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 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.

On a connection between the polylogarithm function and the Bass diffusion model

2008 ◽  
Vol 464 (2099) ◽  
pp. 3081-3088 ◽  
Author(s):  
Pedro Jodrá
Keyword(s):  
Probability Distribution ◽  
Integral Representation ◽  
Closed Form ◽  
Diffusion Model ◽  
Random Variable ◽  
Continuous Random Variable ◽  
Polylogarithm Function ◽  
Probabilistic Formulation ◽  
Bass Diffusion ◽  
Bass Diffusion Model

The purpose of this paper is to establish a connection between the polylogarithm function and a continuous probability distribution. We provide closed-form expressions in terms of the polylogarithm function for all the moments of a continuous random variable related to the Bass diffusion model, which was introduced by Bass and is widely used in marketing science. In addition, a new integral representation of the polylogarithm of order n is achieved from a probabilistic formulation.

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 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.

scholarly journals Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

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