Special properties of Hurwitz series rings

In this paper, we study some properties of the Hurwitz series ring HR (resp. Hurwitz polynomial ring hR), such as the flatness or the faithful flatness of HR / (f) (resp. hR / (f)), the strongly Hopfian property and the radical property of HR (resp. hR). We give some sufficient and necessary conditions for HR / (f) (resp. hR / (f)) to be flat or faithful flat. We also prove that the strongly Hopfian property transfer between R and HR (resp. hR), and some radicals of HR can be determined in terms of those of R, in case R satisfies some additional conditions.     

Basic properties of Hurwitz series rings

This paper studies Hurwitz series rings and Hurwitz polynomial rings. Especially we characterize nilpotent elements, units, McCoy theorem, Noetherian, nonnil-Noetherian and S-Noetherian properties, PS and presimplifiable properties.     

Some combinatorial properties of the Hurwitz series ring

We study some properties and perspectives of the Hurwitz series ring \(H_R[[t]]\), for an integral domain R, with multiplicative identity and zero characteristic. Specifically, we provide a closed form for the invertible elements by means of the complete ordinary Bell polynomials, we highlight some connections with well–known transforms of sequences, and we see that the Stirling transforms are automorphisms of \(H_R[[t]]\). Moreover, we focus the attention on some special subgroups studying their properties. Finally, we introduce a new transform of sequences that allows to see one of this subgroup as an ultrametric dynamic space.     

On some series representations of the Hurwitz zeta function

A variety of infinite series representations for the Hurwitz zeta function are obtained. Particular cases recover known results, while others are new. Specialization of the series representations apply to the Riemann zeta function, leading to additional results. The method is briefly extended to the Lerch zeta function. Most of the series representations exhibit fast convergence, making them attractive for the computation of special functions and fundamental constants.     

On some slowly convergent series involving the Hurwitz zeta-function

We shall extract the essence of the Adamchik–Srivastava generating function method (Analysis (Munich) 18 (1998) 131) by proving the most far-reaching Ramanujan–Yoshimoto formula and by showing that some of the results stated in Srivastava and Choi (Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001) are simple consequences of the above-mentioned formula.     

Pointwise universal trigonometric series

A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to f(z) pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as |n|→∞ for some ε>0, then the series S_a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S_a with as |n|→∞.     

Ramanujan's Eisenstein series and new hypergeometric-like series for

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/π².     

Almost poised basic hypergeometric series

Given a basic hypergeometric series with numerator parametersa ₁,a ₂, ...,a _r and denominator parametersb ₂, ...,b _r, we say it isalmost poised ifb _i, =a ₁ q ^δ,i a _i,δ_i = 0, 1 or 2, for 2 ≤i ≤r. Identities are given for almost poised series withr = 3 andr = 5 when a₁, =q ⁻²ⁿ. Partially supported by N.S.F. Grant No. DMS-8521580.     

On a particular case of series

We present a general formula for a triple product involving four real numbers. As a particular case, we get the sum of a triple product of four odd integers. Some interesting results are recovered. We derive a general formula for more than four odd numbers.     

AbsoluteN qα -summability of the series conjugate to a Fourier series

The object of the paper is to study the absolute N_qα-summability of the series conjugate to a Fourier series, generalising a known result.     

On series by Haar system

The paper considers the series by Haar system \(\sum\limits_{n = 1}^\infty {a_n \chi _n (x)} \), satisfying the conditions \(\sum\limits_{n = 1}^\infty {a_n^2 \chi _n^2 (x)} = \infty \) and a _n χ _n (x) → 0 almost everywhere. Some theorems about correcting a function on sets of arbitrarily small measures are proved.     

Convergence acceleration of orthogonal series

The aim of this paper is to verify efficiency of two acceleration methods for orthogonal series (more strictly, for series defined at the beginning of Section 1). These methods are quite different although they use the same transform of such a series given there. The first method (Section 3) has some features common with Levin’s and Weniger’s methods. It may be profitably used in numerical calculations for a vast class of series. The second one (Sections 4 and 5) is somewhat similar to the Euler–Knopp transform of power series. Also this method is numerically realizable but more important is that for a narrower class of series, including some ones having applications in physics, it gives explicit analytic formulae of their transform.      

On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type

In this paper, we consider certain analogues of Eisenstein seriesinvolving hyperbolic sine and cosine functions. Using a resultof Hurwitz, we prove some evaluation formulas for these seriesat positive integers. We can regard these formulas as analoguesof the famous formulas for certain series given by Cauchy, Ramanujan,Berndt, and so on, as well as those for the Eisenstein seriesgiven by Hurwitz.     

A note on rearrangement of Fourier series

We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge.     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

Cross-validation and the smoothing of orthogonal series density estimators

We describe a class of smoothed orthogonal series density estimates, including the classical sequential-series introduced by [6], Soviet Math. Dokl. 3 1559–1562) and [16], Ann. Math. Statist. 38 1261–1265), and [23], Ann. Statist 9 146–156) two-parameter smoothing. The Bowman-Rudemo method of least-squares cross-validation (1982, Manchester-Sheffield School of Probability and Statistics Research Report 84/AWB/1; 1984, Biometrika 71 353–360; [14], Scand. J. Statist. 9 65–78), is suggested as a practical way of choosing smoothing parameters automatically. Using techniques of [18], Ann. Statist. 12 1285–1297), that method is shown to perform asymptotically optimally in the case of cosine and Hermite series estimators. The same argument may be used for other types of series.     

Correspondence among Eisenstein series of weight 2

There is a known correspondence among modular forms, Jacobi forms and Siegel modular forms of genus 2. In this paper we show this correspondence can be extended to non-holomorphic Eisenstein series, in particular, among , E2,1(τ,z;δ;0), and .