Analytic, Algebraic and Combinatorial studies on continued fractions

连分数的解析、代数和组合研究

• 批准号: 22540005
• 负责人: KOMATSU Takao
• 金额: $ 2.58万
• 依托单位: Hirosaki University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2010
• 资助国家: 日本
• 起止时间: 2010-04-01 至 2014-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22540005/
• 关键词: 数論; 代数学; 組み合わせ論; 連分数; 整数論; 組合せ論; 線形関係式; Leaping convergents

Analytic studies of continued fractions are to obtain exact algebraic expressions for approximation evalucations of various Hurwitz and Tasoev continued fractions, some relations between Fibonacci-zeta functions and continued fractions, and some relations between zeta functions and Cauchy polynomials. Algebraic studies of continued fractions are to obtain independence measures of arithmetic functions and to construct Liouville numbers in non- Archimedean case. Combinatorial studies of continued fractions are to give upper bounds on cyclotomic numbers, and to discover linear recurrence relations associated with multinomial Pascal triangles. In addition, cross-studies of continued fractions are to show some relations among continued fractions, Fibonacci numbers and congruent numbers, and to discover the concept of poly Cauchy numbers and polynomials as some generalizations of Cauchy numbers, which are related with Bernoulli numbers.

连分数的解析研究是为了获得各种Hurwitz和Tasoev连分数的近似求值的精确代数表达式，Fibonacci-zeta函数与连分数之间的一些关系，以及zeta函数与柯西多项式之间的一些关系。连分数的代数研究是为了获得算术函数的独立性测度并构造非阿基米德情况下的刘维尔数。连分数的组合研究是为了给出分圆数的上限，并发现与多项式帕斯卡三角形相关的线性递推关系。此外，连分数的交叉研究是为了揭示连分数、斐波那契数和同余数之间的一些关系，并发现作为柯西数的一些推广的聚柯西数和多项式的概念，与伯努利数有关。

项目成果

HYPERGEOMETRIC CAUCHY NUMBERS

• DOI: 10.1142/s1793042112501473
• 发表时间: 2013-04
• 期刊: International Journal of Number Theory
• 影响因子: 0.7
• 作者: T. Komatsu

Four-term leaping recurrence relations

四项跳跃递推关系

• 发表时间: 2009
• 期刊: 数理解析研究所講究録 1639
• 作者: Takao Komatsu;Carsten Elsner;Iekata Shiokawa;Takao Komatsu;Takao Komatsu
• 通讯作者: Takao Komatsu

Sums of products of Cauchy numbers, including poly-Cauchy numbers

柯西数的乘积和，包括聚柯西数

• 发表时间: 2013
• 期刊: J. Discrete Math.
• 作者: J.C.Eilbeck;S.Matustani;Y.Onishi;Takao Komatsu;Y.Onishi;Takao Komastsu
• 通讯作者: Takao Komastsu

Independence measures of arithmetic functions II

算术函数的独立性测度 II

• 发表时间: 2012
• 期刊: Acta Arith.
• 作者: Takao Komatsu;V. Laohakosol and P. Ruengsinsub
• 通讯作者: V. Laohakosol and P. Ruengsinsub

On the nearest integer of the sum of the reciprocal Fibonacci numbers

关于斐波那契数倒数之和的最接近整数

• 发表时间: 2011
• 期刊: Aportaciones Matematicas Investigacion
• 作者: Takao Komatsu;V. Laohakosol and P. Ruengsinsub;T. Komatsu and S. H. Holliday;T. Komatsu;T. Komatsu
• 通讯作者: T. Komatsu