Pub Date : 2023-04-25DOI: 10.7546/nntdm.2023.29.2.216-225
Vandna Vandna, Mandeep Kaur
Let $N(a_1,a_2,a_3,a_4,a_5;n)$ and $T(a_1,a_2,a_3,a_4,a_5;n)$ count the representations of $n$ as $a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2+a_5x_5^2$ and $a_1X_1(X_1+1)/2+a_2X_2(X_2+1)/2+a_3X_3(X_3+1)/2+a_4X_4(X_4+1)/2+a_5X_5(X_5+1)/2$, respectively, where $a_1,a_2,a_3,a_4,a_5$ are positive integers, $x_1,x_2,x_3,x_4,x_5$ are integers and $n,X_1,X_2,X_3,X_4,X_5$ are nonnegative integers. In this paper, we establish some new relations between $N(a_1,a_2,a_3,a_4,a_5;n)$ and $T(a_1,a_2,a_3,a_4,a_5;n)$. Also, we prove that $T(a_1,a_2,a_3,a_4,a_5;n)$ is a linear combination of $N(a_1,a_2,a_3,a_4,a_5;m)$ and $N(a_1,a_2,a_3,a_4,a_5;m/4)$, where $m=8n+a_1+a_2+a_3+a_4+a_5$, for various values of $a_1,a_2,a_3,$ $a_4,a_5$.
{"title":"Some new relations between T(a1,a2,a3,a4,a5;n) and N(a1,a2,a3,a4,a5;n)","authors":"Vandna Vandna, Mandeep Kaur","doi":"10.7546/nntdm.2023.29.2.216-225","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.216-225","url":null,"abstract":"Let $N(a_1,a_2,a_3,a_4,a_5;n)$ and $T(a_1,a_2,a_3,a_4,a_5;n)$ count the representations of $n$ as $a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2+a_5x_5^2$ and $a_1X_1(X_1+1)/2+a_2X_2(X_2+1)/2+a_3X_3(X_3+1)/2+a_4X_4(X_4+1)/2+a_5X_5(X_5+1)/2$, respectively, where $a_1,a_2,a_3,a_4,a_5$ are positive integers, $x_1,x_2,x_3,x_4,x_5$ are integers and $n,X_1,X_2,X_3,X_4,X_5$ are nonnegative integers. In this paper, we establish some new relations between $N(a_1,a_2,a_3,a_4,a_5;n)$ and $T(a_1,a_2,a_3,a_4,a_5;n)$. Also, we prove that $T(a_1,a_2,a_3,a_4,a_5;n)$ is a linear combination of $N(a_1,a_2,a_3,a_4,a_5;m)$ and $N(a_1,a_2,a_3,a_4,a_5;m/4)$, where $m=8n+a_1+a_2+a_3+a_4+a_5$, for various values of $a_1,a_2,a_3,$ $a_4,a_5$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45647118","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 : 2023-04-16DOI: 10.7546/nntdm.2023.29.2.204-215
J. Kishore, V. Verma
In this paper, we will derive the explicit formulae for Chebyshev polynomials of the third and fourth kind with odd and even indices using the combinatorial method. Similar results are also deduced for their r-th derivatives. Finally, some identities involving Chebyshev polynomials of the third and fourth kind with even and odd indices and Fibonacci polynomials with negative indices are obtained.
{"title":"Some identities involving Chebyshev polynomials, Fibonacci polynomials and their derivatives","authors":"J. Kishore, V. Verma","doi":"10.7546/nntdm.2023.29.2.204-215","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.204-215","url":null,"abstract":"In this paper, we will derive the explicit formulae for Chebyshev polynomials of the third and fourth kind with odd and even indices using the combinatorial method. Similar results are also deduced for their r-th derivatives. Finally, some identities involving Chebyshev polynomials of the third and fourth kind with even and odd indices and Fibonacci polynomials with negative indices are obtained.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48695652","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 : 2023-04-12DOI: 10.7546/nntdm.2023.29.2.195-203
F. R. Alves, R. Vieira, P. Catarino
In the present work, we indicate some matrix properties that allow us to determine new relationships involving telephone numbers and some products that allow us to obtain telephone terms, based on second-order matrices.
