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

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 Br\"and\'en 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.