[seqfan] Cassini transformation of linear recursion of order 3

[Vladimir Shevelev]

Dear SeqFans,
 
I proved the following: 
If a(n+3)=L*a(n+2)+M*a(n+1)+N*a(n), n>=0, a(0)=b, a(1)=c, a(2)=d
and for n>=1, g=a^2(n)-a(n-1)*a(n+1),
then g(n+3)=-M*g(n+2)-L*N*g(n+1)+N^2*g(n).
This identity we call the Cassini transformation of linear recursion of order 3.
In case |g(n)|<=|a(n)|, this allows, using  telesopic summation, to obtain a fast convergent series for lim(a(n+1)/a(n)) as n goes to infinity. Could anyone say whether such a transformation of linear recursion of order 3 is known?
 
Best regards,
Vladimir

 Shevelev Vladimir‎