Sequence Of Rational Functions Research Articles

Exploring general Apéry limits via the Zudilin–Straub t-transform

Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that ‘tweaked’ the sum of the sth powers of the n-th row of Pascal's triangle, getting instead of sequences of numbers, sequences of rational functions, we do the same for general binomial coefficients sums, getting a practically unlimited supply of Apéry limits. While getting what we call ‘major Apéry miracles’, proving irrationality of the associated constants (i.e. the so-called Apéry limits) is very rare, we do get, every time, at least a ‘minor Apéry miracle’ where an explicit constant, defined as an (extremely slowly converging) limit of some explicit sequence, is expressed as an Apéry limit of some recurrence, with some initial conditions, thus enabling a very fast computation of that constant, with exponentially decaying error.

Apparent convergence of Padé approximants for the crossover line in finite density QCD

We propose a novel Bayesian method to analytically continue observables to real baryochemical potential $\mu_B$ in finite density QCD. Taylor coefficients at $\mu_B=0$ and data at imaginary chemical potential $\mu_B^I$ are treated on equal footing. We consider two different constructions for the Pad\'e approximants, the classical multipoint Pad\'e approximation and a mixed approximation that is a slight generalization of a recent idea in Pad\'e approximation theory. Approximants with spurious poles are excluded from the analysis. As an application, we perform a joint analysis of the available continuum extrapolated lattice data for both pseudocritical temperature $T_c$ at $\mu_B^I$ from the Wuppertal-Budapest Collaboration and Taylor coefficients $\kappa_2$ and $\kappa_4$ from the HotQCD Collaboration. An apparent convergence of $[p/p]$ and $[p/p+1]$ sequences of rational functions is observed with increasing $p.$ We present our extrapolation up to $\mu_B\approx 600$ MeV.

On the rapidly convergence in capacity of the sequence of holomorphic functions

In this paper, we are interested in finding sufficient conditions on a Borel set X lying either inside a bounded domain D ? Cn or in the boundary ?D so that if {rm}m?1 is a sequence of rational functions and {fm}m?1 is a sequence of bounded holomorphic functions on D with {fm-rm}m?1 is convergent fast enough to 0 in some sense on X then the convergence occurs on the whole domain D. The main result is strongly inspired by Theorem 3.6 in [3] whether the f fmg is a constant sequence.

Convergence of Sequences of Rational Functions on ℂ n $\mathbb {C}^{n}$

Let \(\mathcal A:=\{\chi _{m}\}_{m \ge 1}\) be a sequence of continuous, real valued functions defined on \([0, \infty )\). We say that a sequence {fm}m ≥ 1 of measurable functions defined on \(X \subset D \subset \mathbb C^{n}\) is convergent in capacity (relative to D) with respect to the weight sequence \(\mathcal A\) to a function f if χm(|f − fm|2) converges to 0 in capacity on X. We are interested in finding conditions (on \(\mathcal A\)) so that every sequence {rm}m ≥ 1 of rational functions on \(\mathbb {C}^{n}\) converges in capacity with respect to \(\mathcal A\) to a holomorphic function f defined on a bounded domain \(D \subset \mathbb {C}^{n}\) provided that the convergence holds only pointwise on a small subset of D.

Vitali's theorem without uniform boundedness

Let {fm}m≥1 be a sequence of holomorphic functions defined on a bounded domain D ⊂ Cn or a sequence of rational functions (1 ≤ deg rm ≤ m) defined on Cn. We are interested infinding sufficient conditions to ensure the convergence of {fm}m≥1 on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence.

A golden iterated map number system: Results and conjectures

We construct a number system for representing numbers in [0, 1] that is based on iterations of an asymmetric tent map that incorporates the golden ratio. It is already known that in this system a number has a periodic representation if and only if it lies in [Formula: see text]. We investigate other aspects of this system such as non-uniquely representable numbers, an inherent semigroup structure, connections with Wythoff's game, related sequences of rational functions, and its connection with an iterative scheme reminiscent of paper folding analysis. Many details of the above-mentioned connections are conjectural.

