scholarly journals Rapidly convergent series representations of symmetric Tornheim double zeta functions

2021 ◽  
Author(s):  
T. Nakamura
Keyword(s):  
Zeta Functions ◽  
Convergent Series ◽  
Double Zeta ◽  
Series Representations

scholarly journals New series representations for the two-loop massive sunset diagram

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
B. Ananthanarayan ◽  
Samuel Friot ◽  
Shayan Ghosh
Keyword(s):  
Perturbation Theory ◽  
Standard Model ◽  
Analytic Continuation ◽  
Chiral Perturbation Theory ◽  
Convergent Series ◽  
Regions Of Interest ◽  
External Momentum ◽  
Chiral Perturbation ◽  
Physical Problems ◽  
Series Representations

Abstract We derive new convergent series representations for the two-loop sunset diagram with three different propagator masses $$m_1,\, m_2$$m1,m2 and $$m_3$$m3 and external momentum p by techniques of analytic continuation on a well-known triple series that corresponds to the Lauricella $$F_C^{(3)}$$FC(3) function. The convergence regions of the new series contain regions of interest to physical problems. These include some ranges of masses and squared external momentum values which make them useful from Chiral Perturbation Theory to some regions of the parameter space of the Minimal Supersymmetric Standard Model. The analytic continuation results presented for the Lauricella series could be used in other settings as well.

Bi-starlike function of complex order associated with double Zeta functions

2014 ◽  
Vol 26 (5-6) ◽  
pp. 1025-1036
Author(s):  
G. Murugusundaramoorthy ◽  
T. Janani
Keyword(s):  
Zeta Functions ◽  
Starlike Function ◽  
Complex Order ◽  
Double Zeta

Bounds of double zeta-functions and their applications

2020 ◽  
Vol 304 (1) ◽  
pp. 15-41
Author(s):  
Debika Banerjee ◽  
T. Makoto Minamide ◽  
Yoshio Tanigawa
Keyword(s):  
Zeta Functions ◽  
Double Zeta

scholarly journals Real zeros of Hurwitz–Lerch zeta and Hurwitz–Lerch type of Euler–Zagier double zeta functions

2015 ◽  
Vol 160 (1) ◽  
pp. 39-50 ◽  
Author(s):  
TAKASHI NAKAMURA
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Real Zeros ◽  
Double Zeta ◽  
Lerch Zeta Function

AbstractLet 0 < a ⩽ 1, s, z ∈ ${\mathbb{C}}$ and 0 < |z| ⩽ 1. Then the Hurwitz–Lerch zeta function is defined by Φ(s, a, z) ≔ ∑∞n = 0zn(n + a)− s when σ ≔ ℜ(s) > 1. In this paper, we show that the Hurwitz zeta function ζ(σ, a) ≔ Φ(σ, a, 1) does not vanish for all 0 < σ < 1 if and only if a ⩾ 1/2. Moreover, we prove that Φ(σ, a, z) ≠ 0 for all 0 < σ < 1 and 0 < a ⩽ 1 when z ≠ 1. Real zeros of Hurwitz–Lerch type of Euler–Zagier double zeta functions are studied as well.

scholarly journals On lognormal random variables: I-the characteristic function

1991 ◽  
Vol 32 (3) ◽  
pp. 327-347 ◽  
Author(s):  
Roy B. Leipnik
Keyword(s):  
Characteristic Function ◽  
Integral Transform ◽  
Zeta Functions ◽  
Random Variable ◽  
Convergent Series ◽  
Hermite Functions ◽  
Lognormal Distributions ◽  
Equation Solving ◽  
Sum Of Products ◽  
Functional Differential

AbstractThe characteristic function of a lognormal random variable is calculated in closed form as a rapidly convergent series of Hermite functions in a logarithmic variable. The series coefficients are Nielsen numbers, defined recursively in terms of Riemann zeta functions. Divergence problems are avoided by deriving a functional differential equation, solving the equation by a de Bruijn integral transform, expanding the resulting reciprocal Gamma function kernel in a series, and then invoking a convergent termwise integration. Applications of the results and methods to the distribution of a sum of independent, not necessarily identical lognormal variables are discussed. The result is that a sum of lognormals is distributed as a sum of products of lognormal distributions. The case of two lognormal variables is outlined in some detail.

On the lacunary convergent series representations for ζ(n)

2018 ◽  
Vol 29 (06) ◽  
pp. 1850038
Author(s):  
Li-Xia Dai ◽  
Hao Pan ◽  
Ji-Zhen Xu
Keyword(s):  
Convergent Series ◽  
Bernoulli Numbers ◽  
Series Representations

We prove some lacunary convergent series representations for [Formula: see text]. For example, we prove that [Formula: see text] where [Formula: see text] are the [Formula: see text]th Bernoulli numbers.

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