iterative b -normal examples from one of Bailey's papers

• To: mathgroup at smc.vnet.net
• Subject: [mg52088] iterative b -normal examples from one of Bailey's papers
• From: Roger Bagula <tftn at earthlink.net>
• Date: Wed, 10 Nov 2004 04:45:35 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```These are some examples that demonstrate what Dr. Bailey
means when he says that Pi is b normal to base 16.
It means that a certain type of iteration is equal probably on the interval
[0,1] ( Like the classical pseudorandom variable)
These b normal formulations are by their definition a Markov type of
chaotic iteration.

(* Bailey's Log[2] b-normal example*)
Clear[x,a,digits]
digits=200
x[n_]:=x[n]=Mod[2*x[n-1]+1/n,1]
x[0]=0
a=Table[N[x[n],digits],{n,0,digits}];
ListPlot[a,PlotJoined->True,PlotRange->All]
b=Sort[Table[N[x[n],digits],{n,0,digits}]];
ListPlot[b,PlotJoined->True,PlotRange->All]
Fit[digits*b,{1,x},x]
(* Log[9/10] example sum*)
Clear[f,a]
f[n_]=1/(n*10^n)
digits=200
a=Table[N[f[n],digits],{n,1,digits}];
b=N[-Apply[Plus,a],digits]
b-N[Log[9/10],digits]
(* second example of b normal sequences: 1/Sqrt[E]*)
Clear[x,a,digits]
digits=200
x[n_]:=x[n]=Mod[4*x[n-1]+(1/(2*n)!)*((4*n+1)/(4*n+2)),1]
x[0]=0
a=Table[N[x[n],digits],{n,0,digits}];
ListPlot[a,PlotJoined->True,PlotRange->All]
b=Sort[Table[N[x[n],digits],{n,0,digits}]];
ListPlot[b,PlotJoined->True,PlotRange->All]
Fit[digits*b,{1,x},x]
N[Sum[(1/(4^n))*(1/(2*n)!)*((4*n+1)/(4*n+2)),{n,0, digits}],digits]
Respectfully, Roger L. Bagula