Theorems on Summation of Series

Summation theorems for multidimensional basic hypergeometric series by determinant evaluations

Summation of Fourier Series

A general summability method of different orthogonal series is given with the help of an integrable function θ. As special cases the trigonometric Fourier, Walsh-, Walsh-Kaczmarz-, Vilenkinand Ciesielski-Fourier series and the Fourier transforms are considered. For each orthonormal system a different Hardy space is introduced and the atomic decomposition of these Hardy spaces are presented. A s...

On the Summation of Certain Iterated Series

The paper gives a unified treatment of the summation of certain iterated series of the form ∑∞ n=1 ∑∞ m=1 an+m, where (an)n∈N is a sequence of real numbers. We prove that, under certain conditions, the double iterated series equals the difference of two single series.

Extension of Some Classical Summation Theorems for the Generalized Hypergeometric Series with Integral Parameter Differences

We derive extensions of the classical summation theorems of Kummer and Watson for the generalized hypergeometric series where r pairs of numeratorial and denominatorial parameters differ by positive integers. The results are obtained with the help of a generalization of Kummer’s second summation theorem for the 2F1 series given recently by Rakha and Rathie [Integral Transforms and Special Funct...

Summation Notation and Series

Corresponding material in the book: Section 12.1, 12.2, 12.3. What students should definitely get: The summation notation and how it works, series, concepts of convergence. The use of telescoping and forward difference operator ideas to sum up series. The use of the integral test and other tests to determine whether a series converges and obtain numerical estimates. Convergence rules for ration...