scholarly journals The Markov-WZ Method

10.37236/1806 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Mohamud Mohammed ◽  
Doron Zeilberger
Keyword(s):  
Infinite Series ◽  
Ad Hoc ◽  
Convergence Acceleration ◽  
Fast Convergence ◽  
New Class ◽  
Wz Theory ◽  
Hypergeometric Type ◽  
Wz Method

Andrei Markov's 1890 method for convergence-acceleration of series bears an amazing resemblance to WZ theory, as was recently pointed out by M. Kondratieva and S. Sadov. But Markov did not have Gosper and Zeilberger's algorithms, and even if he did, he wouldn't have had a computer to run them on. Nevertheless, his beautiful ad-hoc method, when coupled with WZ theory and Gosper's algorithm, leads to a new class of identities and very fast convergence-acceleration formulas that can be applied to any infinite series of hypergeometric type.

scholarly journals On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Arnak Poghosyan
Keyword(s):  
Trigonometric Polynomial ◽  
Numerical Experiments ◽  
Polynomial Interpolation ◽  
Convergence Acceleration ◽  
Fast Convergence ◽  
Exact Constants

We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice.

Binomial sums involving harmonic numbers

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Marian Genčev
Keyword(s):  
Infinite Series ◽  
Rational Functions ◽  
Harmonic Numbers ◽  
Binomial Type ◽  
Hypergeometric Type ◽  
Number Π ◽  
Binomial Sums ◽  
Generalized Harmonic Numbers

AbstractThis paper develops the approach to the evaluation of a class of infinite series that involve special products of binomial type, generalized harmonic numbers of order 1 and rational functions. We give new summation results for certain infinite series of non-hypergeometric type. New formulas for the number π are included.

scholarly journals Accelerated series for universal constants, by the WZ method

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Herbert S. Wilf
Keyword(s):  
Wz Theory ◽  
International Audience ◽  
Wz Method

International audience In this paper, the author presents a method, based on WZ theory, for finding rapidly converging series for universal constants. This method is analogous but different from Amdeberhan and Zeilberger's method.

scholarly journals Analogues of the general theta transformation formula

2013 ◽  
Vol 143 (2) ◽  
pp. 371-399 ◽  
Author(s):  
Atul Dixit
Keyword(s):  
Hypergeometric Function ◽  
Infinite Series ◽  
Confluent Hypergeometric Function ◽  
Möbius Function ◽  
Transformation Formula ◽  
Extended Version ◽  
New Class ◽  
Mobius Function ◽  
By Products

A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and Ferrar and gives, as by-products, transformation formulae of the form F(z, α) = F(iz, β), where αβ = 1. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulae of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization of a conjecture due to Ramanujan, Hardy and Littlewood involving infinite series of the Möbius function.

scholarly journals On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

2021 ◽  
Author(s):  
Yves Guemo Tefo ◽  
Rabia Aktaş ◽  
Iván Area ◽  
Esra Güldoğan Lekesiz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Polynomial Solutions ◽  
New Class ◽  
Polynomial Coefficients ◽  
Sturm Liouville ◽  
Hypergeometric Type

AbstractA new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.

On the partial sums of series of hypergeometric type

1953 ◽  
Vol 49 (3) ◽  
pp. 441-445 ◽  
Author(s):  
R. P. Agarwal
Keyword(s):  
Infinite Series ◽  
Hypergeometric Series ◽  
Partial Sums ◽  
Hypergeometric Type ◽  
The Subject ◽  
Image Position

About twenty years ago a number of results were given expressing the sum of n terms of an ordinary hypergeometric series with unit argument in terms of an infinite series of the type 3F2. The interest in the subject was aroused by a theorem due to Ramanujan, who stated that

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