Research on Quasi-periodic continued fractions in terms of Special functions

• Project/Area Number: 18540006
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Hirosaki University
• Principal Investigator: KOMATSU Takao Hirosaki University, 大学院・理工学研究科, 教授 (70300556)
• Research Collaborator: SHIOKAWA Iekata 慶応大学, 理工学部, 名誉教授 ; BOWMAN Dougalas 北イリノイ大学, 数理科学科, 教授(合衆国) ; ELSNER Carsten ハノーヴァー応用科学大学, 教授(ドイツ) ; LAOHAKOSOL Vichian カセットサート大学, 数学科, 教授(タイ) ; CALDWELL Chris テネシー大学, マーチン校・数学統計学科, 教授(合衆国)
• Project Period (FY): 2006 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,660,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥660,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
• Keywords: 整数論 / 関数論 / 組合せ論 / 代数学 / Continued fractions / Tasoev continued fractions / Hurwitz continued fractions / Hyprgeometric functions / Diophantine Analysis / Leaping convergents / Hypergeometric functions

Research Abstract

Still new types of Hurwitz and Tasoev continued fractions, which belong to quasi-periodic continued fractions, have been found. Several typical relations between quasi-periodic continued fractions and hypergeometric functions, hyperbolic functions, Fibonacci-zeta functions have been discovered. Leaping convergents have been defined from the concept of convergents of quasi-periodic continued fractions. Characteristic properties of leaping convergents have been examined. Leaping convergents have been extensively defined on non-regular continued fractions and on general integer sequences.

Research Products

• [Journal Article] 非アルキメデス附値におけるLiouville数 (2010)
  • Author(s): Takao Komatsu
  • Journal Title: 日本数学会代数学分科会講演アブストラクト Pages: 91-92
• [Journal Article] On convergents formed from Diophantine equations (2009)
  • Author(s): Carsten Elsner, Takao Komatsu, Iekata Shiokawa
  • Journal Title: Glasnik Mat. 44 Pages: 267-284
  • Peer Reviewed
• [Journal Article] Shrinking the period length of quasi-periodic continued fractions (2009)
  • Author(s): Takao Komatsu
  • Journal Title: J. Number Theory 129 Pages: 358-366
  • Peer Reviewed
• [Journal Article] Shrinking the period length of quasi-periodic continued fractions (2009)
  • Author(s): Takao Komatsu
  • Journal Title: Journal of Number Theory 129 Pages: 358-366
  • Peer Reviewed
• [Journal Article] A Diophantine approximation of e^(1/s) in terms of integrals (2009)
  • Author(s): Takao Komatsu
  • Journal Title: Tokyo Journal of Mathematics 32 Pages: 159-176
  • Peer Reviewed
• [Journal Article] Diophantine approximations of tanh, tan, and linear forms of e in terms of integrals (2009)
  • Author(s): Takao Komatsu
  • Journal Title: Revue Roumaine de Mathematiques Pures et Appliquees 54 Pages: 223-242
  • Peer Reviewed
• [Journal Article] On convergents formed from Diophantine equations (2009)
  • Author(s): Takao Komatsu, Carsten Elsner, Iekata Shiokawa
  • Journal Title: Glasnik Matematicki 44 Pages: 267-284
  • Peer Reviewed
• [Journal Article] On continued fraction expansions of Fibonacci and Lucas Dirichlet series (2009)
  • Author(s): Takao Komatsu
  • Journal Title: Fibonacci Quarterly 46/47 Pages: 267-278
  • Peer Reviewed
• [Journal Article] Four-term leaping recurrence relations (2009)
  • Author(s): Takao Komatsu
  • Journal Title: 数理解析研究所講究録 1639 Pages: 1-11
  • Peer Reviewed
• [Journal Article] Diophantine approximations with leaping convergents (2009)
  • Author(s): Takao Komatsu
  • Journal Title: 数理解析研究所講究録 1665 Pages: 67-79
