Pub Date : 2023-08-01DOI: 10.7546/nntdm.2023.29.3.603-619
B. Paudel, Christopher G. Pinner
Let $mathbb Z_n$ denote the cyclic group of order $n$. We show how the group determinant for $G= mathbb Z_n times H$ can be simply written in terms of the group determinant for $H$. We use this to get a complete description of the integer group determinants for $mathbb Z_2 times D_8$ where $D_8$ is the dihedral group of order $8$, and $mathbb Z_2 times Q_8$ where $Q_8$ is the quaternion group of order $8$.
{"title":"The group determinants for ℤ_n × H","authors":"B. Paudel, Christopher G. Pinner","doi":"10.7546/nntdm.2023.29.3.603-619","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.603-619","url":null,"abstract":"Let $mathbb Z_n$ denote the cyclic group of order $n$. We show how the group determinant for $G= mathbb Z_n times H$ can be simply written in terms of the group determinant for $H$. We use this to get a complete description of the integer group determinants for $mathbb Z_2 times D_8$ where $D_8$ is the dihedral group of order $8$, and $mathbb Z_2 times Q_8$ where $Q_8$ is the quaternion group of order $8$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44112397","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 : 2023-08-01DOI: 10.7546/nntdm.2023.29.3.589-597
Krassimir Atanassov
The set $underline{SET}(n)$, generated by an arbitrary natural number $n$, was defined in [3]. There, and in [5,6], some arithmetic functions and arithmetic operators of a modal and topological types are defined over the elements of $underline{SET}(n)$. Here, over the elements of $underline{SET}(n)$ new arithmetic functions are defined and some of their basic properties are studied. Two standard modal topological structures over $underline{SET}(n)$ are described. Perspectives for future research are discussed.
{"title":"Objects generated by an arbitrary natural number. Part 4: New aspects","authors":"Krassimir Atanassov","doi":"10.7546/nntdm.2023.29.3.589-597","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.589-597","url":null,"abstract":"The set $underline{SET}(n)$, generated by an arbitrary natural number $n$, was defined in [3]. There, and in [5,6], some arithmetic functions and arithmetic operators of a modal and topological types are defined over the elements of $underline{SET}(n)$. Here, over the elements of $underline{SET}(n)$ new arithmetic functions are defined and some of their basic properties are studied. Two standard modal topological structures over $underline{SET}(n)$ are described. Perspectives for future research are discussed.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136299722","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 : 2023-07-30DOI: 10.7546/nntdm.2023.29.3.557-563
C. M. D. Fonseca, A. Shannon
Recently, based on the Laplace transform of the characteristic polynomial of the Fibonacci sequence, Deveci and Shannon established a new sequence and analysed some of its properties. They disclosed in particular the odd terms. In this short note, we provide a matricial representation for this sequence as well as one in terms of the Chebyshev polynomials of the second kind. The subsequence of the even terms is also disclosed.
{"title":"On a sequence derived from the Laplace transform of the characteristic polynomial of the Fibonacci sequence","authors":"C. M. D. Fonseca, A. Shannon","doi":"10.7546/nntdm.2023.29.3.557-563","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.557-563","url":null,"abstract":"Recently, based on the Laplace transform of the characteristic polynomial of the Fibonacci sequence, Deveci and Shannon established a new sequence and analysed some of its properties. They disclosed in particular the odd terms. In this short note, we provide a matricial representation for this sequence as well as one in terms of the Chebyshev polynomials of the second kind. The subsequence of the even terms is also disclosed.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44333605","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 : 2023-07-27DOI: 10.7546/nntdm.2023.29.3.538-544
Meriem Tiachachat, M. Mihoubi
In this paper, we discuss a new class of partial Bell polynomials. The first section gives an overview of partial Bell polynomials and their related 2-successive Stirling numbers. In the second section, we introduce the concept of 2-successive partial Bell polynomials. We give an explicit formula for computing these polynomials and establish their generating function. In addition, we derive several recurrence relations that govern the behaviour of these polynomials. Furthermore, we study specific cases to illustrate the applicability and versatility of this new class of polynomials.
