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small typo "in The Book"

• To: mathgroup@smc.vnet.net
• Subject: [mg10556] small typo "in The Book"
• From: Wouter Meeussen <eu000949@pophost.eunet.be>
• Date: Tue, 20 Jan 1998 16:54:00 -0500

Mathematical Functions  >  Combinatorial  >  Fibonacci
 The Fibonacci polynomial F[n,x] is the coefficient of
 t^n in the expansion of  t/(1 - x  t - t^2)

*** the above is ok, but, below, it is wrong : ***

3.2.5   Combinatorial Functions
The Fibonacci polynomials Fibonacci[n, x] appear as the coefficients of 
 t^n in the expansion of
 t/(1 - x  t - t^2)  =  Sum[ F[n,x] t^n /n! ,{n,0,Infinity}]  
                                        ^^^
                                        ^^^ *** check :  ***

Simplify/@Normal[Series[t/(1-x t-t^2),{t,0,6}]] t + t^2*x + t^3*(1 +
x^2) + t^4*x*(2 + x^2) + t^5*(1 + 3*x^2 + x^4) + 
  t^6*x*(3 + 4*x^2 + x^4)

Simplify/@Sum[ Fibonacci[n,x] t^n       ,{n,0,6}]   t + t^2*x + t^3*(1 +
x^2) + t^4*x*(2 + x^2) + t^5*(1 + 3*x^2 + x^4) + 
  t^6*x*(3 + 4*x^2 + x^4)

*** a different matter is whether 
    Sum[Fibonacci[n,x] t^n,{n,0,Infinity}]//FullSimplify
    should evaluate to  t/(1 - x  t - t^2) , and whether or not
    it would be "used" by FullSimplify if given as downvalue to
Fibonacci. ***

wouter.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc@vandemoortele.be
eu000949@pophost.eunet.be