Lacunary Power Series Research Articles

Hilbert's tenth problem for lacunary entire functions of finite order

AbstractIn the context of Hilbert's tenth problem, an outstanding open case is that of complex entire functions in one variable. A negative solution is known for polynomials (by Denef) and for exponential polynomials of finite order (by Chompitaki, Garcia‐Fritz, Pasten, Pheidas, and Vidaux), but no other case is known for rings of complex entire functions in one variable. We prove a negative solution to the analogue of Hilbert's tenth problem for rings of complex entire functions of finite order having lacunary power series expansion at the origin.

Some Lacunary Power Series and Mahler’s Um-numbers in p-adic Domain

In this paper, we consider some lacunar power series in Qp, where p is a prime number. We obtain some results about lacunary power series with rational coefficients for Um-number arguments in Qp. We determined whether the values of the considered lacunar power series belong to a certain field of algebraic numbers or to the set of transcendental numbers in Qp. Also, we find similar results for lacunary power series with p-adic algebraic coefficients.

Minimal varieties of associative algebras and transcendental series

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number [Formula: see text] of minimal varieties of given exponent [Formula: see text] is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit [Formula: see text] exists and can be expressed as the positive solution of an equation [Formula: see text] where [Formula: see text] is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank [Formula: see text]. It follows from classical results on lacunary power series that the generating function of the sequence [Formula: see text], [Formula: see text], is transcendental. With the same approach we construct examples of free graded semigroups [Formula: see text] with the following property. If [Formula: see text] is the number of elements of degree [Formula: see text] of [Formula: see text], then the limit [Formula: see text] exists and is transcendental.

On Mahler’s p-adic S-, T -, and U-numbers

Abstract We consider some lacunary power series with rational coefficients in 𝕈p. We show that under certain conditions these series take transcendental values at non-zero rational number arguments, and we determine the classes of these transcendental values with respect to Mahler’s classification of p-adic numbers.

Lacunary Power Series and 𝑼𝒎-Numbers

AbstractIn this work, the values of certain lacunar power series with rational coefficients for 𝑈𝒎-number arguments were determined to be either in a particular algebraic number field or in the set of transcendental numbers under specific circumstances in the complex numbers field. The result was also applied on some of the lacunary power series with coefficients in an algebraic number field. Roth's theorem which is the essential result in Diophantine approximation to algebraic numbers was used to reach the present results.

-NUMBERS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS

In the field $\mathbb{K}$ of formal power series over a finite field $K$, we consider some lacunary power series with algebraic coefficients in a finite extension of $K(x)$. We show that the values of these series at nonzero algebraic arguments in $\mathbb{K}$ are $U$-numbers.

On countably universal series in the complex plane

ABSTRACTLet be countably many increasing sequences of integers tending to ∞. For a power series , we denote by the nth partial sum of f. Under some condition on , we construct lacunary power series converging in the unit disk , with arbitrarily large lacunes, which satisfy the following property (P): for any countable family of compact sets in the complement of , each of them with connected complement, the set is dense in , endowed with the product topology, inherited from the -norm of the space of all continuous functions on K which are holomorphic in its interior. The set of such countably universal series is shown to be invariant under some summability processes. Our construction also allows us to exhibit power series with Padé approximants enjoying countably universal properties, and to prove that the set of all power series with radius of convergence 1 which satisfy (P) is either void or spaceable. We finally use Ostrowski-gaps to give a very direct proof of two known results: (1) The set of doubly universal Taylor series is non-void only if for some increasing sequence , and (2) The set of frequently universal series is void.

Minimum modulus of lacunary power series and $h$-measure of exceptional sets

Minimum modulus of lacunary power series and $h$-measure of exceptional sets

Analogues of Whittker's theorem for Laplace-Stieltjes integrals

For the maximum of the integrand of Laplace-Stieltjes integral the lower estimates on sequence are found. Using the estimates we obtained analogues of Whittaker's theorem for entire functions given by lacunary power series.

A Numerical Test of Padé Approximation for Some Functions with Singularity

The aim of this study is to examine some numerical tests of Padé approximation for some typical functions with singularities such as simple pole, essential singularity, brunch cut, and natural boundary. As pointed out by Baker, it was shown that the simple pole and the essential singularity can be characterized by the poles of the Padé approximation. However, it was not fully clear how the Padé approximation works for the functions with the branch cut or the natural boundary. In the present paper, it is shown that the poles and zeros of the Padé approximated functions are alternately lined along the branch cut if the test function has branch cut, and poles are also distributed around the natural boundary for some lacunary power series and random power series which rigorously have a natural boundary on the unit circle. On the other hand, Froissart doublets due to numerical errors and/or external noise also appear around the unit circle in the Padé approximation. It is also shown that the residue calculus for the Padé approximated functions can be used to confirm the numerical accuracy of the Padé approximation and quasianalyticity of the random power series.

