Application of the absolute Euler method to some series related to Fourier series and its conjugate series

We study the absolute Euler summability problem of some series associated with Fourier series and its conjugate series generalizing some known results in the literature. Also, it is shown that absolute Euler summability of rth derived Fourier series and rth derived conjugate series can be ensured under local conditions.     

On Taylor series and Kapteyn series of the first and second type

We study the relation between the coefficients of Taylor series and Kapteyn series representing the same function. We compute explicit formulas for expressing one in terms of the other and give examples to illustrate our method.     

Abstract theory of universal series and an application to Dirichlet series

We present an abstract theory of universal series; in particular, we give a necessary and sufficient condition for the existence of universal series of a certain type. Most of the known results can be proved or strengthened by using this condition. We also obtain new results, for example, related to universal Dirichlet series. To cite this article: V. Nestoridis, C. Papadimitropoulos, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Eisenstein series and convolution sums

The Ramanujan Journal - We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the...     

Multiple Eisenstein series and multiple cotangent functions

We construct the multiple Eisenstein series and we show a relation to the multiple cotangent function. We calculate a limit value of the multiple Eisenstein series.     

On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight

We give a Katok-Sarnak type correspondence for Niebur type Poincaré series and Eisenstein series on the three-dimensional hyperbolic space.     

Riordan arrays and generalized Lagrange series

The theory of Riordan arrays studies the properties of formal power series and their sequences. The notion of generalized Lagrange series proposed in the present paper is intended to fill the gap in the methodology of this theory. Generalized Lagrange series appear in it implicitly, as various equalities. No special notation is provided for these series, although particular cases of these series are generalized binomial and generalized exponential series. We give the definition of generalized Lagrange series and study their relationship with ordinary Riordan arrays and, separately, with Riordan exponential arrays.     

Algorithmically random series and Brownian motion

We consider some random series parametrised by Martin-Löf random sequences. The simplest case is that of Rademacher series, independent of a time parameter. This is then extended to the case of Fourier series on the circle with Rademacher coefficients. Finally, a specific Fourier series which has coefficients determined by a computable function is shown to converge to an algorithmically random Brownian motion.     

Special series in Laguerre polynomials and their approximation properties

We consider special series in the classical Laguerre polynomials coinciding in particular cases with the mixed series associated to Laguerre polynomials, introduced by the author previously, as well as Fourier–Sobolev series in Sobolev orthogonal Laguerre polynomials. We address the questions of uniform convergence of these series on a finite segment of the positive half-axis. We study the approximation properties of partial sums of special series on the positive half-axis, with particular attention paid to estimating their Lebesgue function.     

Macdonald difference operators and Harish-Chandra series

We analyse the centralizer of the Macdonald difference operatorin an appropriate algebra of Weyl group invariant differenceoperators. We show that it coincides with Cherednik's commutingalgebra of difference operators via an analog of the Harish-Chandraisomorphism. Analogs of Harish-Chandra series are defined andrealized as solutions to the system of basic hypergeometricdifference equations associated to the centralizer algebra.These Harish-Chandra series are then related to both Macdonaldpolynomials and Chalykh's Baker–Akhiezer functions.     

Radial limits of partial theta and similar series

We study unilateral series in a single variable q where its exponent is an unbounded increasing function and the coefficients are periodic. Such series converge inside the unit disk. Quadratic polynomials in the exponent correspond to partial theta series. We compute limits of those series as the variable tends radially to a root of unity. The proofs use ideas from the q-integral and are elementary.     

Overpartition pairs and two classes of basic hypergeometric series

We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Durfee dissection, as well as certain lattice paths. When further specialized, the series become infinite products, leading to numerous identities for partitions, overpartitions, and overpartition pairs.     

Twisted rational functions and series

We study rings of twisted rational functions and twisted Laurent series over simple artinian rings. We determine the centers of such rings and investigate the structure of subalgebras of these rings. We extend to infinite dimensional division rings and to simple artinian rings results proven in a previous paper for finite dimensional division algebras. We investigate “Galois subrings” of rings of fractions.     

Approximation by fuzzy B-spline series

We study properties concerning approximation of fuzzy-number-valued functions by fuzzy B-spline series. Error bounds in approximation by fuzzy B-spine series are obtained in terms of the modulus of continuity. Particularly simple error bounds are obtained for fuzzy splines of Schoenberg type. We compare fuzzy B-spline series with existing fuzzy concepts of splines.     

The renormalizing series of some integral equations

We consider integral equations for which the perturbation expansion gives a power series in a parameter h whose coefficients are divergent integrals. We eliminate the divergent integrals by introducing a renormalizing Z(t, h) series in the minimal subtraction scheme. We investigate the convergence of the formal Z series in relation to the kernels of the integral equations. We find a relation of the renormalizing series to the Lagrange inversion series and also some other relations.     

Interlacing of Hurwitz series

In this paper, we introduce the notion of interlacing of Hurwitz series. We begin by reviewing some important properties of the ring of Hurwitz series over a commutative ring A of arbitrary characteristic, and we introduce and investigate properties of the maps exp and log. We show that solutions of linear homogeneous differential equations with constant coefficients from the ring A can be described simply as interlacings of solutions of a first order system of differential equations. We give several examples to illustrate this result, and we conclude by defining and investigating properties of trigonometric functions using interlacings of Hurwitz series.     

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.     

Unconditionally Cauchy series and Cesàro summability

In this paper we obtain new characterizations of weakly unconditionally Cauchy series and unconditionally convergent series through Cesàro summability. We study new spaces associated to a series in a Banach space; as a consequence, new characterizations of complete and barrelled normed spaces are proved.     

On uniqueness sets for multiple Walsh series

We study uniqueness sets for multiple Walsh series under ρ-regular (or bounded) convergence in rectangles. We prove that a countable set is a uniqueness set for such a series under this convergence. We construct a class of perfect uniqueness sets for multiple Walsh series under this convergence. We show that the notion of index of a perfect set does not solve the problem of whether this set belongs to the class of uniqueness sets. We note that the results of this paper remain valid for several rearranged multiple Walsh series.