2021/10/14 by Bakir Farhi, Farhi, Bakir
Mathematics · Physics and Astronomy · #11C08 #13F25 #Advanced Mathematical Identities #Arithmetic #Arithmetic progression #Combinatorics #Computer science #Discrete mathematics #FOS: Mathematics #Formal power series #Historical Astronomy and Related Studies #History and Theory of Mathematics #Integer (computer science) #Mathematical analysis #Mathematics #Natural number #Number Theory (math.NT) #Polynomial #Power series #Primary 11B68 #Pure mathematics #Real number #Space (punctuation) #Vector space #math.NT #msc:11B68 #msc:11C08 #msc:13F25
paper · pdf · doi:10.48550/arxiv.2110.07400
published in arXiv (Cornell University) (Cornell University) · 17 pages
arxiv created 2021/10/14 · openalex publication_date 2021/10/14 · arxiv updated 2021/10/15 · openalex created_date 2022/07/18 · openalex updated_date 2026/08/05
The purpose of this paper consists to study the sums of the type P(n) + P(n - d) + P(n - 2 d) + …, where P is a real polynomial, d is a positive integer and the sum stops at the value of P at the smallest natural number of the form (n - k d) (k ∈ ℕ). Precisely, for a given d, we characterize the ℝ-vector space \mathscrEd constituting of the real polynomials P for which the above sum is polynomial in n. The case d = 2 is studied in more details. In the last part of the paper, we approach the problem through formal power series; this inspires us to generalize the spaces \mathscrEd and the underlying results. Also, it should be pointed out that the paper is motivated by the curious formula: n2 + (n - 2)2 + (n - 4)2 + … = (n (n + 1) (n + 2))/(6), due to Ibn al-Banna al-Marrakushi (around 1290).