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In the specific case a = 4 , however, it turns out that by allowing more sophisticated mathematical functions one can get a complete formula: the result after any number of steps t can be written in any of the forms
Sin[2 t ArcSin[ √ x ]] 2
(1 - Cos[2 t ArcCos[1 - 2 x]])/2
(1 - ChebyshevT[2 t , 1 - 2x])/2
where these follow from functional relations such as
Sin[2x] 2 4 Sin[x] 2 (1 - Sin[x] 2 )
ChebyshevT[m n, x] ChebyshevT[m, ChebyshevT[n, x]]
For a = 2 it also turns out that there is a complete formula:
(1 - (1 - 2 x) 2 t )/2
And the same is true for a = -2 :
1/2 - Cos[(1/3) ( π - (-2) t ( π - 3 ArcCos[1/2 - x]))]
In all these examples t enters essentially only in a t . And if one assumes that this is a general feature then one can formally derive for any a the result
1/2 (1 - g[a t InverseFunction[g] [1 - 2x]])
where g is a function that satisfies the functional equation
g[a x] 1 + (a/2) (g[x] 2 - 1)
When a = 4 , g[x] is Cosh[Sqrt[2 x]] .