       Recursive function: What is the proper way to get mathematica to

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• Subject: [mg130237] Recursive function: What is the proper way to get mathematica to
• From: Joao <joaopereira9 at gmail.com>
• Date: Sun, 24 Mar 2013 23:24:30 -0400 (EDT)
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```Hi there

I would appreciate if someone could guide me in this issue. I have a fairly complicated function that is an integral whose integrand contains the initial function. I was trying to write it (pencil and paper) as a recurrence expression, but is not easy. I am also experimenting with some of mathematica's built in commands but nothing has worked so far....most probably because I am not doing it correctly.

The function is:

x[i_, t_] :=
q*Power[Integrate[
Exp[-(r + \[Delta]) (s - t)] Power[
Integrate[(b + X[j, s])^\[Alpha] - b^\[Alpha], {j, 0,
A}], (1 - \[Alpha])/\[Alpha]]*(b + X[i, s])^(\[Alpha] -
1), {s, t, Infinity}], 1/(\[Phi] - 1)];

And:

X[i_, t1_] := (X[i, t] +
Integrate[
Exp[\[Delta] (s - t)] x[i, s], {s, t, t1}]) Exp[-\[Delta] (t1 -
t)];

Basically, X[i_,t1_] is the integration of a differential equation of x[i_,t_]

x[i_,t_] is a flow and X[i_,t1_] the stock.

The goal of all this is to get a nicer expression for x[i,t], which I believe is possible.

The domains for the parameters are:

q > 0; \[Alpha] > 0 && \[Alpha] < 1; b > 0;
r > 0 && r < 1; \[Delta] > 0 && \[Delta] < 1; \[Phi] > 1; A > 0; X[i,t]>0;

My question is: which Mathematica function should I try for this?