Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher Order
We study the formulas for binomial sums of harmonic numbers of higher order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo&...
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2025-01-01
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| author | Takao Komatsu B. Sury |
| author_facet | Takao Komatsu B. Sury |
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| description | We study the formulas for binomial sums of harmonic numbers of higher order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover></mstyle><msubsup><mi>H</mi><mi>k</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msubsup><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac linethickness="0pt"><mi>n</mi><mi>k</mi></mfrac></mstyle></mfenced><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>q</mi><mo>)</mo></mrow><mi>k</mi></msup><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>=</mo><msubsup><mi>H</mi><mi>n</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msubsup><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><msub><mi mathvariant="script">D</mi><mi>r</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>q</mi><mi>j</mi></msup><mi>j</mi></mfrac></mstyle><mspace width="0.166667em"></mspace><mo>.</mo></mrow></semantics></math></inline-formula> Recently, Mneimneh proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">D</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In this paper, we find several different expressions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">D</mi><mi>r</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. |
| format | Article |
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| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-f51287362dd34001b72224c3fb53ac932025-01-24T13:40:10ZengMDPI AGMathematics2227-73902025-01-0113232110.3390/math13020321Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher OrderTakao Komatsu0B. Sury1Faculty of Education, Nagasaki University, Nagasaki 852-8521, JapanStat-Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560059, IndiaWe study the formulas for binomial sums of harmonic numbers of higher order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover></mstyle><msubsup><mi>H</mi><mi>k</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msubsup><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac linethickness="0pt"><mi>n</mi><mi>k</mi></mfrac></mstyle></mfenced><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>q</mi><mo>)</mo></mrow><mi>k</mi></msup><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>=</mo><msubsup><mi>H</mi><mi>n</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></msubsup><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><msub><mi mathvariant="script">D</mi><mi>r</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>q</mi><mi>j</mi></msup><mi>j</mi></mfrac></mstyle><mspace width="0.166667em"></mspace><mo>.</mo></mrow></semantics></math></inline-formula> Recently, Mneimneh proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">D</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In this paper, we find several different expressions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">D</mi><mi>r</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/13/2/321polynomial identitiesharmonic numbersdeterminantBell polynomials |
| spellingShingle | Takao Komatsu B. Sury Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher Order Mathematics polynomial identities harmonic numbers determinant Bell polynomials |
| title | Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher Order |
| title_full | Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher Order |
| title_fullStr | Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher Order |
| title_full_unstemmed | Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher Order |
| title_short | Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher Order |
| title_sort | polynomial identities for binomial sums of harmonic numbers of higher order |
| topic | polynomial identities harmonic numbers determinant Bell polynomials |
| url | https://www.mdpi.com/2227-7390/13/2/321 |
| work_keys_str_mv | AT takaokomatsu polynomialidentitiesforbinomialsumsofharmonicnumbersofhigherorder AT bsury polynomialidentitiesforbinomialsumsofharmonicnumbersofhigherorder |