Pub Date : 2022-09-12DOI: 10.7546/nntdm.2022.28.4.589-592
Ö. Deveci, A. Shannon
In this paper, the Laplace transform and various matrix operations are applied to the characteristic polynomial of the Fibonacci numbers. From this is generated some properties of the Jacobsthal numbers, including triangles where the row sums are known sequences. In turn these produce some new properties.
{"title":"On recurrence results from matrix transforms","authors":"Ö. Deveci, A. Shannon","doi":"10.7546/nntdm.2022.28.4.589-592","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.589-592","url":null,"abstract":"In this paper, the Laplace transform and various matrix operations are applied to the characteristic polynomial of the Fibonacci numbers. From this is generated some properties of the Jacobsthal numbers, including triangles where the row sums are known sequences. In turn these produce some new properties.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41511547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-12DOI: 10.7546/nntdm.2022.28.4.593-602
Ahmet Kaya, Hayrullah Özimamoğlu
In this article, we generalize the well-known Gauss Pell numbers and refer to them as generalized Gauss k-Pell numbers. There are relationships discovered between the class of generalized Gauss k-Pell numbers and the typical Gauss Pell numbers. Also, we generalize the known Gauss Pell polynomials, and call such polynomials as the generalized Gauss k-Pell polynomials. We obtain relations between the class of the generalized Gauss k-Pell polynomials and the typical Gauss Pell polynomials. Furthermore, we provide matrices for the novel generalizations of these numbers and polynomials. After that, we obtain Cassini’s identities for these numbers and polynomials.
{"title":"On a new class of the generalized Gauss k-Pell numbers and their polynomials","authors":"Ahmet Kaya, Hayrullah Özimamoğlu","doi":"10.7546/nntdm.2022.28.4.593-602","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.593-602","url":null,"abstract":"In this article, we generalize the well-known Gauss Pell numbers and refer to them as generalized Gauss k-Pell numbers. There are relationships discovered between the class of generalized Gauss k-Pell numbers and the typical Gauss Pell numbers. Also, we generalize the known Gauss Pell polynomials, and call such polynomials as the generalized Gauss k-Pell polynomials. We obtain relations between the class of the generalized Gauss k-Pell polynomials and the typical Gauss Pell polynomials. Furthermore, we provide matrices for the novel generalizations of these numbers and polynomials. After that, we obtain Cassini’s identities for these numbers and polynomials.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47286960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-20DOI: 10.7546/nntdm.2022.28.3.542-557
Emel Karaca, Fatih Yılmaz
The purpose of this paper is to define and construct new number systems, called the harmonic complex Fibonacci sequences (HCF) and the harmonic hybrid Fibonacci (HHF) sequences. These sequences are defined by inspiring the well-known harmonic and hybrid numbers in literature. We give some fundamental definitions and theorems about these sequences in detail. Moreover, we examine some algebraic properties such as Binet-like-formula, partial sums related to these sequences. Finally, we provide a Maple 13 source code to verify the sequences easily.
{"title":"An introduction to harmonic complex numbers and harmonic hybrid Fibonacci numbers: A unified approach","authors":"Emel Karaca, Fatih Yılmaz","doi":"10.7546/nntdm.2022.28.3.542-557","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.542-557","url":null,"abstract":"The purpose of this paper is to define and construct new number systems, called the harmonic complex Fibonacci sequences (HCF) and the harmonic hybrid Fibonacci (HHF) sequences. These sequences are defined by inspiring the well-known harmonic and hybrid numbers in literature. We give some fundamental definitions and theorems about these sequences in detail. Moreover, we examine some algebraic properties such as Binet-like-formula, partial sums related to these sequences. Finally, we provide a Maple 13 source code to verify the sequences easily.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46155084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-11DOI: 10.7546/nntdm.2022.28.3.533-541
Fouad Bounebirat, M. Rahmani
For a given prime p ≥ 5, let ℤ_p denote the set of rational p-integers (those rational numbers whose denominator is not divisible by p). In this paper, we establish some congruences modulo a prime power p5 on the hyper-sums of powers of integers in terms of Fermat quotient, Wolstenholme quotient, Bernoulli and Euler numbers.
{"title":"Some congruences on the hyper-sums of powers of integers involving Fermat quotient and Bernoulli numbers","authors":"Fouad Bounebirat, M. Rahmani","doi":"10.7546/nntdm.2022.28.3.533-541","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.533-541","url":null,"abstract":"For a given prime p ≥ 5, let ℤ_p denote the set of rational p-integers (those rational numbers whose denominator is not divisible by p). In this paper, we establish some congruences modulo a prime power p5 on the hyper-sums of powers of integers in terms of Fermat quotient, Wolstenholme quotient, Bernoulli and Euler numbers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49618075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-10DOI: 10.7546/nntdm.2022.28.3.525-532
J. Sándor
We continue the study from [1], by studying equations of type $psi(n) = dfrac{k+1}{k} cdot n+a,$ $ain {0, 1, 2, 3},$ and $varphi(n) = dfrac{k-1}{k} cdot n-a,$ $ain {0, 1, 2, 3}$ for $k > 1,$ where $psi(n)$ and $varphi(n)$ denote the Dedekind, respectively Euler's, arithmetical functions.
