Shifted Euler constants and a generalization of Euler-Stieltjes constants

Chatterjee, Tapas; Khurana, Suraj Singh

doi:10.1016/j.jnt.2019.04.001

Cited by 8 publications

(6 citation statements)

References 56 publications

(54 reference statements)

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“…We clarify the underlying reason for identities (2) and (3). Proof of (2) depends on (55), (11) and (68).…”

Section: Elucidation Of Some Identitiesmentioning

confidence: 92%

“…Most of these results have been summarized and elucidated in [12]. There are further generalizations and we refer to [3] and references there given. We may use the Deninger-Meyer method to find Laurent coefficients of a more general class of Dirichlet series including, for example, References [5,47] in terms of principal solutions to a difference equation, which will be conducted elsewhere.…”

Section: Difference Equationsmentioning

confidence: 96%

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Limiting Values and Functional and Difference Equations

2020

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Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. As main results, this involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We establish these by the relation between bases of the Kubert space of functions. Then these expressions are equated with other expressions in terms of special functions introduced by some difference equations, giving rise to analogues of the Lerch-Chowla-Selberg formula. We also state Abelian results which not only yield asymptotic formulas for weighted summatory function from that for the original summatory function but assures the existence of the limit expression for Laurent coefficients.

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“…We clarify the underlying reason for identities (2) and (3). Proof of (2) depends on (55), (11) and (68).…”

Section: Elucidation Of Some Identitiesmentioning

confidence: 92%

Section: Difference Equationsmentioning

confidence: 96%

Limiting Values and Functional and Difference Equations

2020

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“…Most of these results have been summarized and elucidated in [69]. There are further generalizations and we refer to [17] and references there given. We may use the Deninger-Meyer method to find Laurent coefficients of a more general class of Dirichlet series including e.g.…”

Section: Difference Equationsmentioning

confidence: 96%

“…They are referred to as generalized Euler constants or Euler-Stieltjes constants. [5,9,17,23,24,26,27,29], [32,36], [41,46, pp. 71-81], [69], [85], [91] to name a few.…”

Section: Introductionmentioning

confidence: 99%

Limiting Values and Functional and Difference Equations

Wang¹,

Agarwal²,

Kanemitsu³

2020

Preprint

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Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function.This clarifies the second equality in (62). For a general form, cf. (98).Comtet [19, pp. 224-229] is the most informative reference book on Stirling numbers.Preprints (www.preprints.org) | NOT PEER-REVIEWED |

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“…For ∈ N, the th periodic Bernoulli polynomialB (x) is defined bȳ 11) where {x} = x − [x] indicates the fractional part.…”

Section: Introductionmentioning

confidence: 99%

The Boundary Lerch Zeta-Function and Short Character Sums à la Y. Yamamoto

2020

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As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185-195), there have appeared many instances in which only the imaginary part-the odd part-of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q-expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann's fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628-634). We may thus refer to this as the 'Fourier series-boundary q-series', and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275-289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function-and, a fortiori, the modular relation (S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014).

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Shifted Euler constants and a generalization of Euler-Stieltjes constants

Cited by 8 publications

References 56 publications

Limiting Values and Functional and Difference Equations

Limiting Values and Functional and Difference Equations

Limiting Values and Functional and Difference Equations

The Boundary Lerch Zeta-Function and Short Character Sums à la Y. Yamamoto

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