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

wl at particle.uni-karlsruhe.de wl at particle.uni-karlsruhe.de
Mon Feb 25 12:54:02 CET 2013

Hello seq- and phi-fans,
this alternating series appears as eq. (102) in S. Vajda, 
Fibonacci&Lucas Numbers, Ellis Horwood Limited 1989 on p. 103 (with a 
derivation, using eq. (29), which is called Cassini-Identity in 
Graham-Knuth-Patashnik's book). It appears also in the Vajda collection 
of  formulae on p. 183.

Consider instead the more general polynomials identity involving 
Chebyshev S-polynomials (A049310):

sum(1/(S(k,x)*S(k-1,x)),k=1..n) = S(n-1,x)/S(n,x), n >= 1 (n=0 also 
o.k. 0 = S(-1,x)).

This is derived similarly from 1/(S(k,x)*S(k-1,x) = S(k-2,x)/S(k-1,x) - 
S(k-1)/S(k,x) (using the relevant Cassini-Id: -S(N,x)*S(N-2,x) + 
S(N-1,x)^2 = 1, N>=0, and telesopic summation. Then use for 
S(n-1,x)/S(n,x) the n-th continued fraction approximation of 
1/(x-1/(x-1/(x- ..., to study the limit n -> infinity. For the 
Fibonacci case, with S(n,x) = I^n*F(n+1,-I*x)), I the imaginary unit, 
then x -> 1, this works with limit phi. The Pell case uses x=2, 
Vladimir used then x=N.

Greetings Wolfdieter Lang

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