{"title":"The 2-successive partial Bell polynomials","authors":"Meriem Tiachachat, M. Mihoubi","doi":"10.7546/nntdm.2023.29.3.538-544","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.538-544","url":null,"abstract":"In this paper, we discuss a new class of partial Bell polynomials. The first section gives an overview of partial Bell polynomials and their related 2-successive Stirling numbers. In the second section, we introduce the concept of 2-successive partial Bell polynomials. We give an explicit formula for computing these polynomials and establish their generating function. In addition, we derive several recurrence relations that govern the behaviour of these polynomials. Furthermore, we study specific cases to illustrate the applicability and versatility of this new class of polynomials.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71200796","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 : 2023-07-27DOI: 10.7546/nntdm.2023.29.3.545-548
Pascal Ochem
Thue has shown the existence of three types of infinite square-free words over {0,1,2} avoiding the factor 010. They respectively avoid {010,212}, {010,101}, and {010,020}. Also Dejean constructed an infinite $left(tfrac74^+right)$-free ternary word. A word is $d$-directed if it does not contain both a factor of length $d$ and its mirror image. We show that there exist exponentially many $left(tfrac74^+right)$-free 180-directed ternary words avoiding 010. Moreover, there does not exist an infinite $left(tfrac74^+right)$-free 179-directed ternary word avoiding 010.
{"title":"On ternary Dejean words avoiding 010","authors":"Pascal Ochem","doi":"10.7546/nntdm.2023.29.3.545-548","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.545-548","url":null,"abstract":"Thue has shown the existence of three types of infinite square-free words over {0,1,2} avoiding the factor 010. They respectively avoid {010,212}, {010,101}, and {010,020}. Also Dejean constructed an infinite $left(tfrac74^+right)$-free ternary word. A word is $d$-directed if it does not contain both a factor of length $d$ and its mirror image. We show that there exist exponentially many $left(tfrac74^+right)$-free 180-directed ternary words avoiding 010. Moreover, there does not exist an infinite $left(tfrac74^+right)$-free 179-directed ternary word avoiding 010.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44822234","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 : 2023-07-27DOI: 10.7546/nntdm.2023.29.3.549-556
A. P. Camargo
We obtain an asymptotic formula for the sum $tilde{D}_2$ of the divisors of all square-free integers less than or equal to $x$, with error term $O(x^{1/2 + epsilon})$. This improves the error term $O(x^{3/4 + epsilon})$ presented in [7] obtained via analytical methods. Our approach is elementary and it is based on the connections between the function $tilde{D}_2$ and unitary convolutions.
{"title":"The Dirichlet divisor problem over square-free integers and unitary convolutions","authors":"A. P. Camargo","doi":"10.7546/nntdm.2023.29.3.549-556","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.549-556","url":null,"abstract":"We obtain an asymptotic formula for the sum $tilde{D}_2$ of the divisors of all square-free integers less than or equal to $x$, with error term $O(x^{1/2 + epsilon})$. This improves the error term $O(x^{3/4 + epsilon})$ presented in [7] obtained via analytical methods. Our approach is elementary and it is based on the connections between the function $tilde{D}_2$ and unitary convolutions.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47115580","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 : 2023-07-26DOI: 10.7546/nntdm.2023.29.3.525-537
S. Thakur, Pinkimani Goswami, Gautam Chandra Ray
For each positive integer $n$, we assign a digraph $Gamma(n,11)$ whose set of vertices is $Z_n=lbrace 0,1,2, ldots, n-1rbrace$ and there exists exactly one directed edge from the vertex $a$ to the vertex $b$ iff $a^{11}equiv b pmod n$. Using the ideas of modular arithmetic, cyclic vertices are presented and established for $n=3^k$ in the digraph $Gamma(n,11)$. Also, the number of cycles and the number of components in the digraph $Gamma(n,11)$ is presented for $n=3^k,7^k$ with the help of Carmichael’s lambda function. It is proved that for $kgeq 1$, the number of components in the digraph $Gamma(3^k,11)$ is $(2k+1)$ and for $k>2$ the digraph $Gamma(3^k,11)$ has $(k-1)$ non-isomorphic cycles of length greater than $1$, whereas the number of components of the digraph $Gamma(7^k,11)$ is $(8k-3)$.