  • Peer Reviewed
• [Journal Article] Fibonacci-Lucas Zeta functions (2009)
  • Author(s): 小松尚夫
  • Journal Title: 日本数学会代数学分科会講演アブストラクト Pages: 49-50
• [Journal Article] On the residue classes of integer sequences satisfying a linear three-term recurrence formula (2008)
  • Author(s): Carsten Elsner, Takao Komatsu
  • Journal Title: Linear Algebra Appl. 429 Pages: 993-947
  • Peer Reviewed
• [Journal Article] On the residue classes of integer sequences satisfying a linear three-term recurrence formula (2008)
  • Author(s): Carsten Elsner and Takao Komatsu
  • Journal Title: Linear Algebra Appl. 429 Pages: 993-947
  • Peer Reviewed
• [Journal Article] A recurrence formula for leaping convergents of non-regular continued fractions (2008)
  • Author(s): 小松尚夫 Carsten Elsner
  • Journal Title: Linear Algebra and its Applications 428 Pages: 824-833
  • Peer Reviewed
• [Journal Article] Heron's square root approximations (2008)
  • Author(s): 小松尚夫
  • Journal Title: JP Journal of Algebra, Number theory and applications 10 Pages: 81-88
  • Peer Reviewed
• [Journal Article] Tasoev continued fractions with long period (2008)
  • Author(s): 小松尚夫
  • Journal Title: Far East Journal of Mathematical Sciences(FJMS) 28 Pages: 89-121
  • Peer Reviewed
• [Journal Article] On the residue classes of integer sequences satisfying a linear three-term recurrence formula (2008)
  • Author(s): 小松尚夫 Carsten Elsner
  • Journal Title: Linear Algebra and its applications 429 Pages: 933-947
  • Peer Reviewed
• [Journal Article] More on Hurwitz and Tasoev continued fractions (2008)
  • Author(s): 小松尚夫
  • Journal Title: Sarajevo Journal of Mathematics 4 Pages: 155-180
  • Peer Reviewed
• [Journal Article] Four-term leaping recurrence relations (2008)
  • Author(s): 小松尚夫
  • Journal Title: 日本数学会代数学分科会講演アブストラクト Pages: 31-32
• [Journal Article] 連分数の周期縮小 (2008)
  • Author(s): 小松尚夫
  • Journal Title: 日本数学会代数学分科会講演アブストラクト(2008年3月23日〜3月26日, 近畿大学) Pages: 31-32
• [Journal Article] Approximation of values of hyperbolic functions by restricted rationals (2007)
  • Author(s): Carsten Elsner, Takao Komatsu, Iekata Shiokawa
  • Journal Title: J. Theorie des Nombres de Bordeaux 19 Pages: 393-404
  • Peer Reviewed
• [Journal Article] Hurwitz continued fractions with confluent hypergeometric functions (2007)
  • Author(s): Takao Komatsu
  • Journal Title: Czech. Math. J. 57 Pages: 919-932
  • Peer Reviewed
• [Journal Article] Approximation of values of hyperbolic functions by restricted rationals (2007)
  • Author(s): Carsten Elsner, Takao Komatsu and Iekata Shiokawa
  • Journal Title: J. Theorie des Nombres de Bordeaux 19 Pages: 393-404
  • Peer Reviewed
• [Journal Article] A proof of the continued fraction expansion of e^<2/s> (2007)
  • Author(s): Takao Komatsu
  • Journal Title: Integers 7
  • Peer Reviewed
• [Journal Article] Some combinatorial properties of leaping convergents (2007)
  • Author(s): Takao Komatsu
  • Journal Title: Combinatorial Number Theory, Walter de Gruyter Pages: 315-325
  • Peer Reviewed
• [Journal Article] Approximation of values of hyperbolic funtions by restricted rationals (2007)
  • Author(s): Carsten Elsner, Takao Komatsu, Iekata Shiokawa
  • Journal Title: J. Theor. Nombre de Bordeaux 17 Pages: 393-404
  • Peer Reviewed
• [Journal Article] ピタゴラス数による双曲線関数の値の近似 (2007)
  • Author(s): 小松尚夫
  • Journal Title: 日本数学会代数学分科会講演アブストラクト(2007年9月21日〜9月24日, 東北大学) Pages: 109-110
• [Journal Article] Continued fraction of e^2 with confluent hypergeometric functions (2006)
