CoCalc -- 4405.sagews

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var('x,n1,n2,n3,n4,n5')


w=[n1,n2,n3,n4,n5]   #[w-1,w-2,w-3,...]

def phi(x,w):  #returns iteration of the inverse Gauss map along trajectory w
    if (len(w)>1):
        return 1/(w[-1]+phi(x,w[0:-1]))
    else:
        return 1/(w[0]+x)
        
def dphidx(x,w): #returns returns expansion rate along finite trajectory w
    ret=1
    for j in range(len(w)):
        ret*=-1*(phi(x,w[0:j+1]))^2
    return ret
    
def S(x,w):   #returns S_omega as defined in paper for a word of finite length
    ret=0
    for j in range(len(w)):
        ret+=dphidx(x,w[0:j+1])*(-2/phi(x,w[0:j+1]))
    return -retprint

def CF(w):   #returns continuous fraction expansion of w
    if (len(w)>1):
        return 1/(w[0]+CF(w[1:]))
    else:
        return 1/(w[0])

 
show(S(x,w))
show(-2/(x+1/CF(w)))  
g=S(x,w)+2/(x+1/CF(w))

show(g.full_simplify())
show('Thus the two terms above are the same!')

<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2}{n_{1} + x} + \frac{2}{{\left(n_{2} + \frac{1}{n_{1} + x}\right)} {\left(n_{1} + x\right)}^{2}} - \frac{2}{{\left(n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}\right)} {\left(n_{2} + \frac{1}{n_{1} + x}\right)}^{2} {\left(n_{1} + x\right)}^{2}} + \frac{2}{{\left(n_{4} + \frac{1}{n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x

right)} {\left(n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}\right)}^{2} {\left(n_{2} + \frac{1}{n_{1} + x}\right)}^{2} {\left(n_{1} + x\right)}^{2}} - \frac{2}{{\left(n_{5} + \frac{1}{n_{4} + \frac{1}{n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}}}\right)} {\left(n_{4} + \frac{1}{n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}}\right)}^{2} {\left(n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}\right)}^{2} {\left(n_{2} + \frac{1}{n_{1} + x}\right)}^{2} {\left(n_{1} + x\right)}^{2}}

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2}{n_{1} + x + \frac{1}{n_{2} + \frac{1}{n_{3} + \frac{1}{n_{4} + \frac{1}{n_{5}}}}}}

\newcommand{\Bold}[1]{\mathbf{#1}}0

\newcommand{\Bold}[1]{\mathbf{#1}}\verb|Thus|\phantom{x}\verb|the|\phantom{x}\verb|two|\phantom{x}\verb|terms|\phantom{x}\verb|above|\phantom{x}\verb|are|\phantom{x}\verb|the|\phantom{x}\verb|same!|

}}}