On the Eventually Periodic Continued <i>β</i>-Fractions and Their Lévy Constants
In this paper, we consider continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fractions with golden ratio base <inline-formula>...
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MDPI AG
2022-01-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/1/127 |
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| author | Qian Xiao Chao Ma Shuailing Wang |
| author_facet | Qian Xiao Chao Ma Shuailing Wang |
| author_sort | Qian Xiao |
| collection | DOAJ |
| description | In this paper, we consider continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fractions with golden ratio base <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>. We show that if the continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mo>[</mo><mi>β</mi><mo>]</mo></mrow></semantics></math></inline-formula> and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fraction expansion is dense in [<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">c</mi></semantics></math></inline-formula>, +<i>∞</i>), where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">c</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo form="prefix">log</mo><mfrac><mrow><mi>β</mi><mo>+</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>5</mn><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T03:31:32Z |
| format | Article |
| id | doaj.art-14ad4a8946d4499fab2c376cf25e8528 |
| institution | Directory Open Access Journal |
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| language | English |
| last_indexed | 2024-03-10T03:31:32Z |
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| spelling | doaj.art-14ad4a8946d4499fab2c376cf25e85282023-11-23T11:54:34ZengMDPI AGMathematics2227-73902022-01-0110112710.3390/math10010127On the Eventually Periodic Continued <i>β</i>-Fractions and Their Lévy ConstantsQian Xiao0Chao Ma1Shuailing Wang2School of Mathematics, South China University of Technology, Guangzhou 510640, ChinaFaculty of Information Technology, Macau University of Science and Technology, Macau 999078, ChinaSchool of Mathematics, South China University of Technology, Guangzhou 510640, ChinaIn this paper, we consider continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fractions with golden ratio base <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>. We show that if the continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mo>[</mo><mi>β</mi><mo>]</mo></mrow></semantics></math></inline-formula> and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-fraction expansion is dense in [<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">c</mi></semantics></math></inline-formula>, +<i>∞</i>), where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">c</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo form="prefix">log</mo><mfrac><mrow><mi>β</mi><mo>+</mo><mn>2</mn><mo>−</mo><msqrt><mrow><mn>5</mn><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/10/1/127<i>β</i>-integercontinued <i>β</i>-fractionseventually periodicLévy constant |
| spellingShingle | Qian Xiao Chao Ma Shuailing Wang On the Eventually Periodic Continued <i>β</i>-Fractions and Their Lévy Constants Mathematics <i>β</i>-integer continued <i>β</i>-fractions eventually periodic Lévy constant |
| title | On the Eventually Periodic Continued <i>β</i>-Fractions and Their Lévy Constants |
| title_full | On the Eventually Periodic Continued <i>β</i>-Fractions and Their Lévy Constants |
| title_fullStr | On the Eventually Periodic Continued <i>β</i>-Fractions and Their Lévy Constants |
| title_full_unstemmed | On the Eventually Periodic Continued <i>β</i>-Fractions and Their Lévy Constants |
| title_short | On the Eventually Periodic Continued <i>β</i>-Fractions and Their Lévy Constants |
| title_sort | on the eventually periodic continued i β i fractions and their levy constants |
| topic | <i>β</i>-integer continued <i>β</i>-fractions eventually periodic Lévy constant |
| url | https://www.mdpi.com/2227-7390/10/1/127 |
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