Parameterized Finite Binomial Sums

Abstract

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as ∑k=0,k≠nm+n(−1)kn−km+nk=(−1)nm+nn(Hm−Hn), where n,m are non-negative integers and Hn is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums.

Keywords

• binomial sums
• parameterized sums
• harmonic sums
• gamma function
• polygamma function