LINK to Andrew Elvey Price's list of all OEIS sequences with atleast 15 terms that may be Stieltjes moment sequences (as of 25/01/2024).
The Apéry numbers are defined by \( A_n = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{n+k}{k}^2 \). They played a key role in Apéry's 1978 proof of irrationality of \(\zeta(3)\).
Define the Apéry polynomials to be \[ A_n(x) = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{n+k}{k}^2 x^k. \]
Conjecture: The sequence of real numbers \( \left(A_n(x)\right)_{n\geq 0}\) is a Stieltjes moment sequence for all \(x \geq 1\).
The Apéry polynomials are defined by \[ A_n(x) = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{n+k}{k}^2 x^k. \]
Conjecture: The sequence of polynomials \( \left(A_n(1+y)\right)_{n\geq 0}\) is coefficientwise Hankel totally positive in \(y\).
The case \(x=1\) in conjecture P1.1, the Apéry numbers, was proven to be a Stieltjes moment sequence by Edgar, see arXiv:2005.10733.
Conjecture P1.2 was tested up to \(12 \times 12\).
Let \(T\) be a tree with vertex set \([n]\), rooted at the vertex \(1\). An inversion of \(T\) is an ordered pair \((j, k)\) of vertices such that \(j > k\) and the path from \(1\) to \(k\) passes through \(j\). Define the inversion enumerator for trees \[I_n(q) = \sum_{T\in \mathcal{T}_n} q^{\# {\text{inversions}}}\]
Conjecture: The sequence of polynomials \( \left(I_{n+1}(q)\right)_{n\geq 0}\) is coefficientwise Hankel totally positive in \(q\).
\(I_n(0) = (n-1)!\) and \(I_n(1) = n^{n-2}\).
Conjecture P1.3 was tested up to \(10 \times 10\).
The little Domb numbers are defined by \(\tilde{D}_n = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{2k}{k}\) and the little Domb polynomials are defined by \[\tilde{D}_n(x) = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{2k}{k} x^k\]
Conjecture: The sequence of polynomials \( \left(\tilde{D}_n(1+y)\right)_{n\geq 0}\) is coefficientwise Hankel totally positive in \(y\).
The Domb numbers are defined by \(D_n = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{2k}{k} \binom{2n-2k}{n-k}\) and the Domb polynomials are defined by \[D_n(x) = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{2k}{k}\binom{2n-2k}{n-k} x^k\]
Conjecture: The sequence of polynomials \( \left(D_n(1+y)\right)_{n\geq 0}\) is coefficientwise Hankel totally positive in \(y\).
The Boros--Moll polynomials are defined by \[{\rm BM}_n(x) = \sum_{k=0}^{n} \binom{n+k}{k} \binom{2n-2k}{n-k} x^k\]
Conjecture: The sequence of polynomials \( \left({\rm BM}_n(1+y)\right)_{n\geq 0}\) is coefficientwise Hankel totally positive in \(y\).
For any \(x>0\) (Boros & Moll, 1999) showed that the sequence \( {\rm BM}_n(x) \) is a Stieltjes moment sequence
➡️ Continue to Theme 1.7
Problems related to Stieltjes moment sequences and coefficientwise Hankel total positivity in permutation pattern avoidance.
Here the flagship problem is Problem P1.7.1.
➡️ Continue to Theme 1.8
Problems related to Stieltjes moment sequences and total positivity in enumeration of graphs on surfaces.
The generalised Jacobi--Stirling triangle \(\left({\rm JS}_{n,k}\right)_{n,k\geq 0}\) is defined by the following recurrence \[ {\rm JS}_{n,k} = {\rm JS}_{n-1,k-1} + (a k^2 + b k) {\rm JS}_{n-1,k} \] with initial condition \({\rm JS}_{0,0}=1\) and \({\rm JS}_{n,k}=0\) unless \(0\leq k \leq n\).
Let \({\rm JS}_{n}(x) = \sum_{k=0}^n {\rm JS}_{n,k} x^k\) denote its row-generating polynomial.
Conjecture: The sequence of polynomials \( ({\rm JS}_n(x))_{n\geq 0} \) is coefficientwise Hankel totally positive in the indeterminates \(a, b, x\).
Zhu in [Zhu (2014)] and [Zhu (2018)] showed that the sequence \( ({\rm JS}_n(x))_{n\geq 0} \) is coefficientwise log-convex.