$\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 ][1]


  [1]: http://www.ams.org/journals/proc/1960-011-03/S0002-9939-1960-0114769-1/S0002-9939-1960-0114769-1.pdf