Classification of hypergeometric identities for π and other logarithms of algebraic numbers

Chudnovsky, D. V.; Chudnovsky, G. V.

doi:10.1073/pnas.95.6.2744

Cited by 4 publications

(3 citation statements)

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“…Later, several proofs of Euler's formula using various methods have been published by a number of authors, including Refs. [11,20,[33][34][35][36]45,47,50].…”

Section: Discussionmentioning

confidence: 99%

All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells

Belkić

2018

J Math Chem

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All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. Previously, various methods have been used to provide the proofs for the general terms of these two series. Taking n to be any real or complex number, we presently give an alternative proof using the Bell (or exponential) polynomials. The ensuing series is summed up yielding a single, compact, explicit, analytical formula for all the trinomial roots as the confluent Fox-Wright function 1 1. Moreover, we also derive a slightly different, single formula of the trinomial root raised to any power (real or complex number) as another 1 1 function. Further, in this study, the logarithm of the trinomial root is likewise expressed through a single, concise series with the binomial expansion coefficients or the Pochhammer symbols. These findings are anticipated to be of considerable help in various applications of trinomial roots. Namely, several properties of the 1 1 function can advantageously be employed for its implementations in practice. For example, the simple expressions for the asymptotic limits of the 1 1 function at both small and large values of the independent variable can be used to readily predict, by analytical means, the critical behaviors of the studied system in the two extreme conditions. Such limiting situations can be e.g. at the beginning of the time evolution of a system, and in the distant future, if the independent variable is time, or at low and high doses when the independent variable is radiation dose, etc. The present analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation.

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“…Later, several proofs of Euler's formula using various methods have been published by a number of authors, including Refs. [11,20,[33][34][35][36]45,47,50].…”

Section: Discussionmentioning

confidence: 99%

All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells

Belkić

2018

J Math Chem

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“…Remark 2.1º For other trivial summation results see [12], [13] and [2, §1.7], for more complicated cases see [5] or [18]. The methods in all these papers are different or not so much general as the method presented in Theorem 2.…”

Section: Thenmentioning

confidence: 99%

“…The results concerning the summation techniques for the series involving special products or the theory of L e h m e r 's or Gosper's sums and Apéry-like series we can find in [2, §1.7], [3], [5], [6], [16], [17], [20] and [24].…”

Section: Summation Techniquementioning

confidence: 99%

Evaluation of infinite series involving special products and their algebraic characterization

Genčev

2009

Mathematica Slovaca

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Numerical Computations

Trott

2006

The Mathematica GuideBook for Numerics

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Classification of hypergeometric identities for π and other logarithms of algebraic numbers

Cited by 4 publications

References 4 publications

All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells

All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells

Evaluation of infinite series involving special products and their algebraic characterization

Numerical Computations

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