Question: Does maple recognise this as a proper distribution function?

I wonder how do I show with Maple that for


the series

sum(p(x,a,b),x=0..infinity) assuming a>0,b<1,b>-1

converges to 1. I also tried

sum(a*(a+x*b)^(x-1)*(exp(-(a+x*b)))/(x!), x=m..infinity) assuming a>0,b<1,b>-1,a+m*b<=0

but all to no avail. For b=0, Maple shows the series converges and we have the Poisson distribution. For b in (-1,1), the (discrete) density is known as the generalised Poisson distribution. This may be useful:

So how do I show it with Maple that the above seris is a proper distribution? 


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