Pub Date : 2022-10-27DOI: 10.7546/nntdm.2022.28.4.666-676
H. Kim
In this paper, we introduce a new type degenerate Stirling numbers of the second kind and their degenerate Bell polynomials, which is different from degenerate Stirling numbers of the second kind studied so far. We investigate the explicit formula, recurrence relation and Dobinski-like formula of a new type degenerate Stirling numbers of the second kind. We also derived several interesting expressions and identities for bell polynomials of these new type degenerate Stirling numbers of the second kind including the generating function, recurrence relation, differential equation with Bernoulli number as coefficients, the derivative and Riemann integral, so on.
{"title":"New type degenerate Stirling numbers and Bell polynomials","authors":"H. Kim","doi":"10.7546/nntdm.2022.28.4.666-676","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.666-676","url":null,"abstract":"In this paper, we introduce a new type degenerate Stirling numbers of the second kind and their degenerate Bell polynomials, which is different from degenerate Stirling numbers of the second kind studied so far. We investigate the explicit formula, recurrence relation and Dobinski-like formula of a new type degenerate Stirling numbers of the second kind. We also derived several interesting expressions and identities for bell polynomials of these new type degenerate Stirling numbers of the second kind including the generating function, recurrence relation, differential equation with Bernoulli number as coefficients, the derivative and Riemann integral, so on.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43277893","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 : 2022-10-24DOI: 10.7546/nntdm.2022.28.4.635-647
Koustav Banerjee, Manosij Ghosh Dastidar
In this paper we exhibit the box-stacking principle (BSP) in conjunction with Young diagrams to prove generalizations of Stanley’s and Elder’s theorems without even the use of partition statistics in general. We primarily focus on to study Stanley’s theorem in color partition context.
{"title":"Hook type tableaux and partition identities","authors":"Koustav Banerjee, Manosij Ghosh Dastidar","doi":"10.7546/nntdm.2022.28.4.635-647","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.635-647","url":null,"abstract":"In this paper we exhibit the box-stacking principle (BSP) in conjunction with Young diagrams to prove generalizations of Stanley’s and Elder’s theorems without even the use of partition statistics in general. We primarily focus on to study Stanley’s theorem in color partition context.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42662792","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 : 2022-10-24DOI: 10.7546/nntdm.2022.28.4.648-655
M. Bouderbala, Meselem Karras
In this paper, we obtain asymptotic formula on the "hyperbolic" summation begin{equation*} underset{mnleq x}{sum }D_{k}left( gcd left( m,nright) right) text{ }left( kin mathbb{Z}_{geq 2}right), end{equation*} such that $D_{k}left( nright) = dfrac{tau _{k}left( nright) }{tau_{k}^{ast }left( nright) }$, where $tau _{k}left( nright) =!!sumlimits_{n_{1}n_{2}ldots n_{k}=n}!!1$ denotes the Piltz divisor function, and $tau _{k}^{ast }left( nright) $ is the unitary analogue function of $tau _{k}left( nright) $.
{"title":"Asymptotic formula of a “hyperbolic” summation related to the Piltz divisor function","authors":"M. Bouderbala, Meselem Karras","doi":"10.7546/nntdm.2022.28.4.648-655","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.648-655","url":null,"abstract":"In this paper, we obtain asymptotic formula on the \"hyperbolic\" summation begin{equation*} underset{mnleq x}{sum }D_{k}left( gcd left( m,nright) right) text{ }left( kin mathbb{Z}_{geq 2}right), end{equation*} such that $D_{k}left( nright) = dfrac{tau _{k}left( nright) }{tau_{k}^{ast }left( nright) }$, where $tau _{k}left( nright) =!!sumlimits_{n_{1}n_{2}ldots n_{k}=n}!!1$ denotes the Piltz divisor function, and $tau _{k}^{ast }left( nright) $ is the unitary analogue function of $tau _{k}left( nright) $.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45166780","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 : 2022-10-24DOI: 10.7546/nntdm.2022.28.4.656-665
P. Haukkanen
Two subsets P and Q of the set of positive integers is said to form a conjugate pair if each positive integer n possesses a unique factorization of the form n = ab, a ∈ P, b ∈ Q. In this paper we generalize conjugate pairs of sets to the setting of regular convolutions and study associated arithmetical functions. Particular attention is paid to arithmetical functions associated with k-free integers and k-th powers under regular convolution.
{"title":"Arithmetical functions associated with conjugate pairs of sets under regular convolutions","authors":"P. Haukkanen","doi":"10.7546/nntdm.2022.28.4.656-665","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.656-665","url":null,"abstract":"Two subsets P and Q of the set of positive integers is said to form a conjugate pair if each positive integer n possesses a unique factorization of the form n = ab, a ∈ P, b ∈ Q. In this paper we generalize conjugate pairs of sets to the setting of regular convolutions and study associated arithmetical functions. Particular attention is paid to arithmetical functions associated with k-free integers and k-th powers under regular convolution.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46214047","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 : 2022-10-14DOI: 10.7546/nntdm.2022.28.4.617-634
R. Jakimczuk, M. Lalín
We compute asymptotics of the sums of general divisor functions over h-free numbers, h-full numbers and other arithmetically interesting sets and conditions. The main tool for obtaining these results is Perron’s formula.
