scholarly journals On Some Classes of Series Representations for $1/\pi$ and $\pi^2$

2021 ◽  
pp. 176-184
Author(s):  
Hakan KÜÇÜK ◽  
Sezer SORGUN
Keyword(s):  
Series Representations

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals A dominated convergence theorem for Eisenstein series

2021 ◽  
Author(s):  
Johann Franke
Keyword(s):  
Fourier Series ◽  
Dirichlet Series ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Convergence Theorem ◽  
Rational Functions ◽  
New Approach ◽  
Series Representations ◽  
Dominated Convergence ◽  
Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

On a symbolic algebra for Hencky-Prandtl nets

1983 ◽  
Vol 94 (2) ◽  
pp. 341-350
Author(s):  
R. Hill
Keyword(s):  
General Solution ◽  
Boundary Value Problems ◽  
Classical Theory ◽  
Systematic Approach ◽  
Standard Technique ◽  
Field Equations ◽  
Infinite Matrices ◽  
Symbolic Algebra ◽  
Series Representations ◽  
Plane Deformations

AbstractIn the classical theory of plane deformations in isotropic plastic media, the field equations are hyperbolic and the orthogonal families of characteristics are known as Hencky-Prandtl nets. Their distinctive geometry has been given symbolic expression by Collins (1968), in an algebra of infinite matrices associated with canonical series representations of the general solution. This has become the standard technique when investigating boundary-value problems, both analytically and numerically. The basic framework of the algebra is here reorganized and developed. A systematic approach then leads to new identities which are shown to be fundamental in the algebraic hierarchy.

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.

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