Pub Date : 2024-07-22DOI: 10.15330/cmp.16.2.379-390
W. Ramírez, C. Cesarano, S. Wani, S. Yousuf, D. Bedoya
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.
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The paper establishes new convergence domains of branched continued fraction expansions of Horn's hypergeometric function $H_4$ with real and complex parameters. These domains enabled the PC method to establish the analytical extension of analytical functions to their expansions in the studied domains of convergence. A few examples are provided at the end to illustrate this.
本文建立了具有实参数和复参数的霍恩超几何函数 $H_4$ 的分支续分展开的新收敛域。这些域使 PC 方法能够在所研究的收敛域中建立解析函数对其展开的解析扩展。最后提供几个例子来说明这一点。
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.
本文研究了具有正元素和固定分支数的分支连续分数的收敛性和对扰动的相对稳定性问题。元素集 [ （Omega_0 = ( {0,mu _0^{(2)}} ] times [ {nu _0^{(1)}, + infty } ] 的条件是),quad Omega _{i(k)}=[ {mu _k^{(1)},mu _k^{(2)}} ] times [ {nnu _k^{(1)},nnu _k^{(2)}} ],][i(k) in {I_k}, quad k = 1、2,ldots,] 其中 $nu _0^{(1)}>0,$ $0 < mu _k^{(1)} < mu _k^{(2)},$ $0 < nu _k^{(1)} < nu _k^{(2)},$ $k = 1,2,ldots、$ 是一系列收敛和相对稳定的分支续分数扰动的集合序列[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] 已经建立。所得到的条件要求序列的有界性或收敛性，其成员取决于值 $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 ${mu _k}> 0$, ${nu _k}> 0$，$k = 0,1,ldots,$ 那么对扰动的收敛性和稳定性条件是通过其项取决于 $mu _k，$nu _k 值的数列的收敛性来制定的。$ 如果分数偶数层的部分分母受到短缺的扰动，奇数层的部分分母受到过剩的扰动，即部分分母的相对误差在符号上交替变化，那么支化续分数相对抗扰动的条件也就成立了。在所有情况下，我们都得到了因支化续分数元素扰动而产生的近似值相对误差的估计值。
{"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}
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.
{"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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