  • Author(s): Takao Komatsu
  • Journal Title: Liet. Mat. Rink. 46 Pages: 513-531
  • Peer Reviewed
• [Journal Article] Hurwitz and Tasoev continued fractions with long period (2006)
  • Author(s): Takao Komatsu
  • Journal Title: Math. Pannon. 17 Pages: 91-110
• [Journal Article] Some recognizable forms of simple continued fractions, II (2006)
  • Author(s): Takao Komatsu
  • Journal Title: Rev. Roum. Mat. Pures Appl. (RRMPA) 51 Pages: 219-238
• [Journal Article] Tasoev型連分数の有理近似 (2006)
  • Author(s): Takao Komatsu
  • Journal Title: 数理解析研究所講究録 1511 Pages: 166-172
• [Journal Article] On convergents formed by Pythagorean numbers (2006)
  • Author(s): Carsten Elsner, Takao Komatsu, Iekata Shiokawa
  • Journal Title: Diophantine Analysis and Related Fields 2006, in honor of lekata Shiokawa, Seminar on Mathematical Sciences 35 Pages: 59-76
• [Journal Article] More on the continued fractions of e^2 (2006)
  • Author(s): Takao Komatsu
  • Journal Title: Diophantine Analysis and Related Fields 2006, in honor of lekata Shiokawa, Seminar on Mathematical Sciences 35 Pages: 111-124
• [Journal Article] Continued fractions of e^2 with confluent hypergeometric funtions (2006)
  • Author(s): Takao Komatsu
  • Journal Title: Liet. Matem. Rink 46 Pages: 513-531
• [Presentation] 非アルキメデス付置におけるLiouville数 (2010)
  • Author(s): 小松尚夫
  • Organizer: 日本数学会年会代数学分科会
  • Place of Presentation: 慶應義塾大学理工学部
  • Year and Date: 2010-03-26
• [Presentation] Arithmetical properties of Tribonacci numbers (2010)
  • Author(s): Takao Komatsu
  • Organizer: Mini-workshop in Algebraic Combinatorics
  • Place of Presentation: 韓国・浦項工科大学
  • Year and Date: 2010-02-04
• [Presentation] Powers of Sierpinski numbers base $b$ (2009)
  • Author(s): 小松尚夫
  • Organizer: 代数学と計算(AC2009)
  • Place of Presentation: 首都大学東京
  • Year and Date: 2009-12-04
• [Presentation] Arithmetical properties of reciprocal sums of Fibonacci-type numbers (2009)
  • Author(s): Takao Komatsu
  • Organizer: Integers Conference 2009
  • Place of Presentation: 合衆国West Georgia大学
  • Year and Date: 2009-10-14
• [Presentation] Some arithmetical properties of Fibonacci zeta functions (2009)
  • Author(s): Takao Komatsu
  • Organizer: 第26回Journees Arithmetiques(JA2009)
  • Place of Presentation: フランスSaint-Etienne大学
  • Year and Date: 2009-07-07
• [Presentation] Fibonacci-Lucas Zeta functions (2009)
  • Author(s): 小松尚夫
  • Organizer: 日本数学会年会代数学分科会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-27
• [Presentation] Three-term recurrence relations and continued fractions (2009)
  • Author(s): 小松尚夫
  • Organizer: Diophantine Analysis and related fields 2009
  • Place of Presentation: 日本大学駿河台キャンパス
  • Year and Date: 2009-03-03
• [Presentation] Arithmetical properties of Fibonacci Zeta functions (2009)
  • Author(s): Takao Komatsu
  • Organizer: JA2009
  • Place of Presentation: サンテティエンヌ大学 フランス
• [Presentation] Four-term leaping recurrence relations (2008)
  • Author(s): 小松尚夫
  • Organizer: 数理解析研究所研究集会「解析的整数論の新しい展開」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-10-27
• [Presentation] Four-term leaping recurrence relations (2008)
  • Author(s): 小松尚夫
  • Organizer: 日本数学会秋季総合分科会代数学分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-24
• [Presentation] 連分数の周期縮小 (2008)
  • Author(s): 小松尚夫
  • Organizer: 日本数学会年会代数学分科会
  • Place of Presentation: 近畿大学理工学部
  • Year and Date: 2008-03-23
• [Presentation] Continued fraction expansions of Fibonacci zeta functions (2008)