Discrepancy Theorems for the Zero Distribution of Rational Functions

In the theory of zero distribution of polynomials, there are two branches that are closely related to each other. One branch deals with the conditions under which the normalized zero counting measure of a sequence of polynomials converges in the weak-star sense to the equilibrium measure of a Jordan curve. The other consists of discrepancy theorems, where the maximum deviation of these two measures is estimated. There are also theorems on weak-star convergence of the zero distribution of special sequences of rational functions to a measure that depends on the poles of these functions. In this paper, we give appropriate discrepancy theorems for rational functions. These new results also imply theorems about the weak-star convergence of the zero distribution of more general sequences of rational functions.

An Exact Series Representation for the EM Field From a Circular Loop Antenna on a Lossy Half-Space

This letter introduces an analytical approach for the evaluation of the complete integral expressions for the time-harmonic electromagnetic (EM) field components generated by an arbitrarily large thin-wire circular loop antenna lying on the surface of a flat and homogeneous half-space. The method consists of casting the integrands of the field integrals into forms in which the square-root terms, originating branch cuts, are replaced with their rational function representations via Newton's method. Explicit expressions for the fields are then derived by applying the residue theorem. The proposed procedure is exact since previous authors have shown that the sequence of rational functions generated by Newton's method converges quadratically to the square root, even if the independent variable is a complex number. Numerical simulations are performed to confirm the validity of the developed formulation.

Wavelet-type frames for an interval

We construct a sequence of rational functions, which will be called disc wavelets, parametrized by points in the unit disc λk,n(γ)=(1−γ2n)(ωn)k,n∈N, 0≤k<n, where ωn=e2πi/n is the primitive nth root of unity, and obtain a full description, in terms of the parameter γ, of the sets {λk,n(γ)} yielding frames for the space L2(−1,1). This is done using a disc version of the Bargmann transform, which maps L2(−1,1) to the classical spaces of the unit disc (Bergman, Hardy, Dirichlet) and applying Seip’s description of sampling sets in the unit disc. We also describe how to replace the points λk,n(γ) by the orbit of a Fuchsian group Γ, observing that Seip’s density of the orbit of a point contained in the fundamental region of a Fuchsian group Γ is equal to m0, where m0 is the smallest number among the weights of the automorphic forms with a zero in the fundamental region of the group Γ.

Erdős-Turʼan-Type Discrepancy for the Zeros of Rational Functions

Abstract In 1950 P. Erdős and P. Turan published a discrepancy theorem for the zeros of a polynomial. Therein, the maximum deviation of the normalized zero counting measure from the equilibrium measure of the unit circle is estimated. Many other discrepancy theorems and related propositions about weak-star-convergence of the zero distribution of a sequence of polynomials were proved during the last decades. For several years the weak-star-convergence of the zero distribution of a sequence of rational functions is also studied. The main result of this paper is a discrepancy theorem for the zero distribution of a rational function which generalizes and sharpens previous propositions about weak-star-convergence of the zero counting measure of sequences of rational functions and known discrepancy theorems for polynomials.

Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

For any rational functions with complex coefficients $A(z), B(z)$ and $C(z)$, where $A(z)$, $C(z)$ are not identically zero, we consider the sequence of rational functions $H_{m}(z)$ with generating function $\sum H_{m}(z)t^{m}=1/(A(z)t^{2}+B(z)t+C(z))$. We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$, as $n\rightarrow\infty$, counting multiplicities, on certain closed subarcs $J$ of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if $J$ contains the endpoints of $\mathcal{C}$.

An asymptotically consistent approximant method with application to soft- and hard-sphere fluids

A modified Padé approximant is used to construct an equation of state, which has the same large-density asymptotic behavior as the model fluid being described, while still retaining the low-density behavior of the virial equation of state (virial series). Within this framework, all sequences of rational functions that are analytic in the physical domain converge to the correct behavior at the same rate, eliminating the ambiguity of choosing the correct form of Padé approximant. The method is applied to fluids composed of "soft" spherical particles with separation distance r interacting through an inverse-power pair potential, φ = ε(σ∕r)(n), where ε and σ are model parameters and n is the "hardness" of the spheres. For n < 9, the approximants provide a significant improvement over the 8-term virial series, when compared against molecular simulation data. For n ≥ 9, both the approximants and the 8-term virial series give an accurate description of the fluid behavior, when compared with simulation data. When taking the limit as n → ∞, an equation of state for hard spheres is obtained, which is closer to simulation data than the 10-term virial series for hard spheres, and is comparable in accuracy to other recently proposed equations of state. By applying a least square fit to the approximants, we obtain a general and accurate soft-sphere equation of state as a function of n, valid over the full range of density in the fluid phase.

