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I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$. 1. Orthonormality \begin{equation} \int_0^{\infty} e^{-x}x^kL_n^k(x)L_{m}^k(x)dx=\dfrac{(n+k)!}{n!}\delta_{mn} \end{equation} 2. Completeness (?) \begin{equation} \sum_{n=0}^{+\infty}L_n^k(x)L_{n}^k(y)=?\delta(x-y) \end{equation} I am having trouble with the second relation, can anyone give a reference where it is proven or hint for a proof?

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As $L^k_n=x^n/n!+$ lower degree terms, the sequence $L^k_n$, $n=0,1,2,\dots$ can be obtained from $1,x,x^2,x^3,\dots$ by the Gramm-Schmidt orthonormalization process. The completeness is therefore equivalent to the completeness of polynomials in $L^2(\mathbb{R}_+, e^{-x}x^k\,dx)$. – user8268 Jan 19 '12 at 22:08

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