2009/11/12 by F. M. S. Lima, Lima, F. M. S.
Mathematics · #11J81 #11J91 #11M41 #33B15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.0911.2441
openalex publication_date 2009/11/12 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
In a recent work, Gun and co-workers have proposed that ∑n=-∞∞(n+α)-k is a transcendental number for all integer k, k > 1, and α∈ ℚ \backslash ℤ. Here in this work, this proposition is shown to be false whenever k is odd and α is a half-integer. It is also shown that these are the only counterexamples, which allows for a correct reformulation of the original proposition. This leads to a theorem yielding a closed-form expression for the summation of that series, which determines its arithmetic nature. The result is then extended to a sum of polygamma functions and some related zeta series. In view of the recurrent appearance of these series and functions in different areas of mathematics and applications, the closed-form results put forward here could well be included in modern computer algebra systems (CAS).