  • Author(s): Takao Komatsu
  • Organizer: Diophantine Analysis and related fields 2008
  • Place of Presentation: 同志社大学室町キャンパス
  • Year and Date: 2008-03-06
• [Presentation] Three-term recurrence relations and continued fractions (2008)
  • Author(s): Takao Komatsu
  • Organizer: 4th Korea-Japan Workshop on Algebra and Combinatorics
  • Place of Presentation: 浦項工科大学(韓国)
  • Year and Date: 2008-02-02
• [Presentation] Three term recurrence relations and continued fractions (2008)
  • Author(s): Takao Komatsu
  • Organizer: 4th Korea-Japan Workshop on Algebra and Combinatorics
  • Place of Presentation: POSTECH 浦項 韓国
• [Presentation] 三項関係式と整数列 (2007)
  • Author(s): Takao Komatsu
  • Organizer: 第7回「代数学と計算」研究集会(AC2007)
  • Place of Presentation: 首都大学東京国際交流会館
  • Year and Date: 2007-12-07
• [Presentation] A combinatorial proof of the continued fraction expansion ot e (2007)
  • Author(s): Takao Komatsu
  • Organizer: Integers Conference 2007
  • Place of Presentation: 西ジョージア大学(合衆国)
  • Year and Date: 2007-10-24
• [Presentation] Diophantine approximations with leaping convergents (2007)
  • Author(s): Takao Komatsu
  • Organizer: 解析的整数論研究集会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2007-10-18
• [Presentation] Diophantine approximation of e in expressions of integrals (2007)
  • Author(s): Takao Komatsu
  • Organizer: Workshop Number Theory and Ergodic Theory
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2007-09-27
• [Presentation] ピタゴラス数による双曲線関数の値の近似 (2007)
  • Author(s): 小松尚夫
  • Organizer: 日本数学会秋季総合分科会代数学分科会
  • Place of Presentation: 東北大学川内北キャンパス
  • Year and Date: 2007-09-24
• [Presentation] Some combinatorial properties of leaping convergents, II (2006)
  • Author(s): Takao Komatsu
  • Organizer: 12th International Conference on Fibonacci numbers and their applications
  • Place of Presentation: サンフランシスコ州立大学 合衆国
• [Book] 解析的整数論の新しい展開 (2009)
  • Author(s): Takao Komatsu
  • Total Pages: 186
  • Publisher: 京都大学数理解析研究所講究録
• [Book] Advances in Mathematics Research (2009)
  • Author(s): Takao Komatsu, 他11名
  • Total Pages: 345
  • Publisher: Nova Publishers
• [Book] Advances in Mathematics Research, Volume 9 (2009)
  • Author(s): Takao Komatsu, 他12名
  • Total Pages: 345
  • Publisher: Nova Publishers
• [Book] 京都大学数理解析研究所講究録1639「解析的整数論の新しい展開」 (2009)
  • Author(s): 小松尚夫
  • Total Pages: 186
  • Publisher: 京都大学数理解析研究所
• [Book] Diophantine Analysis and Related Fields-DARF 2007/2008 (2008)
  • Author(s): Takao Komatsu
  • Total Pages: 211
  • Publisher: American Institute of Physics
• [Book] Diophantine Analysis and related fields, DARF2007/2008 AIP Conference Proceedings 976 (2008)
  • Author(s): 小松尚夫
  • Total Pages: 211
  • Publisher: American Institute of Physics
• [Book] 初等整数論9章第2版 (2008)
  • Author(s): 小松尚夫
  • Total Pages: 380
  • Publisher: 森北出版
• [Book] Dionhantine Analvsis and Related Fields-DARF2007/2008 (2008)
  • Author(s): Takao Komatsu
  • Total Pages: 211
  • Publisher: American Institute of Physics
• [Book] Diophantine Analysis and Related Fields 2006(Seminar on Mathematical Sciences No. 35) (2006)
  • Author(s): Masanori Katsurada, Takao Komatsu, Hitoshi Nakada
  • Total Pages: 221
  • Publisher: Keio University
• [Book] Diophantine Analysis and Related Fields 2006, in honor of Iekata Shiokawa, Seminar on Mathematical Sciences, No.35 (2006)
  • Author(s): Masanori Katsurada, Takao Komatsu, Hitoshi Nakada
  • Total Pages: 221
  • Publisher: 慶應大学理工学部