I am interested in polynomials $G_n(z)$ defined by the recurrence $$G_{n+1}(z) - 2G_n(z) + (1-nz)G_{n-1}(z)=0$$ for $n\ge1$ with the initial values $G_0(z) = 1$ and $G_1(z) = 1$. The next few values are $G_2(z) = 1+z$, $G_3(z) = 1+4z$, $G_4(z) = 1+10z + 3z^2$. I'd like to find a formula or generating function for these polynomials, which are related to certain P-partitions.
Let $g(x,z) = \sum_{n=0}^\infty G_{n+1}(z)\, x^n/n!$. Then $g=g(x,z)$ satisfies the differential equation $g'' -2g'+g =xzg'+2zg$, with initial conditions $g(0)=1$, where the prime is the derivative with respect to $x$. From this it is not hard to show that $g(x,1) =e^{x^2/2+2x}$.
Maple claims to solve this differential equation in terms of the Kummer $M$ and $U$ functions. (The Kummer $M$ function is just the confluent hypergeometric function $_1F_1$ and the Kummer $U$ function is related.) However the formula that Maple gives involves terms like $M(-1/(2z), 1/2, 2/z)$ that that are singular at $z=0$. Is there a formula for $g(x,z)$ that can be used to compute its coefficients? (I would also be interested in a combinatorial interpretation to the polynomials $G_n(z)$; it's not hard to show that its coefficients are nonnegative.)
If we define $h(x,z) = e^{-x}g(x,z) = \sum_{n=0}^\infty H_{n}(z)\, x^n/n!$ then $h(x,z)$ satisfies the differential equation $h''=(2+x)z h+zxh'$ and $H_n(z)$ satisfies the recurrence $H_{n+2}(z) = z(nH_{n-1}(z) + (n+2) H_n(z))$, with initial values $H_0(z)=1$, $H_1(z)=z$. The polynomials $H_n(z)$ have fewer nonzero coefficients and smaller coefficients than $G_n(z)$, which suggests that they may be simpler: $H_2(z) = 2z$, $H_3(z)=z+3z^2$, $H_4(z) = 10z^2$.