{"title":"Enumeration of cyclic vertices and components over the congruence $a^{11} equiv b pmod n$","authors":"S. Thakur, Pinkimani Goswami, Gautam Chandra Ray","doi":"10.7546/nntdm.2023.29.3.525-537","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.525-537","url":null,"abstract":"For each positive integer $n$, we assign a digraph $Gamma(n,11)$ whose set of vertices is $Z_n=lbrace 0,1,2, ldots, n-1rbrace$ and there exists exactly one directed edge from the vertex $a$ to the vertex $b$ iff $a^{11}equiv b pmod n$. Using the ideas of modular arithmetic, cyclic vertices are presented and established for $n=3^k$ in the digraph $Gamma(n,11)$. Also, the number of cycles and the number of components in the digraph $Gamma(n,11)$ is presented for $n=3^k,7^k$ with the help of Carmichael’s lambda function. It is proved that for $kgeq 1$, the number of components in the digraph $Gamma(3^k,11)$ is $(2k+1)$ and for $k>2$ the digraph $Gamma(3^k,11)$ has $(k-1)$ non-isomorphic cycles of length greater than $1$, whereas the number of components of the digraph $Gamma(7^k,11)$ is $(8k-3)$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43213848","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 : 2023-07-20DOI: 10.7546/nntdm.2023.29.3.503-524
E. Mehraban, M. Hashemi
In this paper, we introduce the generalized balancing sequence and its matrix. Then by using the generalized balancing matrix, we give a coding and decoding method.
本文介绍了广义平衡序列及其矩阵。然后利用广义平衡矩阵给出了一种编码和解码方法。
{"title":"Coding theory on the generalized balancing sequence","authors":"E. Mehraban, M. Hashemi","doi":"10.7546/nntdm.2023.29.3.503-524","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.503-524","url":null,"abstract":"In this paper, we introduce the generalized balancing sequence and its matrix. Then by using the generalized balancing matrix, we give a coding and decoding method.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48767778","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 : 2023-07-15DOI: 10.7546/nntdm.2023.29.3.495-502
R. K. Davala
Let $B_n$ and $C_n$ be the $n$-th balancing and Lucas-balancing numbers, respectively. We consider the Diophantine equations $ax+by=frac{1}{2}(a-1)(b-1)$ and $1+ax+by=frac{1}{2}(a-1)(b-1)$ for $(a,b)$ $in$ $ {(B_n,B_{n+1}),(B_{2n-1},B_{2n+1}), (B_n,C_n),(C_n,C_{n+1})}$ and present the non-negative integer solutions of the Diophantine equations in each case.
{"title":"Solution to a pair of linear, two-variable, Diophantine equations with coprime coefficients from balancing and Lucas-balancing numbers","authors":"R. K. Davala","doi":"10.7546/nntdm.2023.29.3.495-502","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.495-502","url":null,"abstract":"Let $B_n$ and $C_n$ be the $n$-th balancing and Lucas-balancing numbers, respectively. We consider the Diophantine equations $ax+by=frac{1}{2}(a-1)(b-1)$ and $1+ax+by=frac{1}{2}(a-1)(b-1)$ for $(a,b)$ $in$ $ {(B_n,B_{n+1}),(B_{2n-1},B_{2n+1}), (B_n,C_n),(C_n,C_{n+1})}$ and present the non-negative integer solutions of the Diophantine equations in each case.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46068659","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 : 2023-07-11DOI: 10.7546/nntdm.2023.29.3.486-494
Burak Kurt
The main aim of this paper is to introduce and investigate the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials by using monomiality principle and operational methods. We give explicit relations and some identities for the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials.
{"title":"Explicit relations on the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials and numbers","authors":"Burak Kurt","doi":"10.7546/nntdm.2023.29.3.486-494","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.486-494","url":null,"abstract":"The main aim of this paper is to introduce and investigate the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials by using monomiality principle and operational methods. We give explicit relations and some identities for the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42787559","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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