Log-concavity and log-convexity of Stirling columns, and bottom-heaviness of $q$-Stirling numbers

摘要: We prove two conjectures concerning Stirling numbers.  First, we show that every column of the unsigned Stirling numbers of the first kind has a unique transition index from strict log-concavity to strict log-convexity, with equality possible only at the transition index.  We prove the same result for the coefficients of $\prod_{j=0}^{n-1}(x+j+\alpha)$, where $\alpha\geq0$.
Second, we prove a conjecture of Sagan and Swanson by showing that the coefficient sequences of four families of $q$-Stirling numbers are bottom heavy.  More strongly, their coefficients are nonincreasing from the middle to the right endpoint. Using an explicit difference expansion, we show that this stronger property  is preserved under multiplication by $q$-integers. The same method applies to elementary and complete homogeneous symmetric functions evaluated at $q$-integers.