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1

Ergodic properties of random infinite products of nonexpansive mappings

Reich S., Zaslavski A. J.

Journal of Mathematics and Applications

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2017

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Vol. 40

149--159

EN

In this paper we are concerned with the asymptotic behavior of random (unrestricted) infinite products of nonexpansive selfmappings of closed and convex subsets of a complete hyperbolic space. In contrast with our previous work in this direction, we no longer assume that these subsets are bounded. We first establish two theorems regarding the stability of the random weak ergodic property and then prove a related generic result. These results also extend our recent investigations regarding nonrandom infinite products.

2

De la Vallée Poussin Summability, the Combinatorial Sum [wzór] and the de la Vallée Poussin Means Expansion

Ali Z. S.

Journal of Mathematics and Applications

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2017

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Vol. 40

5--20

EN

In this paper we apply the de la Vallee Poussin sum to a combinatorial Chebyshev sum by Ziad S. Ali in [1]. One outcome of this consideration is the main lemma proving the following combinatorial identity: with Re(z) standing for the real part of z we have (wzór). Our main lemma will indicate in its proof that the hypergeometric factors 2F1(1, 1/2 + n; 1 + n; 4); and 2F1(1, 1/2 + 2n; 1 + 2n; 4) are complex, each having a real and imaginary part. As we apply the de la Vallee Poussin sum to the combinatorial Chebyshev sum generated in the Key lemma by Ziad S. Ali in [1], we see in the proof of the main lemma the extreme importance of the use of the main properties of the gamma function. This represents a second important consideration. A third new outcome are two interesting identities of the hypergeometric type with their new Meijer G function analogues. A fourth outcome is that by the use of the Cauchy integral formula for the derivatives we are able to give a dierent meaning to the sum: (wzór). A fifth outcome is that by the use of the Gauss-Kummer formula we are able to make better sense of the expressions (wzór) by making use of the series denition of the hypergeometric function. As we continue we notice a new close relation of the Key lemma, and the de la Vallee Poussin means. With this close relation we were able to talk about P the de la Vallee Poussin summability of the two innite series (wzór). Furthermore the application of the de la Vallee Poussin sum to the Key lemma has created two new expansions representing the following functions: (wzór).

3

Note on some infinite products for π

Kahlig P., Matkowski J.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 2

43--50

EN

After a brief review of (slowly converging) Wallis-type infinite products for π , (faster converging), Dido-type infinite products for π are treated. The notion of “alternating products” facilitates error checking.

4

Associative Omega-products of Traces

Redziejowski R. R.

Fundamenta Informaticae

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2005

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Vol. 67, nr 1-3

175--185

EN

The notion of associative infinite product is applied to traces, resulting in an alternative approach to introducing infinite traces. Four different versions of product are explored, two of them identical to known definitions of infinite trace.

5

On Associative Omega-Products

Rędziejowski R.R.

Fundamenta Informaticae

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2004

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Vol. 60, nr 1-4

333--350

EN

In recent years, a number of classical results connecting rational languages with finite semigroups have been extended to infinite-word languages using the notion of an w-semigroup : a semigroup augmented with an associative infinite product. This paper takes a closer look at the associative infinite product itself. It suggests some improvements and presents a couple of new facts.