$\delta M=\beta(x-y)M_y+\mu x(y-1)M_x+\delta y$, 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 divergent moment ($M_{xx}(1,1)=\infty$). My question is that is there any way to get the asymptotic of $M_{xx}(x,1)$ when $x\rightarrow 1^{-}$? (by asymptotic I mean something like $M_{xx}(x,1)\sim C(1-x)^{-\zeta}$)

Thank you!

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Asymptotic behaviour of the solution to a certain PDE

$\delta M=\beta(x-y)M_y+\mu x(y-1)M_x+\delta y$, 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 divergent moment ($M_{xx}(1,1)=\infty$). My question is that is there any way to get the asymptotic of $M_{xx}(x,1)$ when $x\rightarrow 1^{-}$?

Thank you!