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Full-Text Articles in Physical Sciences and Mathematics
On A Solvable System Of Rational Difference Equations Of Higher Order, Merve Kara, Yasi̇n Yazlik
On A Solvable System Of Rational Difference Equations Of Higher Order, Merve Kara, Yasi̇n Yazlik
Turkish Journal of Mathematics
In this paper, we present that the following system of difference equations $$ x_{n}=\frac{x_{n-k}z_{n-l}}{b_{n}x_{n-k}+a_{n}z_{n-k-l}}, \ y_{n}=\frac{y_{n-k}x_{n-l}}{d_{n}y_{n-k}+c_{n}x_{n-k-l}}, \ z_{n}=\frac{z_{n-k}y_{n-l}}{f_{n}z_{n-k}+e_{n}y_{n-k-l}}, $$ where $n\in \mathbb{N}_{0}$, $k,l\in\mathbb{N}$, the initial values $x_{-i},y_{-i},z_{-i}$ are real numbers, for $i \in \overline{1,k+l}$, and sequences $\left( a_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( b_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( c_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( d_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( e_{n}\right) _{n\in \mathbb{N}_{0}}$ and $\left( f_{n}\right) _{n\in \mathbb{N}_{0}}$ are non-zero real numbers, for all $n\in \mathbb{N}_{0}$, which can be solved in closed form. We describe the forbidden set of the initial values using the obtained formulas and also determine the asymptotic behavior of solutions for the case …
A Series Evaluation Technique Based On A Modified Abel Lemma, John Maxwell Campbell, Marco Cantarini
A Series Evaluation Technique Based On A Modified Abel Lemma, John Maxwell Campbell, Marco Cantarini
Turkish Journal of Mathematics
We introduce a technique for determining infinite series identities through something of a combination of the modified Abel lemma on summation by parts and a method of undetermined coefficients. We succeed in applying our technique in our proving a nontrivial variant of Gauss' hypergeometric identity, giving us an evaluation for a family of ${}_{3}F_{2}(1)$-series with three free parameters, and to establish a ${}_{3}F_{2}(-1)$-variant of Kummer's hypergeometric identity. Also, we apply the technique upon which this article is based to formulate a new and simplified proof of a remarkable series evaluation recently derived by Cantarini via the generalized Clebsch-Gordan integral.