scholarly journals New integral and series representations of the generalized Mathieu series

2008 ◽  
Vol 2 (2) ◽  
pp. 205-212 ◽  
Author(s):  
Zivorad Tomovski
Keyword(s):  
Integral Representations ◽  
Type Series ◽  
Fourier Sine ◽  
Series Representations ◽  
Mathieu Series

By using some recently investigated fourier sine integral representations for the Mathieu type series (see [4]), new integral and series representations are derived here for certain general families of Mathieu type series.

scholarly journals On some properties of the Lüroth-type alternating series representations for real numbers

2001 ◽  
Vol 28 (6) ◽  
pp. 367-373 ◽  
Author(s):  
C. Ganatsiou
Keyword(s):  
Limit Theorems ◽  
Asymptotic Density ◽  
Real Numbers ◽  
Type Series ◽  
Distribution Of The Maximum ◽  
Series Representations

We investigate some properties connected with the alternating Lüroth-type series representations for real numbers, in terms of the integer digits involved. In particular, we establish the analogous concept of the asymptotic density and the distribution of the maximum of the firstndenominators, by applying appropriate limit theorems.

scholarly journals Some new integral representations of generalized Mathieu series and alternating Mathieu series

2010 ◽  
Vol 41 (4) ◽  
pp. 303-312
Author(s):  
Zivorad Tomovski
Keyword(s):  
Integral Representations ◽  
Alpha Beta ◽  
Mathieu Series

The main purpose of this paper is to present a number of new integral representations for the familiar Mathieu series $S_\mu^{(\alpha,\beta )}(r;\{a_k\}_{k=1}^\infty)(r\in R$, $\alpha$, $\beta$, $\mu$, $\{a_k\}_{k=1}^\infty\in R^+)$ [12]  as well as for its alternating version [8,16] when $a_k=\{k^p\}_{k=1}^\infty$, $a_k=\{(k!)^p\}_{k=1}^\infty$, $a_k=\{(\ln k!)^p\}_{k=1}^\infty$ with $p=\gamma$, $\gamma(\mu\alpha-\beta)>1$ and $p=\frac q\alpha$, $\mu -\frac \beta\alpha>q^{-1}$, $q\in N$. 

Series representing transcendental numbers that are not U-numbers

2015 ◽  
Vol 11 (03) ◽  
pp. 869-892
Author(s):  
Emre Alkan
Keyword(s):  
Rational Functions ◽  
Rational Numbers ◽  
Upper Bounds ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Elementary Functions ◽  
Linear Combinations ◽  
Type Series ◽  
Transcendental Numbers ◽  
Mathematical Constants

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures.

scholarly journals Line antiderivations over local fields and their applications

2005 ◽  
Vol 2005 (2) ◽  
pp. 263-309 ◽  
Author(s):  
S. V. Ludkovsky
Keyword(s):  
Laurent Series ◽  
Differential Forms ◽  
Holomorphic Functions ◽  
Integral Representations ◽  
Local Fields ◽  
Cauchy Type ◽  
Series Representations

A non-Archimedean antiderivational line analog of the Cauchy-type line integration is defined and investigated over local fields. Classes of non-Archimedean holomorphic functions are defined and studied. Residues of functions are studied; Laurent series representations are described. Moreover, non-Archimedean antiderivational analogs of integral representations of functions and differential forms such as the Cauchy-Green, Martinelli-Bochner, Leray, Koppelman, and Koppelman-Leray formulas are investigated. Applications to manifold and operator theories are studied.

Generalized Mathieu Series, Associated Integral Representations and Convergence

2021 ◽  
pp. 1-39
Author(s):  
Živorad Tomovski ◽  
Delčo Leškovski ◽  
Stefan Gerhold
Keyword(s):  
Integral Representations ◽  
Mathieu Series

scholarly journals Integral and series representations of q-polynomials and functions: Part I

2018 ◽  
Vol 16 (02) ◽  
pp. 209-281 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Ruiming Zhang
Keyword(s):  
Integral Representation ◽  
Orthogonal Polynomials ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Hermite Polynomials ◽  
Integral Representations ◽  
Moment Problems ◽  
Exponential Functions ◽  
Series Representations ◽  
Indeterminate Moment Problems

By applying an integral representation for [Formula: see text], we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of [Formula: see text]-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include [Formula: see text]-Bessel functions, the Ramanujan function, Stieltjes–Wigert polynomials, [Formula: see text]-Hermite and [Formula: see text]-Hermite polynomials, and the [Formula: see text]-exponential functions [Formula: see text], [Formula: see text] and [Formula: see text]. Their representations are in turn used to derive many new identities involving [Formula: see text]-functions and polynomials. In this paper, we also present contour integral representations for the above mentioned functions and polynomials.

Integral representations and integral transforms of some families of Mathieu type series

2008 ◽  
Vol 19 (7) ◽  
pp. 481-495 ◽  
Author(s):  
Neven Elezović ◽  
H. M. Srivastava ◽  
živorad Tomovski
Keyword(s):  
Integral Transforms ◽  
Integral Representations ◽  
Type Series
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