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I would like to derive the correct Laguerre function from the differential equation but the differential equations seems different from the original one.

What is the correct differential equation and other kinds of equations under the hypergeometric family?

Maple output Kummer, i am not familiar with mathematica, can mathematica do?

The goal is to make target1 and target2 equal

Maple code

s := x;
tau := alpha - 1 - x;
source := s*Diff(y(x),x$2)+tau*Diff(y(x),x)+n*y(x)=0;
sol := dsolve(source,y(x));
target1 :=ztrans(sol,x,z);

target2 := sqrt(1-a^2)/(1-a/z)*((1/z-a)/(1-a/z))^(N-1);
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You might enjoy this book. It can be freely downloaded on that page. Otherwise, have you checked the wikipedia page on Laguerre functions? –  Raskolnikov Jul 17 '12 at 13:43
    
checked wiki, alpha = 0, Maple only output Kummer –  M-Askman Jul 17 '12 at 13:50
    
You seem to have a slight error in your (Sonine-)Laguerre differential equation; the form I know is $$xy^{\prime\prime}+(\alpha+1-x)y^\prime+ny=0$$ –  J. M. Jul 17 '12 at 14:43
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