Addendum to ``Applications of differential algebra to algebraic independence of arithmetic functions'' (Acta Arith. 172 (2016), 149--173)

The Siegel-Shidlovskii method remains one of the basic methods in the theory of transcendental numbers. By using this method, it is possible to prove the transcendence and algebraic independence of the values of entire functions of a certain class (so-called E-functions). A necessary condition for applying this method is that all of the considered functions must constitute a solution of a system of linear differential equations and were algebraically independent over . The question about algebraic independence of the solutions of linear differential equations and systems of such equations is of great importance in differential algebra, analytical theory of differential equations, theory of special functions, and calculus (in the broad sense of the word). As is shown in papers by E. Kolchin, F. Beukers, W.D. Brownawell, and G. Heckman, this question boils down in many instances to verification of the cogredience and contragredience condition. Two systems of 1st order linear homogeneous differential equations with coefficients from are said to be cogredient (or, respectively, contragredient), if for arbitrary fundamental matrices and Ψ of these systems one of the equations Φ=gBΨC, Φ (ΨC)^*=gB, is fulfilled, where C∈GL(C), B∈GL(C(z)), g=g(z) is a function with the condition g^'/g∈C(z), and A^* is the matrix transposed to . The notions of cogredience and contragredience for linear homogeneous differential equations of arbitrary order are defined similarly. Another, more restricted definitions of cogredience and contragredience were in fact used in some papers of the author, devoted to generalized hypergeometric functions. According to these definitions, the function in the presented equalities is the product of a power function and an exponential function of some kind. The conditions for equivalence of these definitions are found.

Download Full-text

Corrigendum to ``Stability aspects of arithmetic functions, II'' (Acta Arith. 139 (2009), 131–146)

Acta Arithmetica ◽

10.4064/aa149-1-6 ◽

2011 ◽

Vol 149 (1) ◽

pp. 83-98 ◽

Cited By ~ 2

Author(s):

Tomasz Kochanek

Keyword(s):

Arithmetic Functions ◽

Acta Arith

Download Full-text

Set-membership identifiability of nonlinear models and related parameter estimation properties

International Journal of Applied Mathematics and Computer Science ◽

10.1515/amcs-2016-0057 ◽

2016 ◽

Vol 26 (4) ◽

pp. 803-813 ◽

Cited By ~ 2

Author(s):

Carine Jauberthie ◽

Louise Travé-MassuyèEs ◽

Nathalie Verdière

Keyword(s):

Mathematical Model ◽

Parameter Estimation ◽

Dynamic System ◽

Nonlinear Models ◽

Differential Algebra ◽

Related Parameter ◽

Set Membership ◽

Estimation Problems ◽

The Mathematical Model

Abstract Identifiability guarantees that the mathematical model of a dynamic system is well defined in the sense that it maps unambiguously its parameters to the output trajectories. This paper casts identifiability in a set-membership (SM) framework and relates recently introduced properties, namely, SM-identifiability, μ-SM-identifiability, and ε-SM-identifiability, to the properties of parameter estimation problems. Soundness and ε-consistency are proposed to characterize these problems and the solution returned by the algorithm used to solve them. This paper also contributes by carefully motivating and comparing SM-identifiability, μ-SM-identifiability and ε-SM-identifiability with related properties found in the literature, and by providing a method based on differential algebra to check these properties.

Download Full-text

The Multi-Dimensional Local Theorem for Additive Arithmetic Functions

Analytic and Probabilistic Methods in Number Theory ◽

10.1515/9783112314234-029 ◽

1992 ◽

pp. 285-292

Keyword(s):

Arithmetic Functions ◽

Additive Arithmetic ◽

Local Theorem

Download Full-text

On the growth and zeros of polynomials attached to arithmetic functions

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ◽

10.1007/s12188-021-00241-3 ◽

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.

Download Full-text