Partial Fraction Expansions for Newton's and Halley's Iterations for Square Roots

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

The Degree of a $q$-Holonomic Sequence is a Quadratic Quasi-Polynomial

A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial, and that the leading term satisfies a linear recursion relation with constant coefficients. Our proof uses differential Galois theory (adapting proofs regarding holonomic $D$-modules to the case of $q$-holonomic $D$-modules) combined with the Lech-Mahler-Skolem theorem from number theory. En route, we use the Newton polygon of a linear $q$-difference equation, and introduce the notion of regular-singular $q$-difference equation and a WKB basis of solutions of a linear $q$-difference equation at $q=0$. We then use the Skolem-Mahler-Lech theorem to study the vanishing of their leading term. Unlike the case of $q=1$, there are no analytic problems regarding convergence of the WKB solutions. Our proofs are constructive, and they are illustrated by an explicit example.

A survey of results on the q-Bernstein polynomials

It is now nearly a century since S. N. Bernstein introduced his well-known polynomials. This paper is concerned with generalizations of the Bernstein polynomials, mainly with the so called q-Bernstein polynomials. These are due to the author of this paper and are based on the q integers. They reduce to the Bernstein polynomials when we put q = I and share the shape-preserving properties of the Bernstein polynomials when q ∈ (0, 1). This paper also describes another earlier generalization of the Bernstein polynomials, a sequence of rational functions that are also based on the q-integers, proposed by A. Lupas, and two even earlier generalizations due to D. D. Stancu. The present author summarizes various results, due to a number of authors, that are concerned with the q-Bernstein polynomials and with Stancu's two generalizations.

Orthogonal rational functions with complex poles: The Favard theorem

Let { φ n } be a sequence of rational functions with arbitrary complex poles, generated by a certain three-term recurrence relation. In this paper we show that under some mild conditions, the rational functions φ n form an orthonormal system with respect to a Hermitian positive-definite inner product.

Recurrence and asymptotics for orthonormal rational functions on an interval

Let μ be a positive bounded Borel measure on a subset I of the real line and = {α 1 , …, α n } a sequence of arbitrary ‘complex’ poles outside I. Suppose {φ 1 , …, φ n } is the sequence of rational functions with poles in orthonormal on I with respect to μ. First, we are concerned with reducing the number of different coefficients in the three-term recurrence relation satisfied by these orthonormal rational functions. Next, we consider the case in which I = [- 1, 1] and μ satisfies the Erdos-Turan condition μ′ > 0 a.e. on I (where μ′ is the Radon-Nikodym derivative of the measure μ with respect to the Lebesgue measure) to discuss the convergence of φ n +1 (x)/φ n (x) as n tends to infinity and to derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation. Finally, we give a strong convergence result for φ n (x) under the more restrictive condition that μ satisfies the Szegő condition (1 - x 2 ) -1/2 log μ′(x) ∈ L 1 ([- 1, 1]).

SCHUR–NEVANLINNA SEQUENCES OF RATIONAL FUNCTIONS

AbstractWe study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

Distribution analogue of the Tumarkin result

We give a distribution analogue of the Tumarkin result that concerns approximation of some functions by sequence of rational functions with given poles. AMS Mathematics Subject Classification (2000): 46F20, 30E25, 32A35.

Some inequalities for alternating Kurepa's function

In this paper we consider alternating Kurepa’s function A(z). We give some recurrent relations for alternating Kurepa’s function via appropriate sequences of rational functions and gamma function. Also we give some inequalities for the real part of alternating Kurepa’s function A(x) for values of argument x> 2. The obtained results are analogous to results from the author’s paper [5].