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ON CONGRUENCES INVOLVING THE GENERALIZED CATALAN NUMBERS AND HARMONIC NUMBERS
Koparal, Sibel,Omur, Nese Korean Mathematical Society 2019 대한수학회보 Vol.56 No.3
In this paper, we prove some congruences involving the generalized Catalan numbers and harmonic numbers modulo $p^2$, one of which is $$\sum\limits_{k=1}^{p-1}k^2B_{p,k}B_{p,k-d}{\equiv}4(-1)^d\{{\frac{1}{3}}d(2d^2+1)(4pH_d-1)-p\({\frac{26}{9}}d^3+{\frac{4}{3}}d^2+{\frac{7}{9}}d+{\frac{1}{2}}\)\}\;(mod\;p^2)$$, where a prime number p > 3 and $1{\leq}d{\leq}p$.
ON CONGRUENCES WITH THE TERMS OF THE SECOND ORDER SEQUENCES {U<sub>kn</sub>} AND {V<sub>kn</sub>}
KOPARAL, SIBEL,OMUR, Nese The Honam Mathematical Society 2018 호남수학학술지 Vol.40 No.3
In this paper, we consider the congruences involving harmonic numbers and the terms of the sequences {$U_{kn}$} and {$V_{kn}$}. For example, for an odd prime number p, $${\sum\limits_{i=1}^{p-1}}H_i{\frac{U_{k(i+m)}}{V^i_k}}{\equiv}{\frac{(-1)^kU_{k(m+1)}}{_pV^{p-1}_k}}(V^p_k-V_{kp})(mod\;p)$$, where $m{\in}{\mathbb{Z}}$ and $k{\in}{\mathbb{Z}}$ with $p{\nmid}V_k$.
ON CONGRUENCES WITH THE TERMS OF THE SECOND ORDER SEQUENCES {U <sub>kn</sub>} AND {V <sub>kn</sub>}
( Sibel Koparal ),( Nese Omur ) 호남수학회 2018 호남수학학술지 Vol.40 No.3
In this paper, we consider the congruences involving harmonic numbers and the terms of the sequences {U <sub>kn</sub>} and {V <sub>kn</sub>}. For example, for an odd prime number p, where m □ Z and k □ Z with p □ V<sub>k</sub>.
On congruences involving the generalized Catalan numbers and harmonic numbers
Sibel Koparal,Nese Omur 대한수학회 2019 대한수학회보 Vol.56 No.3
In this paper, we prove some congruences involving the generalized Catalan numbers and harmonic numbers modulo $p^{2},$ one of which is \begin{align*} \sum\limits_{k=1}^{p-1}k^{2}B_{p,k}B_{p,k-d} \equiv &\ 4\left( -1\right) ^{d}\left\{ \frac{1}{3}d\left( 2d^{2}+1\right) \left( 4pH_{d}-1\right) \right. \\ &\ \left. -p\left( \frac{26}{9}d^{3}+\frac{4}{3}d^{2}+\frac{7}{9}d+\frac{1}{2} \right) \right\}\pmod{p^{2}}, \end{align*} where a prime number $p>3$ and $1\leq d\leq p.$
ON CONGRUENCES WITH THE TERMS OF THE SECOND ORDER SEQUENCES fUkng AND fVkng
Sibel Koparal,Nese Omur 호남수학회 2018 호남수학학술지 Vol.40 No.3
In this paper, we consider the congruences involvingharmonic numbers and the terms of the sequences fUkng and fVkng. For example, for an odd prime number p,Xp 1i=1HiUk(i+m)V ik ( 1)k Uk(m+1)pV p 1k(V pk Vkp) (mod p);where m 2 Z and k 2 Z with p - Vk:
NEW CONGRUENCES WITH THE GENERALIZED CATALAN NUMBERS AND HARMONIC NUMBERS
Elkhiri, Laid,Koparal, Sibel,Omur, Nese Korean Mathematical Society 2021 대한수학회보 Vol.58 No.5
In this paper, we give new congruences with the generalized Catalan numbers and harmonic numbers modulo p<sup>2</sup>. One of our results is as follows: for prime number p > 3, $${\sum\limits_{k=(p+1)/2}^{p-1}}\;k^2B_{p,k}B_{p,k-(p-1)/2}H_k{\equiv}(-1)^{(p-1)/2}\(-{\frac{521}{36}}p-{\frac{1}{p}}-{\frac{41}{12}}+pH^2_{3(p-1)/2}-10pq^2_p(2)+4\({\frac{10}{3}}p+1\)q_p(2)\)\;(mod\;p^2),$$ where q<sub>p</sub>(2) is Fermat quotient.
Two Variants of the Reciprocal Super Catalan Matrix
Kilic, Emrah,Omur, Nese,Koparal, Sibel,Ulutas, Yucel Turker Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.2
In this paper, we define two kinds variants of the super Catalan matrix as well as their q-analoques. We give explicit expressions for LU-decompositions of these matrices and their inverses.
Keskinruzgar, Aydin,Cankal, Dilek Aynur,Koparal, Mahmut,Simsek, Ali,Karadag, Ayse Sevgi The Korean Dental Society of Anesthsiology 2019 Journal of Dental Anesthesia and Pain Medicine Vol.19 No.1
Objective: Arthrocentesis is a minimally invasive surgical procedure that is used to alleviate the symptoms of temporomandibular joint (TMJ) disorders. The aim of this study was to investigate the effect of arthrocentesis on the blood supply to the retinal structures. Materials and Methods: Arthrocentesis was performed on 20 patients with TMJ disorders, and choroidal thickness (CT) in patients was measured to evaluate retinal blood circulation. The blood volume of the retinal structures was evaluated ipsilaterally before and after arthrocentesis, and these measurements were then compared with measurements obtained from the contralateral side. Results: Before arthrocentesis, there were no differences in retinal blood volumes between the ipsilateral and contralateral sides (P = 0.96). When ipsilateral CT measurements taken before and after arthrocentesis were compared, retinal blood supply was found to have significantly decreased after arthrocentesis (P = 0.04). When contralateral CT measurements taken before and after arthrocentesis were compared, retinal blood supply was also found to have decreased after arthrocentesis, but not significantly (P = 0.19). Conclusion: The solution of local anesthesia with epinephrine applied before the arthrocentesis procedure was found to reduce the blood volume of the retinal structures. To the best of our knowledge, this is the first study that has investigated the blood volume of the retinal structures following arthrocentesis.