[seqfan] Re: Cassini transformation of linear recursion of order 3

[Andrew N W Hone]

Dear Vladimir, 

I don't know to what extent this is "known" but such transformations involving polynomial 
functions of the terms of a linear recurrence have nice properties. 

For instance, if the sequence (a_n) satisfies a linear recurrence of order N with constant coefficients, then 

g_n = P(a_n, a_{n+1}, ... , a_{n+k-1}) 

also satisfies a linear recurrence with constant coefficients, for any polynomial P in k variables (where w.l.o.g. 
we can assume that k is at most N). With a bit of work one could also get an upper bound on 
the order of the recurrence for (g_n). 

One can easily prove this for sequences defined over the complex numbers, since (generically) one 
can write the terms of the first sequence as a sum 
 
a_n = \sum_j   C_j x_j^n 

over the roots x_j of the characteristic polynomial of the recurrence for a_n, and then from the polynomial 
P one has an expression of the form 

g_n = \sum_k D_k y_k^n, 

where the y_k are formed from certain products of x_j. Then g_n satisfies 

F(S) g_n = 0 

where F(S) = \prod _k (S-y_k) and S is the shift operator (sending n -> n+1). 

With a bit more work one could come up with a more general proof that would work 
with recurrences defined over any field, or over a ring. 

Your particular choice of g_n is very nice: it is a 2 x 2 determinant (a discrete Wronskian, or Casorati 
determinant). 

Best wishes,
Andy   

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at bgu.ac.il]
Sent: 05 March 2013 03:21
To: seqfan at list.seqfan.eu
Subject: [seqfan] Cassini transformation of linear recursion of order 3

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

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