{"id":"hexagon/2611.00019","pdf":"https://hexagonmath.org/pdf/2611.00019v1","url":"https://hexagonmath.org/abs/2611.00019v1","tags":["math.GM","msc:11B37","Ap\\'ery limits","Franel numbers","Franel’s recurrence problem","binomial power sums"],"score":0,"title":"Minimal recurrences for binomial power sums","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Khai-Hoan Nguyen-Dang"],"reviews":0,"subject":"math.NT","summary":"For every positive integer $r$, we prove that the least order of a polynomial-coefficient recurrence for $F_r(n)=\\sum_{k=0}^n\\binom nk^r$ is $\\lfloor(r+1)/2\\rfloor$. This establishes the unrestricted minimal-order assertion arising from Franel's work of 1894--1895. We also determine the least order $\\lfloor r/2\\rfloor$, for $r\\geq2$, of the alternating sums $(-1)^n\\sum_{k=0}^{2n}(-1)^k\\binom{2n}{k}^r$. The lower bounds follow from real moment representations: every nonzero singularity of the representing measure forces a root of the leading characteristic polynomial of the recurrence. A…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}