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It is well-known, as follows from the Stirling’s approximation n!∼2πn(n/e)nn! \sim \sqrt {2\pi n{{\left( {n/e} \right)}^n}}, that n!/n→1/en\root n \of {n!/n \to 1/e}. A generalization of this limit is (11s· 22s· · · nns)1/ns+1 · n−1/(s+1) → e−1/(s+1)2 which was established by N. Schaumberger in 1989 (see [8]). The aim of this work is to establish a new generalization that is in fact an improvement of Schaumberger’s formula for a general sequence An of positive real numbers. All of the results are applied to some well-known sequences in mathematics, for example, for the prime numbers sequence and the sequence of perfect powers.

Let pn be the n th prime number. In this note, we study strictly increasing sequences of positive integers An such that the limit limn→∞ (A1A2 · · · An)1/pn = e holds. This limit formula is in fact a generalization of some previously known results. Furthermore, some other generalizations are established.

In this paper we present a class of convergent infinite products ∏ k = 1 ∞ H k = lim k → ∞ (H k H k − 1 ⋯ H 1) of 3 × 3 hollow matrices of the form H k := [ 0 a k a k b k 0 b k c k c k 0 ] , (0 < a k , b k , c k < 1 / 2). These sequences of hollow matrices are obtained from the Ando-Li-Mathias matrix geometric mean of 2 × 2 positive definite matrices. We further provide a three dimensional analytic manifold and an analytic self-map realizing such sequences. [ABSTRACT FROM AUTHOR]

It is well-known, as follows from the Stirling's approximation n ! ∼ 2 π n (n / e) n , that n ! / n → 1 / e n . A generalization of this limit is (11s· 22s· · · nns)1/ns+1· n−1/(s+1)→ e−1/(s+1)2 which was established by N. Schaumberger in 1989 (see [8]). The aim of this work is to establish a new generalization that is in fact an improvement of Schaumberger's formula for a general sequence An of positive real numbers. All of the results are applied to some well-known sequences in mathematics, for example, for the prime numbers sequence and the sequence of perfect powers. [ABSTRACT FROM AUTHOR]

The aim of this study was to present a simple method for finding the asymptotic relations for products of elements of some positive real sequences. The main reason to carry out this study was the result obtained by Alzer and Sandor concerning an estimation of a sequence of the product of the first k primes.

In this paper, we discuss the application of the author's d ~ (m) transformation to accelerate the convergence of infinite series ∑ n = 1 ∞ a n when the terms an have asymptotic expansions that can be expressed in the form a n ∼ (n !) s / m exp ∑ i = 0 m q i n i / m ∑ i = 0 ∞ w i n γ − i / m as n → ∞ , s integer. We discuss the implementation of the d ~ (m) transformation via the recursive W-algorithm of the author. We show how to apply this transformation and how to assess in a reliable way the accuracies of the approximations it produces, whether the series converge or they diverge. We classify the different cases that exhibit unique numerical stability issues in floating-point arithmetic. We show that the d ~ (m) transformation can also be used efficiently to accelerate the convergence of infinite products ∏ n = 1 ∞ (1 + v n) , where v n ∼ ∑ i = 0 ∞ e i n − t / m − i / m as n → ∞ , t ≥ m + 1 an integer. Finally, we give several numerical examples that attest the high efficiency of the d ~ (m) transformation for the different cases. [ABSTRACT FROM AUTHOR]

This note summarizes results that were obtained by the author in his habilitation thesis concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions u of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a line (following Hitchin and Bobenko ) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data of such Cauchy data is answered. Finally a Jacobi variety for the spectral curve is constructed, and this is used to study the asymptotic behavior of the spectral data corresponding to actual simply periodic solutions of the sinh-Gordon equation on strips of positive height. [ABSTRACT FROM AUTHOR]

In this paper, we provide some new continued fraction approximation and inequalities of the Somos quadratic recurrence constant, using its relation with the generalized Euler constant. [ABSTRACT FROM AUTHOR]

