How do I make my TI-89 evaluate a recursive function? On my TI-89 I can assign variables recursively such as:
$1\to x$ returns 1
$x+1 \to x$ returns 2
$x+1 \to x$ returns 3
etc.
How could I do functions the same way:
$x \to f(x)$ returns Done
$2\cdot f(x)+x \to f(x)$ returns Done
but
$f(x)$ now returns Error: Memory&lt; br />

I think that it is because it is trying to define $f(x)$ in terms of itself but each iteration I just want it to evaluate $f(x)$ and return the result to the function in terms of $x$ therefore $f(x)$ would return $3x$.
Someone might wonder why I don't just evaluate this one by hand but I am trying to understand the concept so I can apply it to more complex recursive formulas such as $$p_{k+1}(x)=2p_k(x-1)+\frac{1-2p_k(-1)}{k!}(1-x)(2-x)\cdots(k-x) $$
CORRECTED

I rebooted my old Ti92 and wrote you a little program for the symbolic evaluation of :
$$p_{k+1}(x)=2p_k(x-1)+\frac{1-2p_k(-1)}{k!}(1-x)(2-x)\cdots(k-x)$$

I had too to shift the $k$ index ($k\to k-1$) and add the indispensable initial condition (arbitrary $1$ for $k\le 1$).

The resulting function is (use  [APPS] [7] [3] to create it!) :

p(k,xx)
Func
   Local a,i
   If k&lt;= 1 Then
      Return 1
   Else
      1-&gt;a
      For i, 1, k-1
         a*(i-xx)/i-&gt;a
      EndFor
      Return 2*p(k-1,xx-1)+(1-2*p(k-1,-1))*a
   EndIf
EndFunc


and you will get (use the keys $\,[\diamond]$ [Home]  for the input) :

$\displaystyle p(3,x)\qquad\qquad\qquad \frac{x^2}2+\frac x2+1$       

Note that in my first version I obtained "circular definition" errors and could call only $\,p_k(x)$ instead of the wished $\,p_k(x-1)$. The simple replacement of the function parameter $x$ by $xx$ solved both problems as explained in the TI Knowledge Base.

Hoping this made all this useful,
As far as I know, the TI-89 cannot find closed forms for recursive functions. Try Wolfram|Alpha for that. However, we can easily graph recursively defined sequences.

Click MODE and switch Graph... to SEQUENCE. Then go to the plotter, and input your recursive function. For example, u1 = u1(n-1) + u1(n-2) with initial conditions of ui1 = {0,1}.

From there you may graph your sequence and extract a table of values, which can be used as data to find the best fitting curve, for example.