Proofs of Two Positivity Conjectures of Guo
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914607069659136 |
|---|---|
| author | Liu, Ji-Cai |
| author_facet | Liu, Ji-Cai |
| contents | We prove two positivity conjectures proposed by Guo for alternating sums and factorial ratios built from Gaussian coefficients. The first result proves the positivity of the odd $q$-super Catalan numbers \[
C_{m,n}(q)=\frac{[2m+1]![2n]!}{[m+n+1]![m]![n]!}. \] The proof uses the positivity theorem of Warnaar and Zudilin for the usual $q$-super Catalan numbers, together with two recurrences obtained from a double application of the $q$-Chu--Vandermonde summation. The second result proves Guo's conjectural strengthening of his alternating-sum positivity theorem, replacing the exponent coefficient $2r-1$ by every odd coefficient $2b-1$, $1\leq b\leq r$. Its proof combines a $q\mapsto q^{-1}$ reciprocity with a finite deletion recurrence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28012 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proofs of Two Positivity Conjectures of Guo Liu, Ji-Cai Combinatorics Number Theory 05A10, 05A30, 33D15 We prove two positivity conjectures proposed by Guo for alternating sums and factorial ratios built from Gaussian coefficients. The first result proves the positivity of the odd $q$-super Catalan numbers \[ C_{m,n}(q)=\frac{[2m+1]![2n]!}{[m+n+1]![m]![n]!}. \] The proof uses the positivity theorem of Warnaar and Zudilin for the usual $q$-super Catalan numbers, together with two recurrences obtained from a double application of the $q$-Chu--Vandermonde summation. The second result proves Guo's conjectural strengthening of his alternating-sum positivity theorem, replacing the exponent coefficient $2r-1$ by every odd coefficient $2b-1$, $1\leq b\leq r$. Its proof combines a $q\mapsto q^{-1}$ reciprocity with a finite deletion recurrence. |
| title | Proofs of Two Positivity Conjectures of Guo |
| topic | Combinatorics Number Theory 05A10, 05A30, 33D15 |
| url | https://arxiv.org/abs/2605.28012 |