Polynomial Identities for Binomial Sums of Harmonic Numbers of Higher Order

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&...

Full description

Saved in:
Bibliographic Details
Main Authors: Takao Komatsu, B. Sury
Format: Article
Language:English
Published: MDPI AG 2025-01-01
Series:Mathematics
Subjects:
polynomial identities
harmonic numbers
determinant
Bell polynomials
Online Access:https://www.mdpi.com/2227-7390/13/2/321
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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>.
ISSN:2227-7390

Similar Items