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Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

Let n = 1 a n be a convergent series of positive real numbers. L. Olivier proved that if the sequence ( a n ) is non-increasing, then lim n n a n = 0 . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having lim n n a n = 0 ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the -convergence, that is a convergence according to an ideal of subsets of . Again, Olivier’s theorem is a consequence of...

Some identities involving differences of products of generalized Fibonacci numbers

Curtis Cooper (2015)

Colloquium Mathematicae

Melham discovered the Fibonacci identity F n + 1 F n + 2 F n + 6 - F ³ n + 3 = ( - 1 ) F . He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and W = p W n - 1 + q W n - 2 and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: W n + 1 W n + 2 W n + 6 - W ³ n + 3 = e q n + 1 ( p ³ W n + 2 - q ² W n + 1 ) . There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: F F n + 4 F n + 5 - F ³ n + 3 = ( - 1 ) n + 1 F n + 6 . We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is W W n + 4 W n + 5 - W ³ n + 3 = e q ( p ³ W n + 4 - q W n + 5 ) .

Some identities of degenerate special polynomials

Dae San Kim, Taekyun Kim (2015)

Open Mathematics

In this paper, by considering higher-order degenerate Bernoulli and Euler polynomials which were introduced by Carlitz, we investigate some properties of mixed-type of those polynomials. In particular, we give some identities of mixed-type degenerate special polynomials which are derived from the fermionic integrals on Zp and the bosonic integrals on Zp.

Some interpretations of the ( k , p ) -Fibonacci numbers

Natalia Paja, Iwona Włoch (2021)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider two parameters generalization of the Fibonacci numbers and Pell numbers, named as the ( k , p ) -Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the ( k , p ) -Fibonacci numbers.

Some new transformations for Bailey pairs and WP-Bailey pairs

James Mc Laughlin (2010)

Open Mathematics

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.

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