Applications of the Cosecant and Related Numbers

Abstract

Power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a power series expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c k , converge rapidly to zero as k → ∞. It is then shown how recent advances in obtaining meaningful values from divergent series can be modified to determine exact numerical results from the asymptotic series derived from the Laplace transform of the power series expansion for t csc(at). Next the power series expansion for secant is derived in terms of related coefficients known as the secant numbers d k . These numbers are related to the Euler numbers and can also be evaluated by numerous recurrence relations, some of which involve the cosecant numbers. The approaches used to obtain the power series expansions for these fundamental trigonometric functions in addition to the methods used to evaluate their coefficients are employed in the derivation of power series expansions for integer powers and arbitrary powers of the trigonometric functions. Recurrence relations are of limited benefit when evaluating the coefficients in the case of arbitrary powers. Consequently, power series expansions for the Legendre-Jacobi elliptic integrals can only be obtained by the partition method for a power series expansion. Since the Bernoulli and Euler numbers give rise to polynomials from exponential generating functions, it is shown that the cosecant and secant numbers gives rise to their own polynomials from trigonometric generating functions. As expected, the new polynomials are related to the Bernoulli and Euler polynomials, but they are found to possess far more interesting properties, primarily due to the convergence of the coefficients. One interesting application of the new polynomials is the re-interpretation of the Euler-Maclaurin summation formula, which yields a new regularisation formula.V. Kowalenko ( ) Mathematics Subject Classification (2010) Primary 11A25 · 11A99 · 11B37 · 11B75 · 11B83 · 11P82 · 11P84 · 30B10 · 30E15 · 41A58 · 65B10 · 65B15 · 65Q30 · 68R05 · 68R99 · 97N40 · 97N70 · Secondary 11B68 · 11M99 · 30E15 · 41-04 · 65S05

On a Finite Sum Involving Inverse Powers of Cosines

“…[2], the cosecant function raised to any arbitrary power ρ can be expressed as a power series given by…”

“…There the concept is defined as the removal of the infinity in the remainder of a divergent series so as to make it summable. The coefficients c ρ,k in Equivalence (6) can be evaluated by the powerful and versatile partition method for a power series expansion [2,4,8]. In particular, in Ref.…”

“…Table 1 displays the coefficients up to k = 8 obtained by evaluating (7). When ρ = 1, the coefficients reduce to a special infinite set of fractions known as the cosecant numbers [2]. These numbers, which converge rapidly to zero as k → ∞, not only possess many interesting properties, but they also arise in a vast number of applications.…”

“…We can avoid using (11) to (13) in order to evaluate S m,1 by turning instead to the coefficients p k in the power series expansion for 1/(1 − cos x) given in Theorem 5 of Ref. [2]. There it is found that…”

“…[2] the p k are referred to as the cosecant-squared coefficients because they are generated by the lhs of Equivalence (15). In actual fact, the coefficients c 2,k as defined in Equivalence (6) represent the true cosecant-squared coefficients, while the p k are related to the latter by the fact that p k = 2 −2k c 2,k .…”