scholarly journals Reflection Identities of Harmonic Sums of Weight Four

Universe ◽  
2019 ◽  
Vol 5 (3) ◽  
pp. 77 ◽  
Author(s):  
Alexander Prygarin
Keyword(s):  
Analytic Continuation ◽  
Linear Combination ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Bfkl Equation ◽  
Color Singlet ◽  
Pole Structure

In attempt to find a proper space of function expressing the eigenvalue of the color-singlet BFKL equation in N = 4 SYM, we consider an analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The resulting meromorphic functions have pole singularities at negative integers. We derive the reflection identities for harmonic sums at weight four decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight four, which represents the main result of the paper. We also discuss how other trilinear and quadlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

scholarly journals The analytic structure of the BFKL equation and reflection identities of harmonic sums at weight five

2019 ◽  
Vol 34 (11) ◽  
pp. 1950064 ◽  
Author(s):  
Mohammad Joubat ◽  
Alexander Prygarin
Keyword(s):  
Analytic Continuation ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Functional Form ◽  
Bfkl Equation ◽  
Analytic Structure ◽  
Color Singlet ◽  
Nonlinear Reflection ◽  
Pole Structure ◽  
Entire Complex

We analyze the structure of the eigenvalue of the color-singlet Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation in N[Formula: see text]=[Formula: see text]4 SYM in terms of the meromorphic functions obtained by the analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The meromorphic functions we discuss have pole singularities at negative integers and take finite values at all other points. We derive the reflection identities for harmonic sums at weight five decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight five which presents the main result of the paper. We show how the reflection identities can be used to restore the functional form of the next-to-leading eigenvalue of the color-singlet BFKL equation in N[Formula: see text]=4[Formula: see text]SYM, i.e. we argue that it is possible to restore the full functional form on the entire complex plane provided one has information how the function looks like on just two lines on the complex plane. Finally we discuss how nonlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

ANALYTIC CONTINUATION OF NONANALYTIC VECTOR-VALUED EISENSTEIN SERIES

2009 ◽  
Vol 05 (05) ◽  
pp. 805-830
Author(s):  
KAREN TAYLOR
Keyword(s):  
Analytic Continuation ◽  
Eisenstein Series ◽  
Complex Plane ◽  
Cusp Form ◽  
Vector Valued

In this paper, we introduce a vector-valued nonanalytic Eisenstein series appearing naturally in the Rankin–Selberg convolution of a vector-valued modular cusp form associated to a monomial representation ρ of SL(2,ℤ). This vector-valued Eisenstein series transforms under a representation χρ associated to ρ. We use a method of Selberg to obtain an analytic continuation of this vector-valued nonanalytic Eisenstein series to the whole complex plane.

Entire solutions of certain type of non-linear differential equations

2020 ◽  
Vol 70 (1) ◽  
pp. 87-94
Author(s):  
Bo Xue
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Nonlinear Differential Equations ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Linear Differential Equations ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential

AbstractUtilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| > |α2| > |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

scholarly journals The uniqueness of meromorphic functions in k-punctured complex plane

2017 ◽  
Vol 15 (1) ◽  
pp. 724-733 ◽  
Author(s):  
Hong Yan Xu ◽  
San Yang Liu
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Finite Sets

Abstract The main purpose of this paper is to investigate the uniqueness of meromorphic functions that share two finite sets in the k-punctured complex plane. It is proved that there exist two sets S1, S2 with ♯S1 = 2 and ♯S2 = 5, such that any two admissible meromorphic functions f and g in Ω must be identical if EΩ(Sj, f) = EΩ(Sj, g)(j = 1,2).

scholarly journals The problem of missing terms in term by term integration involving divergent integrals

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160567 ◽  
Author(s):  
Eric A. Galapon
Keyword(s):  
Analytic Continuation ◽  
Complex Plane ◽  
Finite Part ◽  
Stieltjes Transform ◽  
Complex Valued

Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity conditions, the problem of missing terms for the incomplete Stieltjes transform, ∫ 0 a f ( x ) ( ω + x ) − 1   d x , and the Stieltjes transform itself, ∫ 0 ∞ f ( x ) ( ω + x ) − 1   d x , is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex-valued function f ( z )( ω + z ) −1 , with the divergent integrals arising from term by term integration interpreted as finite part integrals.

On the value distribution of f2f(k)

2005 ◽  
Vol 78 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Xiaojun Huang ◽  
Yongxing Gu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Multiple Zeros ◽  
Transcendental Meromorphic Function

AbstractIn this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

scholarly journals On Certain Subclasses of Meromorphic Functions with Positive and Fixed Second Coefficients Involving the Liu-Srivastava Linear Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
N. Magesh ◽  
N. B. Gatti ◽  
S. Mayilvaganan
Keyword(s):  
Linear Operator ◽  
Linear Combination ◽  
Hypergeometric Function ◽  
Meromorphic Functions ◽  
Univalent Functions ◽  
Extreme Points ◽  
Generalized Hypergeometric Function ◽  
Coefficient Estimates ◽  
Convex Linear Combination ◽  
Meromorphic Univalent Functions

We introduce and study a subclass ΣP(γ,k,λ,c) of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class ΣP(γ,k,λ,c) by fixing the second coefficient. Further, it is shown that the class ΣP(γ,k,λ) is closed under convex linear combination.

The distribution of finite values of meromorphic functions with deficient poles

2002 ◽  
Vol 132 (2) ◽  
pp. 311-317
Author(s):  
J. K. LANGLEY
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Nevanlinna Deficiency

We prove the existence of unbounded open subsets S of the complex plane with the following property. If f is a function transcendental and meromorphic in the plane, the poles of which have positive Nevanlinna deficiency, then f takes every finite value, with at most one exception, infinitely often in the complement of S.

scholarly journals On Lucas-balancing zeta function

2018 ◽  
Vol 22 (1) ◽  
pp. 65-74
Author(s):  
Debismita Behera ◽  
Utkal Keshari Dutta ◽  
Prasanta Kumar Ray
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Complex Plane ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Riemann Zeta

In the present study a new modication of Riemann zeta function known as Lucas-balancing zeta function is introduced. The Lucas-balancing zeta function admits its analytic continuation over the whole complex plane except its poles. This series converges to a fixed rational number − ½ at negative odd integers. Further, in accordance to Dirichlet L-function, the analytic continuation of Lucas-balancing L-function is also discussed.

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