scholarly journals On the growth and zeros of polynomials attached to arithmetic functions

2021 ◽  
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser
Keyword(s):  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Fourier Coefficients ◽  
Arithmetic Functions ◽  
Zeros Of Polynomials ◽  
Simple Complex ◽  
Hook Length ◽  
Moderate Growth ◽  
Growth Properties ◽  
Sum Of Divisors

AbstractIn this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and $$0<h(n) \le h(n+1)$$ 0 < h ( n ) ≤ h ( n + 1 ) . We put $$P_0^{g,h}(x)=1$$ P 0 g , h ( x ) = 1 and $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ P n g , h ( x ) : = x h ( n ) ∑ k = 1 n g ( k ) P n - k g , h ( x ) . As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind $$\eta $$ η -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

On linear relations for special L-values over certain totally real number fields

2021 ◽  
pp. 1-18
Author(s):  
Seiji Kuga
Keyword(s):  
Real Number ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Arithmetic Functions ◽  
Hilbert Modular Form ◽  
Linear Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
Totally Real

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special [Formula: see text]-values over certain totally real number fields.

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

scholarly journals Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 354 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Jiří Patera
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Bell Polynomials ◽  
The Third ◽  
Weight Lattice ◽  
Hybrid Character ◽  
Polynomial Family

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

ON THE FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

2016 ◽  
Vol 234 ◽  
pp. 1-16
Author(s):  
SIEGFRIED BÖCHERER ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Main Tool ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Growth Properties

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

Quasi-Hereditary Endomorphism Algebras

1995 ◽  
Vol 38 (4) ◽  
pp. 421-428 ◽  
Author(s):  
V. Dlab ◽  
P. Heath ◽  
F. Marko
Keyword(s):  
Lie Algebras ◽  
Algebraic Groups ◽  
Simple Complex ◽  
Highest Weight ◽  
Endomorphism Algebras ◽  
Hereditary Algebras ◽  
Highest Weight Categories

AbstractQuasi-hereditary algebras were introduced by Cline-Parshall-Scott (see [CPS] or [PS]) to deal with highest weight categories which occur in the study of semi-simple complex Lie algebras and algebraic groups. In fact, the quasi-hereditary algebras which appear in these applications enjoy a number of additional properties. The objective of this brief note is to describe a class of lean quasi-hereditary algebras [ADL] which possess such typical characteristics. A study of these questions originated in collaboration with C. M. Ringel (see [DR]).

scholarly journals ON LIE INDUCTION AND THE EXCEPTIONAL SERIES

2005 ◽  
Vol 04 (06) ◽  
pp. 707-737 ◽  
Author(s):  
JAN E. GRABOWSKI
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Point Of View ◽  
Lie Bialgebras ◽  
Simple Complex ◽  
Single Node ◽  
Finite Dimensional ◽  
Dynkin Diagrams ◽  
Induction Process ◽  
And Algebra

Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9.

scholarly journals On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

2010 ◽  
Vol 13 ◽  
pp. 357-369 ◽  
Author(s):  
Simon M. Goodwin ◽  
Gerhard Röhrle ◽  
Glenn Ubly
Keyword(s):  
Lie Algebras ◽  
Nilpotent Element ◽  
Coadjoint Orbit ◽  
Dimensional Representation ◽  
Simple Complex ◽  
Enveloping Algebras ◽  
Minimal Dimension ◽  
Pbw Theorem ◽  
Associated Variety ◽  
Modular Lie Algebras

AbstractWe consider the finiteW-algebraU(𝔤,e) associated to a nilpotent elemente∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem forU(𝔤,e), we verify a conjecture of Premet, thatU(𝔤,e) always has a 1-dimensional representation when 𝔤 is of typeG2,F4,E6orE7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal inU(𝔤) whose associated variety is the coadjoint orbit corresponding to e.

ON SEMIREGULAR NILPOTENT ORBITS IN SIMPLE LIE ALGEBRAS OF TYPE D

2002 ◽  
Vol 01 (03) ◽  
pp. 341-356 ◽  
Author(s):  
BENOÎT ARBOUR ◽  
DRAGOMIR Ž. ĐOKOVIĆ
Keyword(s):  
Linear Combination ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Nilpotent Orbits ◽  
Connected Component ◽  
Real Form ◽  
Simple Complex ◽  
Type D ◽  
Explicit Formulae

We derive explicit formulae for the characteristics H(k) of the semiregular nilpotent orbits Dn(ak) of the simple complex Lie algebra [Formula: see text] of type Dn. These formulae express H(k) as an integral linear combination of a basis of the Cartan subalgebra [Formula: see text] of [Formula: see text]. For that purpose we use several suitable bases of [Formula: see text] consisting of coroots. We also construct several explicit standard triples (E, H, F) with H = H(k), and E, F ∈ Dn(ak). Similar triples are constructed also for each connected component of the intersection of the orbit Dn(ak) with the split real form [Formula: see text] and the real form [Formula: see text] of [Formula: see text].

scholarly journals On certain upper bounds for the sum of divisors function σ(n)

2015 ◽  
Vol 7 (2) ◽  
pp. 265-277
Author(s):  
József Sándor ◽  
Lehel István Kovács
Keyword(s):  
Upper Bounds ◽  
Arithmetic Functions ◽  
Sum Of Divisors

Abstract Upper bounds for σ(n) are provided in terms of other arithmetic functions as ϕ(n), d(n), ψ(n), P(n), etc. Comparision of older results are given, too.

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