{"title":"A note on telephone numbers and their matrix generators","authors":"F. R. Alves, R. Vieira, P. Catarino","doi":"10.7546/nntdm.2023.29.2.195-203","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.195-203","url":null,"abstract":"In the present work, we indicate some matrix properties that allow us to determine new relationships involving telephone numbers and some products that allow us to obtain telephone terms, based on second-order matrices.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48963802","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 : 2023-04-03DOI: 10.7546/nntdm.2023.29.2.185-194
P. Haukkanen
We study existence of a solution of the arithmetical equation $fast h = g$ in $f,$ where $fast h$ is the Dirichlet convolution of arithmetical functions $f$ and $h,$ and derive an explicit expression for the solution. As applications we obtain expressions for the Möbius function $mu$ and the so-called totients. As applications we also present our results on the arithmetical equation $fast h = g$ in the language of Cauchy convolution and further deconvolution in discrete linear systems.
{"title":"Quotients of arithmetical functions under the Dirichlet convolution","authors":"P. Haukkanen","doi":"10.7546/nntdm.2023.29.2.185-194","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.185-194","url":null,"abstract":"We study existence of a solution of the arithmetical equation $fast h = g$ in $f,$ where $fast h$ is the Dirichlet convolution of arithmetical functions $f$ and $h,$ and derive an explicit expression for the solution. As applications we obtain expressions for the Möbius function $mu$ and the so-called totients. As applications we also present our results on the arithmetical equation $fast h = g$ in the language of Cauchy convolution and further deconvolution in discrete linear systems.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43395865","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 : 2023-03-01DOI: 10.7546/nntdm.2023.29.1.98-129
P. Shiue, A. Shannon, Shen C. Huang, Jorge E. Reyes
A generalized Computation procedure for construction of the Ramanujan-type from a given general cubic equation and a cosine Ramanujan-type identity is developed from detailed analyses of the properties of Ramanujan-type cubic equations. Examples are provided together with cubic Shevelev sums.
{"title":"A generalized computation procedure for Ramanujan-type identities and cubic Shevelev sum","authors":"P. Shiue, A. Shannon, Shen C. Huang, Jorge E. Reyes","doi":"10.7546/nntdm.2023.29.1.98-129","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.98-129","url":null,"abstract":"A generalized Computation procedure for construction of the Ramanujan-type from a given general cubic equation and a cosine Ramanujan-type identity is developed from detailed analyses of the properties of Ramanujan-type cubic equations. Examples are provided together with cubic Shevelev sums.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47476002","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 : 2023-03-01DOI: 10.7546/nntdm.2023.29.1.154-170
Yasemin Alp
Different number systems have been studied lately. Recently, many researchers have considered the hybrid numbers which are generalization of the complex, hyperbolic and dual number systems. In this paper, we define the hybrid hyper-Fibonacci and hyper-Lucas numbers. Furthermore, we obtain some algebraic properties of these numbers such as the recurrence relations, the generating functions, the Binet’s formulas, the summation formulas, the Catalan’s identity, the Cassini’s identity and the d’Ocagne’s identity.
{"title":"Hybrid hyper-Fibonacci and hyper-Lucas numbers","authors":"Yasemin Alp","doi":"10.7546/nntdm.2023.29.1.154-170","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.154-170","url":null,"abstract":"Different number systems have been studied lately. Recently, many researchers have considered the hybrid numbers which are generalization of the complex, hyperbolic and dual number systems. In this paper, we define the hybrid hyper-Fibonacci and hyper-Lucas numbers. Furthermore, we obtain some algebraic properties of these numbers such as the recurrence relations, the generating functions, the Binet’s formulas, the summation formulas, the Catalan’s identity, the Cassini’s identity and the d’Ocagne’s identity.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48124108","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 : 2023-03-01DOI: 10.7546/nntdm.2023.29.1.171-180
K. Atanassov
The set $underline{SET}(n)$ generated by an arbitrary natural number $n$, was defined in [3]. There, and in [4], some arithmetic functions and arithmetic operators of a modal type are defined over the elements of $underline{SET}(n)$. Here, over the elements of $underline{SET}(n)$ arithmetic operators of a topological type are defined and some of their basic properties are studied. Perspectives for future research are discussed.
