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 Enumerative Combinatorics, vol.1, 2nd ed.
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A polynomial recurrence involving partial derivativesDefine 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) of Enumerative Combinatorics, vol.1, 2nd ed.
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