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How to convert recurrence into generating function?


there is an Mexiner recurrence equation,
but both generatingfunction and convert not convert into generating
function as stated in paper.

How to recurrence into generating function correctly?

GeneratingFunction[c*(n+beta)*P(n+1)/(-1+c)+(c*(x+beta)/(1-c)-x)*P(n)
+n*P(n-1)/(-1+c) = 0,n,z]
GeneratingFunction[c*(n+beta)*P(n+1)/(-1+c)+(c*(x+beta)/(1-c)-x)*P(n)
+n*P(n-1)/(-1+c),n,z]

Expand((c*(n+beta)*P[n+1]/(-1+c)+(c*(x+beta)/(1-c)-x)*P[n]+n*P[n-1]/
(-1+c))*z^n/n!)

Convert[sum[z^n*c*P[n+1]*n/(n!*(-1+c))+z^n*c*P[n+1]*beta/(n!*(-1+c))
+z^n*P[n]*c*x/(n!*(1-c))+z^n*P[n]*c*beta/(n!*(1-c))-z^n*P[n]*x/n!
+z^n*n*P[n-1]/(n!*(-1+c)),{n,0,infinity}],z]
Convert[sum[(c*(n+beta)*P[n+1]/(-1+c)+(c*(x+beta)/(1-c)-x)*P[n]
+n*P[n-1]/(-1+c))*z^n/n!,{n,0,infinity}],z]



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