{"title":"On certain rational perfect numbers, II","authors":"J. Sándor","doi":"10.7546/nntdm.2022.28.3.525-532","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.525-532","url":null,"abstract":"We continue the study from [1], by studying equations of type $psi(n) = dfrac{k+1}{k} cdot n+a,$ $ain {0, 1, 2, 3},$ and $varphi(n) = dfrac{k-1}{k} cdot n-a,$ $ain {0, 1, 2, 3}$ for $k > 1,$ where $psi(n)$ and $varphi(n)$ denote the Dedekind, respectively Euler's, arithmetical functions.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41378423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-10DOI: 10.7546/nntdm.2022.28.3.517-524
H. M. Nagesh, V. R. Girish
For a simple graph $G$, a vertex labeling $phi:V(G) rightarrow {1, 2,ldots,k}$ is called $k$-labeling. The weight of an edge $xy$ in $G$, written $w_{phi}(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_{phi}(xy)=phi(x)+phi(y)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, $w_{phi}(e) neq w_{phi}(f)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this paper, we find the exact value of edge irregularity strength of line graph of comb graph $P_n bigodot K_1$ for $n=2,3,4$; and determine the bounds for $n geq 5$. Also, the edge irregularity strength of line cut-vertex graph of $P_n bigodot K_1$ for $n=2$; and determine the bounds for $n geq 3$.
{"title":"On edge irregularity strength of line graph and line cut-vertex graph of comb graph","authors":"H. M. Nagesh, V. R. Girish","doi":"10.7546/nntdm.2022.28.3.517-524","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.517-524","url":null,"abstract":"For a simple graph $G$, a vertex labeling $phi:V(G) rightarrow {1, 2,ldots,k}$ is called $k$-labeling. The weight of an edge $xy$ in $G$, written $w_{phi}(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_{phi}(xy)=phi(x)+phi(y)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, $w_{phi}(e) neq w_{phi}(f)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this paper, we find the exact value of edge irregularity strength of line graph of comb graph $P_n bigodot K_1$ for $n=2,3,4$; and determine the bounds for $n geq 5$. Also, the edge irregularity strength of line cut-vertex graph of $P_n bigodot K_1$ for $n=2$; and determine the bounds for $n geq 3$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45383256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-04DOI: 10.7546/nntdm.2022.28.3.500-506
S. Janphaisaeng, T. Srichan, Pinthira Tangsupphathawat
In this paper, we study asymptotic behaviour of the sum $sum_{nleq N}{f}Big(lfloor n^c rfloorBig),$ where $f(n)=sum_{d^2mid n}g(d)$ under three different types of assumptions on $g$ and $1& < c < 2$.
在本文中，我们研究了在$g$和$1&
{"title":"Average value of some certain types of arithmetic functions with Piatetski-Shapiro sequences","authors":"S. Janphaisaeng, T. Srichan, Pinthira Tangsupphathawat","doi":"10.7546/nntdm.2022.28.3.500-506","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.500-506","url":null,"abstract":"In this paper, we study asymptotic behaviour of the sum $sum_{nleq N}{f}Big(lfloor n^c rfloorBig),$ where $f(n)=sum_{d^2mid n}g(d)$ under three different types of assumptions on $g$ and $1& < c < 2$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45750624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-04DOI: 10.7546/nntdm.2022.28.3.507-516
A. Shannon, E. Özkan
This paper builds on Roettger and Williams’ extensions of the primordial Lucas sequence to consider some relations among difference equations of different orders. This paper utilises some of their second and third order recurrence relations to provide an excursion through basic second order sequences and related third order recurrence relations with a variety of numerical illustrations which demonstrate that mathematical notation is a tool of thought.
{"title":"Some aspects of interchanging difference equation orders","authors":"A. Shannon, E. Özkan","doi":"10.7546/nntdm.2022.28.3.507-516","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.507-516","url":null,"abstract":"This paper builds on Roettger and Williams’ extensions of the primordial Lucas sequence to consider some relations among difference equations of different orders. This paper utilises some of their second and third order recurrence relations to provide an excursion through basic second order sequences and related third order recurrence relations with a variety of numerical illustrations which demonstrate that mathematical notation is a tool of thought.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41970024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-04DOI: 10.7546/nntdm.2022.28.3.491-499
S. Arolkar
In this paper, B-Tribonacci polynomials which are extensions of Fibonacci polynomials are defined. Some identities relating B-Tribonacci polynomials and their derivatives are established.
{"title":"On the derivatives of B-Tribonacci polynomials","authors":"S. Arolkar","doi":"10.7546/nntdm.2022.28.3.491-499","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.491-499","url":null,"abstract":"In this paper, B-Tribonacci polynomials which are extensions of Fibonacci polynomials are defined. Some identities relating B-Tribonacci polynomials and their derivatives are established.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42092645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-03DOI: 10.7546/nntdm.2022.28.3.477-490
Y. Soykan, Inci Okumuş, E. Taşdemir
In this paper, we define and investigate the generalized Pisano sequences and we deal with, in detail, two special cases, namely, Pisano and Pisano–Lucas sequences. We present Binet’s formulas, generating functions and Simson’s formulas for these sequences. Moreover, we give some identities and matrices associated with these sequences. Furthermore, we show that there are close relations between Pisano and Pisano–Lucas numbers and modified Oresme, Oresme–Lucas and Oresme numbers.
{"title":"Generalized Pisano numbers","authors":"Y. Soykan, Inci Okumuş, E. Taşdemir","doi":"10.7546/nntdm.2022.28.3.477-490","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.477-490","url":null,"abstract":"In this paper, we define and investigate the generalized Pisano sequences and we deal with, in detail, two special cases, namely, Pisano and Pisano–Lucas sequences. We present Binet’s formulas, generating functions and Simson’s formulas for these sequences. Moreover, we give some identities and matrices associated with these sequences. Furthermore, we show that there are close relations between Pisano and Pisano–Lucas numbers and modified Oresme, Oresme–Lucas and Oresme numbers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48363921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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