For n∈N∪{0}, let yₙ satisfy $xy_n\prime\prime+(1-x)y_n\prime+ny_n=0$ and yₙ(0) = 1. Which weight w gives $\int_0^\infty y_p(x)y_q(x)w(x)dx=0$ for p ≠ q?
The regular polynomial solutions are the Laguerre polynomials. Multiplying the ODE by $e^{-x}$ gives the self-adjoint form $(xe^{-x}y_n\prime)\prime+ne^{-x}y_n=0$. Multiply equations for indices p,q by the other solution and subtract, then integrate. Boundary terms vanish, leaving $(p-q)\int_0^\infty e^{-x}y_py_qdx=0$. Thus the weight is $e^{-x}$, B.
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