Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have the following recurrence relation, and I need to obtain generating function and an analytic solution. How to go about with it?

$$ f(N,M) = 0, N < M\\ f(N+1,M+1) = 2f(N,M) + (N-1)f(N,M+1), N > 0, M > 0\\ f(1,1) = 1\\ f(N,1) = 0, N > 1\\ $$

PS: this is not homework. A problem I encountered during a discussion.


share|improve this question
add comment

1 Answer 1

Hint: Denote $F(x,y)=\sum_{N,M\ge 1}f_{N,M}x^Ny^M$. If I'm not mistaken in calculations, then $$ F=1+2xyF-xF+ x^2\frac{\partial F}{\partial x}. $$

share|improve this answer
+1. Thank you, Boris. Any further simplification please? –  Qiang Li Jul 4 '13 at 16:30
No proposals. I think one should decide this differential equation for fixed $y$. –  Boris Novikov Jul 4 '13 at 16:44
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.