Generating function, recurrence relations, differential relations (Q1318424)

If the generating relation \(G(x,w)= \sum_{i\geq 0} p_ n(x) w^ n\) for a family of polynomials \(p_ n\) is given, recurrence relations or differential recurrence relations for the polynomials are easily derivable. The inverse problem is studied and it is shown that the generating function can be derived from the relations known for the polynomial \(p_ n\). A theorem is first proved for Sheffer polynomials and then, in the general case, where a function \(G\) satisfies the equation \[ a(x,w) {{\partial G} \over {\partial \omega}}= b(x,w) G(x,w), \] where \(a(x,w)= \sum_ 0^{s+1} a_ i(x) \omega^ i\), \(b(x,w)= \sum_ 0^ s b_ i(x) \omega^ i\), it is shown that \(p_ n\) satisfies the recurrence relation \[ a_ 0(x) (n+1)p_{n+1}= \sum_ 0^ s (b_ i(x)- (n-i) a_{i+1})p_{n-i} \] and the converse is also proved. Two examples showing how the generating function is recoverable from the recurrence relations satisfied by them are also given.