[seqfan] Re: A very fast convergent alternating series for phi

[Vladimir Shevelev]

Finally, I found the most wide generalization of Catalan's (maybe, more exactly, Cassini's)
identity for the considered recursion. Let, in general, 
a(0)=b,a(1)=c, for n>=2, a(n)=N*a(n-1)+M*a(n-2).               (1)
Then for n>=1,
 a^2(n) - a(n-1)*a(n+1) = (-M)^(n-1)*(c^2-N*b*c-M*b^2).  (2)
Proof. The 3 first terms of the sequence are b, c, N*c+M*b.  (3)
Thus for n=1, a^2(1) - a(0)*a(2) =c^2-N*b*c-M*b^2. It is the base of induction.
Suppose (2) holds for every sequence of form (1), where b,c are arbitrary parameters and prove it for n:=n+1. Then the inductive supposition is true for sequence (1) with b:=c, c:=N*c+M*b. However, the n-th term of this sequence coincides
with (n+1)-th term of (1). Therefore, we have a^2(n+1) - a(n)*a(n+2) = (-M)^(n-1)*((N*c+M*b)^2-N*c*(N*c+M*b)-M*c^2)=(-M)^n*(c^2-N*b*c-M*b^2). (End)
Usual Cassini's identity is obtained in case b=0,c=1,N=M=1. It is interesting that every of known proofs of proper Cassini's identity (see http://www.proofwiki.org/wiki/Cassini's_Identity) are much longer than above proof of the considered generalization. 
Note that the identity yields that c^2-N*b*c-M*b^2=0, iff {a(n)} is a geometric progression (e.g., if b=1,c=4,N=3,M=4, we obtain 1,4,16,64,...). Thus we can obtain all solutions of this Diophantine equation.
Now we can generalize the formulas with b=0,c=1. In particular,
a(n+1)= (N*a(n)  +  sqrt((N^2+4*M)*a^2(n) + 4*(-1)^n)*M^n*(N*b*c + M*b^2-c^2)))/2;
sum{k=1,...,n}(-M)^(k-1)/(a(k)*a(k+1)) = (a(n)/a(n+1)-b/c)/(c^2-N*b*c-M*b^2),
 if c not equals 0 and {a(n)} is not a geometric progression; if here n goes to infinity, we obtain series
sum{k>=1}(-M)^(k-1)/(a(k)*a(k+1)) = (1/phi(M,N) - b/c)/(c^2-N*b*c-M*b^2), where 
phi_(N,M)=(N+sqrt(N^2+4*M))/2.
 
 
Best regards,
Vladimir



----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Monday, February 25, 2013 0:03
Subject: [seqfan] Re: A very fast convergent alternating series for phi
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> Thank you, Richard, for interesting references and Alexander for 
> an important question!
>  
> >Could you generalize to
> >a(n)=N*a(n-1)+M*a(n-2).
> >?
> 
> >Regards,
> >ARP
> 
>  
> Here is an answer on Alexander's question.
>  
> For positive integers M,N, consider sequence  
> a(0)=0,a(1)=1, for n>=2, a(n)=N*a(n-1)+M*a(n-2). Then
> (1) Expression a(n+1) via a(n): a(n+1) = (N*a(n) + 
> sqrt((N^2+4*M)*a^2(n) + 4*(-1)^n)*M^n))/2;
> (2) a^2(n+1) - a(n)*a(n+2) = (-M)^n (Catalan-like formula);
> (3) sum{k=1,...,n}(-M)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);
> (4) sum{k>=1}(-M)^(k-1)/(a(k)*a(k+1)) = 1/phi_(N, M), where 
> phi_(N,M)=(N+sqrt(N^2+4*M))/2;(5)  a(n)/a(n+1) = 
> 1/phi_(N,M) + r(n), where |r(n)| < M^n/(a(n+1)*a(n+2)).
> Thus r(n)=O((4*M/(N+sqrt(N^2+4*M))^2))^n).
> However, it is easy to verify that function 
> f_N(x)=(4*x/(N+sqrt(N^2+4*x))^2 is increasing for x>=1.  
> Therefore, the smallest r(n) we obtain in the case M=1.
>  
> Best regards,
> Vladimir
> 
> 
> 
> ----- Original Message -----
> From: mathar <mathar at mpia-hd.mpg.de>
> Date: Sunday, February 24, 2013 21:55
> Subject: [seqfan] Re: A very fast convergent alternating series 
> for phi
> To: seqfan at seqfan.eu
> 
> > In response to http://list.seqfan.eu/pipermail/seqfan/2013-
> > February/010846.html
> > vs> 
> > vs> (4)sum{k>=1}(-1)^(k-1)/(a(k)*a(k+1)) = 1/phi_N, where 
> > phi_N=(N+sqrt(N^2+4))/2 is a "metallic" ratio (for N=1-golden 
> > ratio, for N=2-silver ratio, etc.);  
> > vs> ...
> > 
> > one should point to the applicable literature on the subject of
> > summing inverse products of terms from the 2-term recurrences:
> > 
> > Gert Almkvist, A solution to a tantalizing problem, Fib. 
> Quart. 
> > (1986) 316, 
> > http://www.fq.math.ca/Scanned/29-3/andre-jeannin1.pdf
> > 
> > Blagoj S. Popov, Summation of reciprocal series of numerical 
> > functions of second order, Fib. Quart. 24 (1) (1986) 17-21, 
> > http://www.fq.math.ca/Scanned/24-1/popov.pdf
> > 
> > R. Andre-Jeannin, Summation of certain reciprocal series 
> related 
> > to Fibonacci and Lucas numbers, Fig. Quart. 29 (1991) 200, 
> > http://www.fq.math.ca/Scanned/29-3/andre-jeannin1.pdf
> > 
> > R. S. Melham, A generalization of a result of Andre-Jeannin 
> > concering summation of reciprocals, Proguliae Mathematica 57 
> (1) 
> > (2000) p. 45
> > http://www.emis.ams.org/journals/PM/57f1/pm57f104.pdf
> > 
> > F. Zhao, Notes on reciprocal series related to Fibonacci and 
> > Lucas Numbers, Fib. Quart. 39 (5) (2001) 392 , 
> > http://www.fq.math.ca/Scanned/39-5/zhao1.pdf
> > 
> > N. Omur, On reciprocal series of generalized Fibonacci numbers 
> > with subscripts in arithmetic progression, Discr. Dyn. Nat. 
> Soc. 
> > 2012 #684280 http://dx.doi.org/10.1155/2012/684280
> > 
> > RJM
> > 
> > _______________________________________________
> > 
> > Seqfan Mailing list - http://list.seqfan.eu/
> > 
> 
>  Shevelev Vladimir‎
> 
> _______________________________________________
> 
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> 

 Shevelev Vladimir‎