$\delta M=\beta(x-y)M_y+\mu x(y-1)M_x+\delta y$, where $M(x,y)=\sum_{n=0}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}x^ny^k}}$ is the generating function for a certain probability distribution $\{s_{n,k}\}$ (the exact formula for $s_{n,k}$ is unknown), and $\delta$, $\beta$, $\mu$ are all constants.
The problem comes out of a probability model, and data show that the distribution should have a finite mean and infinite second moment ($M_{xx}(1,1)=\infty$). My question is that is there any way (or any theory on it) to get the asymptotic of $M_{xx}(x,1)$ when $x\rightarrow 1^{-}$? (by asymptotic I mean something like $M_{xx}(x,1)\sim C(1-x)^{-\zeta}$)
The background for this problem is here: Asymptotic behaviour of a two-dimensional recurrence relation
Thank you!