A Factorization of the Log-Concavity Operator for Pascal Determinantal Arrays and Their Infinite Row-Wise Log-Concavity
Hossein Teimoori Faal, Hasan Khodakarami
TL;DR
The paper addresses the problem of establishing infinite row-wise log-concavity for the Pascal determinantal arrays $\\operatorname{PD}_k$ across all $k \\ge 1$ by identifying a precise algebraic structure for the log-concavity operator. The main method proves the exact factorization $\\mathcal{L}(\\operatorname{PD}_k) = \\operatorname{PD}_{k-1} \\Had \\operatorname{PD}_{k+1}$ and a Hadamard-submultiplicativity inequality $\\mathcal{L}(A \\Had X) \\ge \\mathcal{L}(A) \\Had \\mathcal{L}(X)$, enabling induction on both $k$ and the iteration depth. The key contributions are a uniform algebraic proof that every row of every $\\PD_k$ with $k \\ge 1$ is infinitely log-concave (extending Brändén's result for $\\PD_1$), together with log-convexity in $k$ and determinantal Hadamard inequalities. The work reveals a rigid multiplicative structure tied to the star-of-David rule in totally positive kernels and suggests natural $q$-analogues and extensions to other determinantal families.
Abstract
We study the Pascal determinantal arrays $\PD_k$, whose entries $\PD_k(i,j)$ are the $k\times k$ minors of the lower-triangular Pascal matrix $P=( \binom{a}{b} )_{a,b\ge 0}$. We prove an exact factorization of the row-wise log-concavity operator: \[ \LC(\PD_k)=\PD_{k-1}\Had\PD_{k+1}, \] where $\LC(a)_j=a_j^2-a_{j-1}a_{j+1}$ and $\Had$ denotes the Hadamard (entrywise) product. This identity is established by an elementary algebraic manipulation implicitly based on the idea of start of David rule. We further prove a general inequality asserting that the log-concavity operator is submultiplicative under Hadamard products of log-concave arrays: $\LC(A\Had X)\ge\LC(A)\Had\LC(X)$. Combining the factorization with this inequality yields a uniform algebraic proof that every row of every array $\PD_k$ ($k\ge 1$) is infinitely log-concave, extending the celebrated theorem of Brändén for the particular case of Pascal's triangle ($\PD_1$) to the entire determinantal hierarchy. Applications include the log-convexity of $\{\PD_k(i,j)\}_{k\ge 0}$ in the determinantal order $k$ and a family of determinantal Hadamard inequalities.