defining recursively a function with multiple if conditions

Question

I am trying to recursively define a function which satisfies the following system of equations and which depends on two parameters $n$ and $l$,

$$ \begin{align} A(x, n, l) &= F[ A(x,n-1,l) ]\\ A(x, n, l) &= x^l \mbox{ if } l = n \\ A(x, n, l) &= 0 \mbox{ if } l < n \\ \end{align} $$

In order to do this I used the following code (here I make the expression for $F$ explicit):

A[x_, n_, l_]:=
  Boole[n > l]((1 - x)/(1 + x^l) * (Sum[A[x, n-1-i, l] * x^i, {i,0,l-1}])
  + x^l/(1 + x^l)) + Boole[n == l] * x ^ l

The function defined in this way, $A(x, n, l)\;$, gives the probability that tossing a coin $n$ times gives exactly one run of $l$ consecutive $1$s, where $x$ is the probability getting $1$ on single toss and $(1-x)$ is the probability getting $0$ on single toss.

When I try to plot one of these functions, for instance,

Plot[A[x, 7, 7], {x, 0, 1}]

I encounter the following error:

RecursionLimit::reclim: Recursion depth of 256 exceeded.

Is my code wrong? Is there an other way to recursively define a function, taking into account the fact that the third condition must be satisfied for all the values $l<n$, where $l$ and $n$ are not specified at the moment of the definition.

Answer 1

I am sorry, but have hard time understanding your code with the use of Bool.

This does the same thing as you have, using Piecewise which is easier to understand

Also, hard to read l as variable name, it looks like 1. It is not a good idea to use l for variable. Also, not good idea to user UpperCase A. Changed to f.

f[x_, n_, r_] :=
 Piecewise[{
   {(1 - x)/(1 + x^r)*(Sum[f[x,n-1-i,r]*x^i,{i, 0, r - 1}])+x^r/(1 + x^r),n>r},
   {x^r, n == r},
   {0, True}
   }]

Plot[f[x, 7, 7], {x, 0, 1}]

Answer 2

Here is an alternate way of structuring the definition of your function that avoids the use of If completely by employing a combination of conditional expression evaluation and argument pattern matching:

j[x_, n_, l_] := 0 /; l < n
j[x_, l_, l_] := x^l
j[x_, n_, l_] := (1-x)/(1 + x^l)*(Sum[j[x, n-1-i,l]*x^i, {i,0,l-1}]) + x^l/(1+x^l)

And plotted:

Plot[j[x, 7, 7], {x, 0, 1}]