I'm looking for a reference to a proof of formulas 6.26 and 6.27 in Concrete Mathematics: $\def\sone#1#2{\left[#1\atop #2\right]} \def\stwo#1#2{\left\{#1\atop #2\right\}} $ $$ \stwo{n}{n-m} = \sum_k \binom{m-n}{m+k}\binom{m+n}{n+k} \sone{m+k}{k} $$
$$\sone{n}{n-m} = \sum_k \binom{m-n}{m+k}\binom{m+n}{n+k} \stwo{m+k}{k}$$
$\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
