A polynomial recurrence involving partial derivatives - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:31:45Z http://mathoverflow.net/feeds/question/87801 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87801/a-polynomial-recurrence-involving-partial-derivatives A polynomial recurrence involving partial derivatives Richard Stanley 2012-02-07T15:13:50Z 2012-02-18T00:22:26Z <p>Define recursively polynomials $f_n(a,b)$ by $$f_0(a,b)=1,\ \ f_n(0,b)=0\ \mathrm{for}\ n>0$$ $$\frac{\partial}{\partial a}f_n(a,b) = f_{n-1}(b-a,1-a).$$ For instance, $$f_1(a,b) = a,\ \ f_2(a,b) = \frac 12(2ab-a^2)$$ $$f_3(a,b) = \frac 16(a^3-3a^2-3ab^2+6ab).$$ Is there a nice'' solution to this recurrence, e.g., a formula for the generating function $\sum_{n\geq 0}f_n(a,b)x^n/n!$? What I am really interested in is $f_n(1,1)$. For the motivation, see the solution to Exercise~4.56(d) (pg. 645) of <a href="http://math.mit.edu/~rstan/ec/ec1.pdf" rel="nofollow"><em>Enumerative Combinatorics</em></a>, vol.1, 2nd ed.</p> http://mathoverflow.net/questions/87801/a-polynomial-recurrence-involving-partial-derivatives/88771#88771 Answer by Bob Terrell for A polynomial recurrence involving partial derivatives Bob Terrell 2012-02-17T21:23:55Z 2012-02-18T00:22:26Z <p>There seems to be a PDE for $g(a,b,x)=\sum_{n\ge0}f_n(a,b)x^n$, which can be thought of as a boundary value problem in the triangle $0\lt a\lt b\lt1$. $$g_{aab}+g_{abb}+x^3g=0$$ ($x$ is a parameter and subscripts are derivatives) with boundary values $g(0,b,x)=1$, $g_a(a,a,x) = x$, and $g_{ab}(a,1,x) = x^2$. This comes from iterating the $f_n$ recurrence, after Pietro's remarks that $(a,b)\to(b-a,1-a)$ has period 3 suggested looking at third derivatives. Does that determine $g$ uniquely, nicely? I don't know yet. [Edit: I wrongly wrote $g$ at first using $\frac{x^n}{n!}$.]</p>