What is "difference equation"

n. 1 (context mathematics English) An equation involving an ordered sequence or real numbers ($left\{a\_nright\}\_\{n=1\}^infty$) and some number if it's differences, where the ''first difference'' is defined as $Delta(a\_n)\; =\; a\_\{n+1\}\; -\; a\_n,$ and the ''k^th difference'' is defined recursively as $Delta^k(a\_n)\; =\; Delta^\{k-1\}(a\_\{n+1\})\; -\; Delta^\{k-1\}(a\_n),$. 2 (context mathematics English) more broadly, any recurrence relation