The theory of Barnes beta probability distributions is advanced and related to the Riemann xi function. The scaling invariance, multiplication formula, and Shintani factorization of Barnes multiple gamma functions are reviewed using the approach of Ruijsenaars and shown to imply novel properties of Barnes beta distributions. The applications are given to the meromorphic extension of the Selberg integral as a function of its dimension and the scaling invariance of the underlying probability distribution. This probability distribution in the critical case is described and conjectured to be the distribution of the derivative martingale. The Jacobi triple product is interpreted probabilistically resulting in an approximation of the Riemann xi function by the Mellin transform of the logarithm of a limit of Barnes beta distributions. [ABSTRACT FROM AUTHOR]

We give the first (to the best of our knowledge) closed-form expression for sin(p/7) in terms of ordinary arithmetical operations and root extractions on real rational numbers. [ABSTRACT FROM PUBLISHER]

In this paper, we obtain some new continued fraction approximations and inequalities of the Somos' quadratic recurrence constant, using its relation with the generalized Euler's constant. [ABSTRACT FROM AUTHOR]

Using the system theory notion of state-space realization of matrix-valued rational functions, we describe the Ruelle operator associated with wavelet filters. The resulting realization of infinite products of rational functions have the following four features: (1) It is defined in an infinite-dimensional complex domain. (2) Starting with a realization of a single rational matrix-function $$M$$ , we show that a resulting infinite product realization obtained from $$M$$ takes the form of an (infinite-dimensional) Toeplitz operator with the symbol that is a reflection of the initial realization for $$M$$ . (3) Starting with a subclass of rational matrix functions, including scalar-valued ones corresponding to low-pass wavelet filters, we obtain the corresponding infinite products that realize the Fourier transforms of generators of $$\mathbf L_2(\mathbb R)$$ wavelets. (4) We use both the realizations for $$M$$ and the corresponding infinite product to obtain a matrix representation of the Ruelle-transfer operators used in wavelet theory. By 'matrix representation' we refer to the slanted (and sparse) matrix which realizes the Ruelle-transfer operator under consideration. [ABSTRACT FROM AUTHOR]

Let D be an open disk of radius ≤1 in $\mathbb{C}$, and let ( ϵ) be a sequence of ±1. We prove that for every analytic function $f: D \to \mathbb{C}$ without zeros in D, there exists a unique sequence ( α) of complex numbers such that $f(z) = f(0)\prod_{n=1}^{\infty} (1+\epsilon_{n}z^{n})^{\alpha_{n}}$ for every z∈ D. From this representation we obtain a numerical method for calculating products of the form ∏ f(1/ p) provided f(0)=1 and f′(0)=0; our method generalizes a well-known method of Pieter Moree. We illustrate this method on a constant of Ramanujan $\pi^{-1/2}\prod _{p~\text{prime}} \sqrt{p^{2}-p}\ln(p/(p-1))$. From the properties of the exponents α, we obtain a proof of the following congruences, which have been the subject of several recent publications motivated by some questions of Arnold: for every n× n integral matrix A, every prime number p, and every positive integer k we have $\operatorname{tr} A^{p^{k}} \equiv\operatorname{tr} A^{p^{k-1}} { \hbox {\rm { (mod\ $p^{k}$) }}}$. [ABSTRACT FROM AUTHOR]

In this paper we develop a novel approach to construct non-stationary subdivision schemes with a tension control parameter which can reproduce functions in a finite-dimensional subspace of exponential polynomials. The construction process is mainly implemented by solving linear systems for primal and dual subdivision schemes respectively, which are based on different parameterizations. We give the theoretical basis for the existence, uniqueness, and refinement rules of schemes proposed in this paper. The convergence and smoothness of the schemes are analyzed as well. Moreover, conics reproducing schemes are analyzed based on our theory, and a new idea that the tensor parameter ωk of the schemes can be adjusted for conics generation is proposed. [ABSTRACT FROM AUTHOR]

Two subdivision schemes with Hermite data on ℤ are studied. These schemes use 2 or 7 parameters respectively depending on whether Hermite data involve only first derivatives or include second derivatives. For a large region in the parameter space, the schemes are convergent in the space of Schwartz distributions. The Fourier transform of any interpolating function can be computed through products of matrices of order 2 or 3. The Fourier transform is related to a specific system of functional equations whose analytic solution is unique except for a multiplicative constant. The main arguments for these results come from Paley-Wiener-Schwartz theorem on the characterization of the Fourier transforms of distributions with compact support and a theorem of Artzrouni about convergent products of matrices. [ABSTRACT FROM AUTHOR]