Let $\mathbb { Z } ^ { + }$ denote the set of positive integers. We want to find all functions $g : \mathbb { Z } ^ { + } \rightarrow \mathbb { Z } ^ { + }$ such that the following equation holds for any $m, n$ in $\mathbb { Z } ^ { + }$. $$g ( n + m ) = g ( n ) + n m ( n + m ) + g ( m )$$ Prove that $g ( n )$ must be of the form $\sum _ { i = 0 } ^ { d } c _ { i } n ^ { i }$ and find the precise necessary and sufficient condition(s) on $d$ and on the coefficients $c _ { 0 } , \ldots , c _ { d }$ for $g$ to satisfy the required equation.
Let $\mathbb { Z } ^ { + }$ denote the set of positive integers. We want to find all functions $g : \mathbb { Z } ^ { + } \rightarrow \mathbb { Z } ^ { + }$ such that the following equation holds for any $m, n$ in $\mathbb { Z } ^ { + }$.
$$g ( n + m ) = g ( n ) + n m ( n + m ) + g ( m )$$
Prove that $g ( n )$ must be of the form $\sum _ { i = 0 } ^ { d } c _ { i } n ^ { i }$ and find the precise necessary and sufficient condition(s) on $d$ and on the coefficients $c _ { 0 } , \ldots , c _ { d }$ for $g$ to satisfy the required equation.