Minimal recurrences for binomial power sums
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…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2611.00019
curl -X POST /api/v1/records/hexagon/2611.00019/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2611.00019/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'