intrinsic routines in Mathematica. In fact, they represent polynomials in ρ of degree k, where k is the power of z.In addition, though related to the Bernoulli numbers, they possess more properties and do not diverge like the former. The partition method for a power series expansion has the advantage that it yields the k-behaviour of the highest order coefficients. Thus, general formulas for such coefficients are derived by considering the properties of the highest part partitions. It is then shown how the generalized cosecant numbers are related to the specific symmetric polynomials that arise from summing over quadratic powers of integers. Consequently, integral values of the Hurwitz zeta function for even powers are expressed for the first time ever in terms of ratios of the generalized cosecant numbers.

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Programming the Partition Method for a Power Series Expansion

Cited by 2 publications

References 18 publications

Generalized cosecant numbers and trigonometric inverse power sums

Generalized cosecant numbers and trigonometric inverse power sums

Generalized Cosecant Numbers and the Hurwitz Zeta Function

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