{"title":"Objects generated by an arbitrary natural number. Part 3: Standard modal-topological aspect","authors":"K. Atanassov","doi":"10.7546/nntdm.2023.29.1.171-180","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.171-180","url":null,"abstract":"The set $underline{SET}(n)$ generated by an arbitrary natural number $n$, was defined in [3]. There, and in [4], some arithmetic functions and arithmetic operators of a modal type are defined over the elements of $underline{SET}(n)$. Here, over the elements of $underline{SET}(n)$ arithmetic operators of a topological type are defined and some of their basic properties are studied. Perspectives for future research are discussed.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46666891","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 : 2023-03-01DOI: 10.7546/nntdm.2023.29.1.130-136
József Sándor
In papers [3] and [5] we have studied certain equations and inequalities involving the arithmetical functions varphi(n) and d(n). In this paper we will consider some other equations. Some open problems will be stated, too.
{"title":"On certain equations and inequalities involving the arithmetical functions φ(n) and d(n) – II","authors":"József Sándor","doi":"10.7546/nntdm.2023.29.1.130-136","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.130-136","url":null,"abstract":"In papers [3] and [5] we have studied certain equations and inequalities involving the arithmetical functions varphi(n) and d(n). In this paper we will consider some other equations. Some open problems will be stated, too.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135185321","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 : 2023-03-01DOI: 10.7546/nntdm.2023.29.1.181-184
J. A. Dris, Immanuel Tobias San Diego
In [2], the authors proposed a theorem which they recently found out to contradict Chen and Luo’s results [1]. In the present paper, we provide the correct form of this theorem.
{"title":"Corrigendum to: “Some modular considerations regarding odd perfect numbers – Part II” [Notes on Number Theory and Discrete Mathematics, 2020, Vol. 26, No. 3, 8–24]","authors":"J. A. Dris, Immanuel Tobias San Diego","doi":"10.7546/nntdm.2023.29.1.181-184","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.181-184","url":null,"abstract":"In [2], the authors proposed a theorem which they recently found out to contradict Chen and Luo’s results [1]. In the present paper, we provide the correct form of this theorem.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48220599","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 : 2023-03-01DOI: 10.7546/nntdm.2023.29.1.147-153
Umme Salma, H. M. Nagesh, D. Prahlad
Let $G$ be a simple graph. A vertex labeling $psi:V(G) rightarrow {1, 2,ldots,alpha}$ is called $alpha$-labeling. For an edge $uv in G$, the weight of $uv$, written $z_{psi}(uv)$, is the sum of the labels of $u$ and $v$, i.e., $z_{psi}(uv)=psi(u)+psi(v)$. A vertex $alpha$-labeling is said to be an edge irregular $alpha$-labeling of $G$ if for every two distinct edges $a$ and $b$, $z_{psi}(a) neq z_{psi}(b)$. The minimum $alpha$ for which the graph $G$ contains an edge irregular $alpha$-labeling is known as the edge irregularity strength of $G$ and is denoted by $es(G)$. In this paper, we find the exact value of edge irregularity strength of different cases of firefly graph $F_{s,t,n-2s-2t-1}$ for any $s geq 1, t geq 1, n-2s-2t-1 geq 1 $.
{"title":"A note on edge irregularity strength of firefly graph","authors":"Umme Salma, H. M. Nagesh, D. Prahlad","doi":"10.7546/nntdm.2023.29.1.147-153","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.147-153","url":null,"abstract":"Let $G$ be a simple graph. A vertex labeling $psi:V(G) rightarrow {1, 2,ldots,alpha}$ is called $alpha$-labeling. For an edge $uv in G$, the weight of $uv$, written $z_{psi}(uv)$, is the sum of the labels of $u$ and $v$, i.e., $z_{psi}(uv)=psi(u)+psi(v)$. A vertex $alpha$-labeling is said to be an edge irregular $alpha$-labeling of $G$ if for every two distinct edges $a$ and $b$, $z_{psi}(a) neq z_{psi}(b)$. The minimum $alpha$ for which the graph $G$ contains an edge irregular $alpha$-labeling is known as the edge irregularity strength of $G$ and is denoted by $es(G)$. In this paper, we find the exact value of edge irregularity strength of different cases of firefly graph $F_{s,t,n-2s-2t-1}$ for any $s geq 1, t geq 1, n-2s-2t-1 geq 1 $.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43349816","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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