A Factorization of the Log-Concavity Operator for Pascal Determinantal Arrays and Their Infinite Row-Wise Log-Concavity

We study the Pascal determinantal arrays ${\text{PD}}_k$, whose entries ${\text{PD}}_k(i,j)$ are the $k\times k$ minors of the lower-triangular Pascal matrix $P=(\left(\genfrac{}{}{0px}{}{a}{b}\right){)}_{a,b\ge 0}$. We prove an exact factorization of the row-wise log-concavity operator: $$\text{LC}({\text{PD}}_k)={\text{PD}}_{k-1}\text{Had}{\text{PD}}_{k+1},$$ where $\text{LC}(a{)}_j=a_j^2-a_{j-1}{a}_{j+1}$ and $\text{Had}$ 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: $\text{LC}(A\text{Had}X)\ge \text{LC}(A)\text{Had}\text{LC}(X)$. Combining the factorization with this inequality yields a uniform algebraic proof that every row of every array ${\text{PD}}_k$ ($k\ge 1$) is infinitely log-concave, extending the celebrated theorem of Brändén for the particular case of Pascal’s triangle (${\text{PD}}_1$) to the entire determinantal hierarchy. Applications include the log-convexity of $\{{\text{PD}}_k(i,j){\}}_{k\ge 0}$ in the determinantal order $k$ and a family of determinantal Hadamard inequalities.