$\def\sone#1#2{\left[#1\atop #2\right]} \def\stwo#1#2{\left\{#1\atop #2\right\}} $

These formulas can be proved by Lagrange interpolation, using the fact that $\stwo{n}{n-m}$ and $\sone{n}{n-m}$ are polynomials in $n$ of degree $2m$. See , H. W. Gould, The Lagrange interpolation formula and Stirling numbers, Proc. Amer. Math. Soc. 11 (1960), 421–425

$\def\sone#1#2{\left[#1\atop #2\right]} \def\stwo#1#2{\left\{#1\atop #2\right\}} $

These formulas can be proved by Lagrange interpolation, using the fact that $\stwo{n}{n-m}$ and $\sone{n}{n-m}$ are polynomials in $n$ of degree $2m$. See H. W. Gould, The Lagrange interpolation formula and Stirling numbers, Proc. Amer. Math. Soc. 11 (1960), 421–425