Search engine for discovering works of Art, research articles, and books related to Art and Culture
Javascript must be enabled to continue!
Zeros of modular forms and Faber polynomials
View through CrossRef
AbstractWe study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near vertical lines, with the zeros of a weight form lying at height approximately . This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree .
Title: Zeros of modular forms and Faber polynomials
Description:
AbstractWe study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number of finite zeros in the fundamental domain.
We show that for large weight the zeros of these forms cluster near vertical lines, with the zeros of a weight form lying at height approximately .
This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain.
Our method uses the Faber polynomials.
We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree .
Related Results
Complementary Romanovski-Routh polynomials and their zeros
Complementary Romanovski-Routh polynomials and their zeros
The efficacy of numerical methods like integral estimates via Gaussian quadrature formulas depends on the localization of the zeros of the associated family of orthogonal polynomia...
On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight
On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight
Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered....
Truncated-Exponential-Based Appell-Type Changhee Polynomials
Truncated-Exponential-Based Appell-Type Changhee Polynomials
The truncated exponential polynomials em(x) (1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known pol...
Orthogonality of quasi-orthogonal polynomials
Orthogonality of quasi-orthogonal polynomials
A result of P?lya states that every sequence of quadrature formulas Qn(f)
with n nodes and positive Cotes numbers converges to the integral I(f) of
a continuous function f pr...
Krein–Sobolev Orthogonal Polynomials II
Krein–Sobolev Orthogonal Polynomials II
In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials {Kn}n=0∞—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space...
On the Study of Families of Linearized Polynomials over Finite Fields
On the Study of Families of Linearized Polynomials over Finite Fields
Linearized polynomials are gaining attention from many researchers because of their applications in the field of coding theory, cryptography and finite geometry. The linearized pol...
Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function
Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function
The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and ...
Symmetric $*$-polynomials on $\mathbb C^n$
Symmetric $*$-polynomials on $\mathbb C^n$
$*$-Polynomials are natural generalizations of usual polynomials between complex vector spaces. A $*$-polynomial is a function between complex vector spaces $X$ and $Y,$ which is a...