New Harmonic Number Series
Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous qua...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-03-01
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| Series: | AppliedMath |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2673-9909/5/1/21 |
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| Summary: | Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous quadratic Euler sum originally discovered in 1995 by Borwein and Borwein. In addition, the following series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mfenced separators="" open="(" close=")"><mfrac linethickness="0pt"><mrow><mi>n</mi><mo>+</mo><mi>z</mi></mrow><mi>n</mi></mfrac></mfenced></mrow></mfrac></mstyle><mo>,</mo><mspace width="1.em"></mspace><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mfenced separators="" open="(" close=")"><mfrac linethickness="0pt"><mrow><mi>n</mi><mo>+</mo><mi>z</mi></mrow><mi>n</mi></mfrac></mfenced></mrow></mfrac></mstyle><mo>,</mo><mspace width="1.em"></mspace><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mrow><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mfenced separators="" open="(" close=")"><mfrac linethickness="0pt"><mrow><mi>n</mi><mo>+</mo><mi>z</mi></mrow><mi>n</mi></mfrac></mfenced></mrow></mfrac></mstyle></mrow></semantics></math></inline-formula>, as well as the harmonic and odd harmonic number series associated with them are evaluated. |
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| ISSN: | 2673-9909 |