User:Eviktorb/sandbox

Discrete Laguerre functions
The discrete Laguerre functions are a set of functions on $$\ell^2[0,\infty)$$. The functions are orthonormal with respect to the inner product $$\langle w, v \rangle_d = \sum_{t=0}^\infty w(t)v(t)$$. Let $$p \in (0, 1)$$ denote the Laguerre parameter. This parameter defines the poles of the functions, and acts as a time scaling constant. In the time domain the $$j$$:th discrete Laguerre function may be expressed as $$l_j(t) = p^{\frac{t-j}{2}}\sqrt{1-p}\sum_{l=0}^j(-1)^l\binom{t}{l}\binom{j}{l}p^{j-l}(1-p)^l.$$In the z-domain, the $$j$$:th discrete Laguerre function can be expressed as $$L_j(z) = \frac{\sqrt{1-p}z}{z-\sqrt{p}}\left(\frac{1-\sqrt{p}z}{z-\sqrt{p}}\right)^j.$$