W. Ramírez, C. Cesarano, S. Wani, S. Yousuf, D. Bedoya
{"title":"About properties and the monomiality principle of Bell-based Apostol-Bernoulli-type polynomials","authors":"W. Ramírez, C. Cesarano, S. Wani, S. Yousuf, D. Bedoya","doi":"10.15330/cmp.16.2.379-390","DOIUrl":"https://doi.org/10.15330/cmp.16.2.379-390","url":null,"abstract":"This article investigates the properties and monomiality principle within Bell-based Apostol-Bernoulli-type polynomials. Beginning with the establishment of a generating function, the study proceeds to derive explicit expressions for these polynomials, providing insight into their structural characteristics. Summation formulae are then derived, facilitating efficient computation and manipulation. Implicit formulae are also examined, revealing underlying patterns and relationships. Through the lens of the monomiality principle, connections between various polynomial aspects are elucidated, uncovering hidden symmetries and algebraic properties. Moreover, connection formulae are derived, enabling seamless transitions between different polynomial representations. This analysis contributes to a comprehensive understanding of Bell-based Apostol-Bernoulli-type polynomials, offering valuable insights into their mathematical nature and applications.","PeriodicalId":502864,"journal":{"name":"Carpathian Mathematical Publications","volume":"21 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141815369","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}
{"title":"Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements","authors":"V.R. Hladun, D.I. Bodnar, R.S. Rusyn","doi":"10.15330/cmp.16.1.16-31","DOIUrl":"https://doi.org/10.15330/cmp.16.1.16-31","url":null,"abstract":"In the paper, the problems of convergence and relative stability to perturbations of a branched continued fraction with positive elements and a fixed number of branching branches are investigated. The conditions under which the sets of elements [Omega_0 = ( {0,mu _0^{(2)}} ] times [ {nu _0^{(1)}, + infty } ),quad Omega _{i(k)}=[ {mu _k^{(1)},mu _k^{(2)}} ] times [ {nu _k^{(1)},nu _k^{(2)}} ],][i(k) in {I_k}, quad k = 1,2,ldots,] where $nu _0^{(1)}>0,$ $0 < mu _k^{(1)} < mu _k^{(2)},$ $0 < nu _k^{(1)} < nu _k^{(2)},$ $k = 1,2,ldots,$ are a sequence of sets of convergence and relative stability to perturbations of the branched continued fraction [frac{a_0}{b_0}{atop+}sum_{i_1=1}^Nfrac{a_{i(1)}}{b_{i(1)}}{atop+}sum_{i_2=1}^Nfrac{a_{i(2)}}{b_{i(2)}}{atop+}ldots{atop+} sum_{i_k=1}^Nfrac{a_{i(k)}}{b_{i(k)}}{atop+}ldots] have been established. The obtained conditions require the boundedness or convergence of the sequences whose members depend on the values $mu _k^{(j)},$ $nu _k^{(j)},$ $j=1,2.$ If the sets of elements of the branched continued fraction are sets ${Omega _{i(k)}} = ( {0,{mu _k}} ] times [ {{nu _k}, + infty } )$, $i(k) in {I_k}$, $k = 0,1,ldots,$ where ${mu _k} > 0$, ${nu _k} > 0$, $k = 0,1,ldots,$ then the conditions of convergence and stability to perturbations are formulated through the convergence of series whose terms depend on the values $mu _k,$ $nu _k.$ The conditions of relative resistance to perturbations of the branched continued fraction are also established if the partial numerators on the even floors of the fraction are perturbed by a shortage and on the odd ones by an excess, i.e. under the condition that the relative errors of the partial numerators alternate in sign. In all cases, we obtained estimates of the relative errors of the approximants that arise as a result of perturbation of the elements of the branched continued fraction.","PeriodicalId":502864,"journal":{"name":"Carpathian Mathematical Publications","volume":" 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140391137","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}
{"title":"Comparative growth of an entire function and the integrated counting function of its zeros","authors":"I. Andrusyak, P. Filevych","doi":"10.15330/cmp.16.1.5-15","DOIUrl":"https://doi.org/10.15330/cmp.16.1.5-15","url":null,"abstract":"Let $(zeta_n)$ be a sequence of complex numbers such that $zeta_ntoinfty$ as $ntoinfty$, $N(r)$ be the integrated counting function of this sequence, and let $alpha$ be a positive continuous and increasing to $+infty$ function on $mathbb{R}$ for which $alpha(r)=o(log (N(r)/log r))$ as $rto+infty$. It is proved that for any set $Esubset(1,+infty)$ satisfying $int_{E}r^{alpha(r)}dr=+infty$, there exists an entire function $f$ whose zeros are precisely the $zeta_n$, with multiplicities taken into account, such that the relation $$ liminf_{rin E, rto+infty}frac{loglog M(r)}{log rlog (N(r)/log r)}=0 $$ holds, where $M(r)$ is the maximum modulus of the function $f$. It is also shown that this relation is best possible in a certain sense.","PeriodicalId":502864,"journal":{"name":"Carpathian Mathematical Publications","volume":"18 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140430741","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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