We start with the following recurrence relation for complex coefficients $c_{n,m}$: $$i\dot{c}_{n,m}(t) = \sqrt{(n+1)(n+2)(m-1)m}c_{n-2,m+2} + \sqrt{n(n-1)(m+1)(m+2)}c_{n+2,m-2}$$ where $\dot{c}_{n,m}$ denotes time derivative and $n,m = 0,1,2,...$.
In order to solve this equation and find the time evolution of $c_{n,m}$ (we assume that at $t=0$ we know values of $c_{n,m}$) we define a generation function: $$G(x,y,t) = \sum\limits_{n,m=0}^{\infty}x^{n}y^{m}\alpha_{n,m}c_{n,m}(t)$$ with some unknown coefficient $\alpha_{n,m}$ that can be adjusted manually. Recurrence relation can be rewritten in terms of differential equation for $G$: $$i\partial_{t}G(x,y,t) = (y^2\partial_{x}^2 + x^2\partial_{y}^2)G(x,y,t)$$ if we set $$\alpha_{n,m} = \alpha_{n+2,m-2}\sqrt{\frac{(n+1)(n+2)}{m(m-1)}}$$ We can assume that $G(x,y,0)$ is known and converges in the domain $[0,1] \times [0,1]$ for any $t$.
Is there a way to find function $G(x,y,t)$ that is not represented by infinite series? This is a special kind of Schrödinger equation.