I have been using the Gram–Schmidt procedure to generate the discrete Laguerre polynomials, over the domain m = 0 ... +Inf, using an exponentially decaying weighting function w(m) = exp(sig*m), where sig
... OK, on further investigation I have found that the roots are the same, it's just that there is a sign change for the odd functions in the sequence between my defn (hlk) and the "standard" (std) defn, i.e. (-1)^k*P_hlk(m;k) = P_std(m;k). This explains why the orthonormality conditions are satisfied by both.
So I guess the definitions are the same for all intents and purposes.
Hugh,
I would think that the usual way to define classed of orthonormal polynomials if via differential or difference equations with specific initial conditions. I'm not an expert on this, but I would guess that the discrete kind arises from the continuous kind from a discretisation of the underlying differential equation, e.g. using a finite difference approximation to the differential oeprator. From memory, Abramovitch and Stegun has continuous but not discrete, so that won't help you.
Hi Graham, I think my definition via orthonormalization is OK. But now I am starting to wonder about the difference equation that they might satisfy. So I guess you could say I have a solution looking for a problem. But in the meantime, I'm happy to use them, simply because they form an orthonormal set, which means I can do (recursive, stable) discounted least-squares estimation without needing to do a matrix inversion.
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