2023/01/23 by Necdet Batır, Batir, Necdet, Sezer Sorgunand Sevda Atpinar +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Primary 05A10 #Secondary 05A19
paper · pdf · doi:10.48550/arxiv.2301.09587
openalex publication_date 2023/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. ∑k=0n\binomαn-k\binomβ+kkxk amp;=∑k=0n(-1)n+k\binomβ-α+nn-k\binomβ+kk(x+1)k, where n is a non-negative integer and α and β are complex numbers, which are not negative integers. Our approach is based on a particularly interesting combination of the Taylor theorem and the Wilf-Zeilberger algorithm. We also generalize a combinatorial identity due to Alzer and Kouba, and offer a new binomial sum identity. Furthermore, as applications, we give many harmonic number sum identities. As examples, we prove that Hn=(1)/(2)∑k=1n(-1)n+k\binomnk\binomn+kkHk and ∑k=0n\binomnk2HkHn-k=\binom2nn ((H2n-2Hn)2+Hn(2)-H2n(2)).