The Non-relativistic Charged Bose Gas in a Magnetic Field II. Quantum Properties

Kowalenko, Victor

doi:10.1006/aphy.1999.5909

Cited by 9 publications

(12 citation statements)

References 35 publications

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“…· · · n i !. Multiplying the multinomial factor with the assigned values yields the contribution to A k from the partition as given by (6). By summing all the contributions from the partitions one obtains the final value of A k .…”

Section: Resultsmentioning

confidence: 99%

“…According to (6), for the partition involving the path (1,5), (1,4), (2,2) and (0, 2) in Fig. 1, we have l 1 = 1, n 1 = 2, l 2 = 2 and n 2 = 2 with i ranging from 1 to 2 in the product.…”

Section: Derivationmentioning

confidence: 99%

“…The first algorithm is based on a graphical method introduced in [1,6]. In this method tree diagrams are constructed for each A k .…”

Section: Derivationmentioning

confidence: 99%

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Properties and Applications of the Reciprocal Logarithm Numbers

Kowalenko

2008

Acta Appl Math

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115

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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.

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Section: Resultsmentioning

confidence: 99%

“…According to (6), for the partition involving the path (1,5), (1,4), (2,2) and (0, 2) in Fig. 1, we have l 1 = 1, n 1 = 2, l 2 = 2 and n 2 = 2 with i ranging from 1 to 2 in the product.…”

Section: Derivationmentioning

confidence: 99%

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Properties and Applications of the Reciprocal Logarithm Numbers

Kowalenko

2008

Acta Appl Math

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115

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“…In obtaining (80) we have used the power series for exp(xt), which is convergent for all values of x and t . However, we have also used Equivalence (1), which according to the regularized result given by (12) is divergent when t 2 /4π 2 ≤ −1 and convergent when t 2 /4π 2 > −1. Thus the same conditions apply to Equivalence (78), which means that we can write it as ∞ k=0 a k (x)t k = te xt e t −1 , t 2 > −4π 2 and ∀x, ≡ te xt e t −1 , t 2 ≤ −4π 2 and ∀x.…”

Section: Bernoulli Polynomialsmentioning

confidence: 99%

“…As discussed in Refs. [1,11,12], the tree diagram for the a k is constructed by drawing branch lines to all pairs of numbers that can be summed to k, where the first number in the tuple is an integer less than or equal to [k/2]. For example, the tree diagram for a 6 possesses branch lines to (0, 6), (1,5), (2,4) and (3,3).…”

Section: Bernoulli Numbersmentioning

confidence: 99%

Generalizing the Reciprocal Logarithm Numbers by Adapting the Partition Method for a Power Series Expansion

Kowalenko

2008

Acta Appl Math

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Recently, a novel method based on the coding of partitions was used to determine a power series expansion for the reciprocal of the logarithmic function, viz. z/ ln(1 + z).Here we explain how this method can be adapted to obtain power series expansions for other intractable functions. First, the method is adapted to evaluate the Bernoulli numbers and polynomials. As a result, new integral representations and properties are determined for the former. Then via another adaptation of the method we derive a power series expansion for the function z s / ln s (1 + z), whose polynomial coefficients A k (s) are referred to as the generalized reciprocal logarithm numbers because they reduce to the reciprocal logarithm numbers when s = 1. In addition to presenting a general formula for their evaluation, this paper presents various properties of the generalized reciprocal logarithm numbers including general formulas for specific values of s, a recursion relation and a finite sum identity. Other representations in terms of special polynomials are also derived for the A k (s), which yield general formulas for the highest order coefficients. The paper concludes by deriving new results involving infinite series of the A k (s) for the Riemann zeta and gamma functions and other mathematical quantities.

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Introduction

Kowalenko¹

2017

The Partition Method for a Power Series Expansion

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The Non-relativistic Charged Bose Gas in a Magnetic Field II. Quantum Properties

Cited by 9 publications

References 35 publications

Properties and Applications of the Reciprocal Logarithm Numbers

Properties and Applications of the Reciprocal Logarithm Numbers

Generalizing the Reciprocal Logarithm Numbers by Adapting the Partition Method for a Power Series Expansion

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