Special Values and Complex Integral Representation of L-Functions

2014 ◽  
pp. 139-153
Author(s):  
Tomoyoshi Ibukiyama ◽  
Masanobu Kaneko
Keyword(s):  
Integral Representation ◽  
Special Values ◽  
Complex Integral

scholarly journals On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 451 ◽  
Author(s):  
Dae Kim ◽  
Taekyun Kim ◽  
Cheon Ryoo ◽  
Yonghong Yao
Keyword(s):  
Integral Representation ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Finite Product ◽  
Bernoulli Numbers And Polynomials ◽  
Special Values ◽  
Invariant Integrals ◽  
Evaluation Problem ◽  
Double Integrals

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

scholarly journals An Integral Representation for the Polylogarithm Function and some Special Values

2014 ◽  
Vol 8 ◽  
pp. 6-10
Author(s):  
Edigles Guedes ◽  
Raja Rama Gandhi
Keyword(s):  
Integral Representation ◽  
Polylogarithm Function ◽  
Special Values

We proved a new integral representation for the polylogarithm function.

scholarly journals On higher regulators of Siegel threefolds II: the connection to the special value

2017 ◽  
Vol 153 (5) ◽  
pp. 889-946 ◽  
Author(s):  
Francesco Lemma
Keyword(s):  
Integral Representation ◽  
Automorphic Representations ◽  
Motivic Cohomology ◽  
Special Values ◽  
Special Value ◽  
Cuspidal Automorphic Representations

We establish a connection between motivic cohomology classes over the Siegel threefold and non-critical special values of the degree-four $L$-function of some cuspidal automorphic representations of $\text{GSp}(4)$. Our computation relies on our previous work [On higher regulators of Siegel threefolds I: the vanishing on the boundary, Asian J. Math. 19 (2015), 83–120] and on an integral representation of the $L$-function due to Piatetski-Shapiro.

scholarly journals Integral Representation and Explicit Formula at Rational Arguments for Apostol–Tangent Polynomials

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 35
Author(s):  
Cristina B. Corcino ◽  
Roberto B. Corcino ◽  
Baby Ann A. Damgo ◽  
Joy Ann A. Cañete
Keyword(s):  
Fourier Series ◽  
Integral Representation ◽  
Zeta Function ◽  
Explicit Formula ◽  
Summation Formula ◽  
Fourier Series Expansion ◽  
Residue Theorem ◽  
Lerch Zeta Function ◽  
Complex Integral ◽  
Rational Arguments

The Fourier series expansion of Apostol–tangent polynomials is derived using the Cauchy residue theorem and a complex integral over a contour. This Fourier series and the Hurwitz–Lerch zeta function are utilized to obtain the explicit formula at rational arguments of these polynomials. Using the Lipschitz summation formula, an integral representation of Apostol–tangent polynomials is also obtained.

scholarly journals Multi-Integral Representations for Associated Legendre and Ferrers Functions

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1598
Author(s):  
Howard S. Cohl ◽  
Roberto S. Costas-Santos
Keyword(s):  
Integral Representation ◽  
Legendre Function ◽  
Order Parameters ◽  
Integral Representations ◽  
Special Values ◽  
Representation Formulas ◽  
Integration Formulas ◽  
Parameter Values ◽  
Associated Legendre Function

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.

INFLUENCE OF LIVE WEIGHT IN CULTIVATION ON THE DURATION OF PRODUCTIVE USE OF COWS OF UKRAINIAN BLACK-AND-WHITE DAIRY BREED OF DIFFERENT BLOODLINE ON THE HOLSTEIN BREED

2020 ◽  
Vol 11 (2) ◽  
pp. 16-27
Author(s):  
L. M. Danets ◽  
I. V. Tkachova ◽  
V. P. Shablia
Keyword(s):  
Average Duration ◽  
Live Weight ◽  
Black And White ◽  
Holstein Breed ◽  
Complex Integral ◽  
The Cost ◽  
State Enterprises ◽  
Dairy Breed ◽  
Integral Feature ◽  
A Share

The duration of productive use is a complex integral feature and is determined by both genetic and paratypical factors. This feature should ensure maximum milk productivity of animals, economic efficiency of dairy farms and generally limit the cost of raising and keeping cows. Research conducted in the experimental farms of state enterprises "Kutuzovka", Kharkov district of Kharkov region using cow’s Ukrainian black-and-white dairy breed (4038 cows). We studied the duration of productive use of cows divided into gradations according to the conditional bloodline for the Holstein breed, depending on the weight in the control age periods of cultivation (at birth, at 6, 12 and 18 months). The maximum value of the duration of productive use was recorded in the group of cows with a share of conditional blood for the Holstein breed up to 30% inclusive (the average duration of productive use is 2.77 lactations). The highest indicator of the duration of productive use of the studied animals was 4.09 lactations. Among cows with a share of conditional blood for the Holstein breed up to 30% longer produced those born with a weight of more than 40 kg (4.09 lactations), at 6 months of age weighed 100-149 kg, at 12 months – 200-249 kg, at 18 months of age – 350-399 kg. Among animals with a share of bloodline of 31-60 %, those that had a weight at the age of 6 months produced the longest: 155-190 kg (3.17 lactations), at 12 months – 250-299 kg (2.98 lactations), at 18 months – 350-399 kg (3.06 lactations). In the most numerous gradation with the share of bloodline for the Holstein breed 61-90 %, the longest productive use was characterized by animals born with a weight of 30-39 kg (2.12 lactation), at 6 months of age they weighed more than 200 kg (3.29 lactation), at 12 months-300-349 kg (3.40 lactation), at 18 months – 400-449 kg (2.82 lactation). The average duration of productive use of cows of this grade is quite low – in the range of 2.82-3.29 lactations. The highest degree of influence on the duration of productive use in this gradation of animals was recorded by the live weight factor at 6 months of age (η2 = 10.8). Сcomparative assessment of the cows gradation with a share of conditional blood for the Holstein breed of 91 % or more found that the longest productive use was characterized by cows that had a live weight at birth of 30-39 kg (1.93 lactation), in 6 months – 155-199 kg (2.73 lactation), in 12 months – 250-299 kg (2.87 lactation) and in 18 months – 350-399 kg (2.62 lactation). The degree of influence of live weight at this age on the duration of productive use of cows is insignificant, except for live weight at 6 months of age (η2 = 11.8).

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