Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the reciprocal of the logarithmic function ln(1 + z), whose coefficients represent an infinite set of fractions. These numbers, which are called reciprocal logarithm numbers and are denoted by A k , converge to zero as k → ∞. Several properties of the numbers are then obtained including recursion relations and their relationship with the Stirling numbers of the first kind. Also appearing here are several applications including a new representation for Euler's constant known as Hurst's formula and another for the logarithmic integral. From the properties of the A k it is found that a term of ζ(2) cannot be eliminated by the remaining terms in Hurst's formula, thereby indicating that Euler's constant is irrational. Finally, another power series expansion for the reciprocal of arctangent is developed by adapting the preceding material.

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

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We investigate the magnetized Bose gas at temperatures above pair threshold. New magnetization laws are obtained for a wide range of field strengths, and the gas is shown to exhibit the Meissner effect. Some related results for the Fermi gas, a relativistic paramagnet, are also discussed.PACS numbers: 05.30.JpAs one of the fundamental quantum systems, the Bose gas has provided insight into a variety of exotic physical phenomena, from liquid 4 He superfluidity to superconductivity [1]. Schafroth [2] showed how the nonrelativistic Bose gas exhibits the Meissner effect, that is, total expulsion of an external magnetic field, and thereafter the Bose gas has played a role in the understanding of superconductivity in metals. More recently, in a seminal work Haber and Weldon [3,4] have developed the statistical mechanics of the relativistic Bose gas with no external fields, and applied this to a study of spontaneous symmetry breaking [5].It is now apparent that astrophysics and cosmology provide venues where high temperatures and large magnetic fields play a significant role. For example, white dwarfs, neutron stars, and supernovae [6] are examples of exotic stellar objects where fields and temperatures can be of the order of the mass scale of their constituent particles. Furthermore, it has been suggested that fields of the order of -10 23 G [7], and possibly -10 33 G [8] existed at the electroweak phase transition, where temperatures were ~ 10 15 K. Given this, and that the mass of the pion, for example, is in field/temperature units ~ (10 18 G)/(10 12 K), and that of the electron is