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recurrencemaximahermitepolynomials

In Maxima, how do I express the recurrence for the hermite polynomials?

I'm trying to tell maxima about a recurrence relation for the hermite polynomials:

My first expression is like this:

phi[0]:exp(-1/2*x^2);

phi[1]:sqrt(2)*x*phi[0];

wxplot2d([phi[0],phi[1]], [x,-5,5]);

So far so good, but I'd like now to define all the others by:

phi[n]:sqrt(2/n)*x*phi[n-1] - sqrt((n-1)/n)*phi[n-2];

This just bombs (stack overflow). What do I really want to say so that

wxplot2d(phi[10], [x,-5,5]) will give me a sensible picture?

Solution

  • There's more than one way to handle this. Here is one way that works.

    (%i2) phi[n](x) := sqrt(2/n)*x*phi[n-1](x) - sqrt((n-1)/n)*phi[n-2](x) $
(%i3) phi[0] : lambda ([x], exp(-1/2*x^2)) $
(%i4) phi[1] : lambda ([x], sqrt(2)*x*phi[0](x)) $
(%i5) phi[0];
(%o5) lambda([x],exp((-1)/2*x^2))
(%i6) phi[1];
(%o6) lambda([x],sqrt(2)*x*phi[0](x))
(%i7) phi[2];
(%o7) lambda([x],sqrt(2)*x^2*%e^-(x^2/2)-%e^-(x^2/2)/sqrt(2))
(%i8) phi[3];
(%o8) lambda([x],
         sqrt(2)*x*(sqrt(2)*x^2*%e^-(x^2/2)-%e^-(x^2/2)/sqrt(2))/sqrt(3)
          -2*x*%e^-(x^2/2)/sqrt(3))
(%i9) phi[10];
       <very large expression here>
(%i10) plot2d (%, [x, -5, 5]);
       <nice plot appears>

    This makes use of so-called array functions. For any integer n, phi[n] is a lambda expression (unnamed function).

    Note that this only works for literal integers (e.g., 0, 1, 2, 3, ...). If you need to work with phi[n] where n is a symbol, we can look for a different approach.