2017/04/09 by Ce Xu, Xu, Ce
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1704.03515
openalex publication_date 2017/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p,p1,…,pm be positive integers with p1≤ p2≤⋯≤ pm and x∈ [-1,1), define the so-called Euler type sums S_p1p2 ⋯ pm,p( x ), which are the infinite sums whose general term is a product of harmonic numbers of index n, a power of n-1 and variable xn, by Sp1 p2 ⋯ pm, p(x) := ∑n = 1^∞ \fracHn(p1) Hn(p2) ⋯ Hn(pm) np xn (m∈ ℕ := \1,2,3,…\), where Hn(p) is defined by the generalized harmonic number. Extending earlier work about classical Euler sums, we prove that whenever p+p1+⋯+pm ≤ 5, then all sums S_p1p2 ⋯ pm,p( 1/2) can be expressed as a rational linear combination of products of zeta values, polylogarithms and log(2). The proof involves finding and solving linear equations which relate the different types of sums to each other.