Analyticity of quantum states in one-dimensional tight-binding model

Analytical complexity of quantum wavefunction whose argument is extended into the complex plane provides an important information about the potentiality of manifesting complex quantum dynamics such as time-irreversibility, dissipation and so on. We examine Pade approximation and some complementary methods to investigate the complex-analytical properties of some complex quantum states such as impurity states, Anderson-localized states and localized states of Harper model. The impurity states can be characterized by simple poles of the Pade approximation, and the localized states of Anderson model and Harper model can be characterized by an accumulation of poles and zeros of the Pade approximated function along a critical border, which implies a natural boundary (NB). A complementary method based on shifting the expansion-center is used to confirm the existence of the NB numerically, and it is strongly suggested that the both Anderson-localized state and localized states of Harper model have NBs in the complex extension. Moreover, we discuss an interesting relationship between our research and the natural boundary problem of the potential function whose close connection to the localization problem was discovered quite recently by some mathematicians. In addition, we also examine the usefulness of the Pade approximation for numerically predicting the existence of NB by means of two typical examples, lacunary power series and random power series.

Estimates of a spectrum of the integral means for lacunary series

We obtain an upper estimate for the integral means spectrum of conformal mappings generated by the Hadamard lacunary power series with a lacunarity index 2.

Radial boundary values of lacunary power series

We strengthened MacLane's theorem concerning radial boundary values of lacunary power series.

On the path properties of a lacunary power series

A power series f(z) which converges in D = {|z| < 1} maps the radii [0, ζ) onto paths Γ(ζ), ζ ∈ T = ∂D. These are studied under several aspects in the case of the special lacunary series f(z) = z+ z2 + z4 + z8 + . . .. First the Γ(ζ) are considered as random functions on the probability space (T,B,mes /2π), where B is the σ-algebra of Borel sets and mes the Lebesgue measure. Then analytical properties of the Γ(ζ) are discussed which hold on subsets A of T with Hausdorff dimension 1 in spite of mesA = 0. Furthermore, estimates of the derivative of f and of the arc length of sections of the Γ(ζ) are given. Finally, these results are used to derive connections between the distribution of critical points of f and the overall behaviour of the paths.

SOME GENERALIZED LACUNARY POWER SERIES WITH ALGEBRAIC COEFFICIENTS FOR MAHLER'S $U-$NUMBER ARGUMENTS

In this work, we show that under certain conditions the values of some generalized lacunary power series with algebraic coefficients for Mahler's $U_{m}-$number arguments belong to either a certain algebraic number field or $\bigcup_{i=1}^{t} U_{i}$ in Mahler's classification of the complex numbers, where $t$ denotes a positive rational integer dependent on the coefficients of the given series and on the argument. Moreover, the obtained results are adapted to the field $\mathbb{Q}_{p}$ of $p-$adic numbers.

On the algebraic independence of certain Mahlerʼs U-numbers

Abstract We examine the algebraic independence over Q of certain Mahlerʼs U-numbers appearing as the values of certain generalized lacunary power series at non-zero algebraic number arguments.

Lacunary formal power series and the Stern–Brocot sequence

Let $F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-\lambda_n}$ be a real lacunary formal power series, where $\varepsilon_n = 0, 1$ and $\lambda_{n+1}/\lambda_n > 2$. It is known that the denominators $Q_n(X)$ of the convergents of its continued fraction expansion are polynomials with coefficients $0, \pm 1$, and that the number of nonzero terms in $Q_n(X)$ is the $n$th term of the Stern-Brocot sequence. We show that replacing the index $n$ by any 2-adic integer $\omega$ makes sense. We prove that $Q_{\omega}(X)$ is a polynomial if and only if $\omega \in {\mathbb Z}$. In all the other cases $Q_{\omega}(X)$ is an infinite formal power series, the algebraic properties of which we discuss in the special case $\lambda_n = 2^{n+1} - 1$.

On some lacunary power series with algebraic coefficients and Mahler’s U-numbers

In this work, we consider some generalized lacunary power series with algebraic coefficients from a certain algebraic number field K of degree m and show that under some conditions these series take values belonging to either the algebraic number field K or ⋃i=1mUi in Mahler’s classification of complex numbers for some Liouville number arguments.

On the existence of lacunary power series analytically continuable and bounded outside sectors

It is shown that for a set of positive integers Λ, a Müntz-type condition is necessary and sufficient for the existence of a lacunary power series f(z) = ∑ν∈Λ f ν/z ν which defines a function analytic and bounded in the complement of a sector of given angle and with vertex at the origin. The main tool consists in interpolation of the coefficients f ν of such functions by certain functions of exponential type in the right half-plane.

On the localization of singularities of Lacunar power series

The main result of this paper is Theorem 4 on the existence of singularities of Lacunary power series on prescribed open boundary arcs of the circle of convergence. The influence of the gaps on the length of these arcs is expressed in terms of newly introduced integral densities. Theorem 4 contains the known Fabry–Pólya theorem on gaps, describing the closed arcs having singularities, and suggests its extensions, using additional information on the coefficients of the power series. An essential step in its proof is Theorem 2, providing necessary and sufficient conditions (in terms of the so called “Coefficient Functions”) for the analytic continuation of power series across fixed open boundary arcs.