{"title":"Asymptotics of sums of divisor functions over sequences with restricted factorization structure","authors":"R. Jakimczuk, M. Lalín","doi":"10.7546/nntdm.2022.28.4.617-634","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.617-634","url":null,"abstract":"We compute asymptotics of the sums of general divisor functions over h-free numbers, h-full numbers and other arithmetically interesting sets and conditions. The main tool for obtaining these results is Perron’s formula.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41584308","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 : 2022-10-14DOI: 10.7546/nntdm.2022.28.4.603-616
Nabil Tahmi, Abdallah Derbal
Let f_r: mathbb{N}^r longrightarrow mathbb{C} be an arithmetic function of r variables, where rgeq 2. We study multiple Dirichlet series defined by begin{equation*} D(f_r,s_1,ldots,s_r)=sumlimits_{substack{n_1,ldots,n_r=1 (n_1,ldots,n_r)=1}}^{+infty}frac{f_r(n_1,ldots,n_r)}{n_1^{s_1}cdots n_r^{s_r}}, end{equation*} where f_r(n_1,ldots,n_r)=f(n_1)cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.
{"title":"Some multiple Dirichlet series of completely multiplicative arithmetic functions","authors":"Nabil Tahmi, Abdallah Derbal","doi":"10.7546/nntdm.2022.28.4.603-616","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.603-616","url":null,"abstract":"Let f_r: mathbb{N}^r longrightarrow mathbb{C} be an arithmetic function of r variables, where rgeq 2. We study multiple Dirichlet series defined by begin{equation*} D(f_r,s_1,ldots,s_r)=sumlimits_{substack{n_1,ldots,n_r=1 (n_1,ldots,n_r)=1}}^{+infty}frac{f_r(n_1,ldots,n_r)}{n_1^{s_1}cdots n_r^{s_r}}, end{equation*} where f_r(n_1,ldots,n_r)=f(n_1)cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42450954","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 : 2022-09-27DOI: 10.7546/nntdm.2022.28.3.575-580
M. Bouderbala, Meselem Karras
The main purpose of this paper is to define a new additive arithmetic function related to a fixed integer kgeq 1 and to study some of its properties. This function is given by begin{equation*} f_{k}left( 1right) =0text{ and }f_{k}left( nright) =sum_{p^{alpha}parallel n}left( k,alpha right) , end{equation*} such that (a, b) denotes the greatest common divisor of the integers a and b.
{"title":"On a new additive arithmetic function related to a fixed integer","authors":"M. Bouderbala, Meselem Karras","doi":"10.7546/nntdm.2022.28.3.575-580","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.575-580","url":null,"abstract":"The main purpose of this paper is to define a new additive arithmetic function related to a fixed integer kgeq 1 and to study some of its properties. This function is given by begin{equation*} f_{k}left( 1right) =0text{ and }f_{k}left( nright) =sum_{p^{alpha}parallel n}left( k,alpha right) , end{equation*} such that (a, b) denotes the greatest common divisor of the integers a and b.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49198921","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 : 2022-09-27DOI: 10.7546/nntdm.2022.28.3.564-574
N. Ömür, S. Koparal, Ö. Duran, K. Südemen
In this paper, we get new identities involving Bernoulli, Daehee and Stirling numbers, and their representations by using p-adic integrals and combinatorial techniques.
{"title":"Identities involving some special numbers and polynomials on p-adic integral","authors":"N. Ömür, S. Koparal, Ö. Duran, K. Südemen","doi":"10.7546/nntdm.2022.28.3.564-574","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.564-574","url":null,"abstract":"In this paper, we get new identities involving Bernoulli, Daehee and Stirling numbers, and their representations by using p-adic integrals and combinatorial techniques.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42075529","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 : 2022-09-27DOI: 10.7546/nntdm.2022.28.3.581-588
A. I. Vijaya Shankar
S. Ramanujan recorded theta function identities of different levels in the unorganized pages of his second notebook and the lost notebook. In this paper, we prove level 6 and level 10 theta function identities by using Eisenstein series identities.
S. Ramanujan在他的第二本笔记本和丢失的笔记本的杂乱无章的页面上记录了不同层次的θ函数恒等式。本文利用爱森斯坦级数恒等式证明了6级和10级函数恒等式。
{"title":"Eisenstein series of level 6 and level 10 with their applications to theta function identities of Ramanujan","authors":"A. I. Vijaya Shankar","doi":"10.7546/nntdm.2022.28.3.581-588","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.581-588","url":null,"abstract":"S. Ramanujan recorded theta function identities of different levels in the unorganized pages of his second notebook and the lost notebook. In this paper, we prove level 6 and level 10 theta function identities by using Eisenstein series identities.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"136 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71200735","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 : 2022-09-19DOI: 10.7546/nntdm.2022.28.3.558-563
K. Atanassov
The set Set(n), generated by an arbitrary natural number n, was defined in [2] and some arithmetic functions, defined over its elements are introduced in an algebraic aspect. Here, over the elements of Set(n), two arithmetic functions similar to the modal type of operators are defined and some of their basic properties are studied.
{"title":"Objects generated by an arbitrary natural number. Part 2: Modal aspect","authors":"K. Atanassov","doi":"10.7546/nntdm.2022.28.3.558-563","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.3.558-563","url":null,"abstract":"The set Set(n), generated by an arbitrary natural number n, was defined in [2] and some arithmetic functions, defined over its elements are introduced in an algebraic aspect. Here, over the elements of Set(n), two arithmetic functions similar to the modal type of operators are defined and some of their basic properties are studied.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49228286","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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