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Studying for a final and came accross this problem in the textbook. Considering I have no idea how to even start im a bit scared :). Any explanation would be greatly appreciated.

problem: If f(x)is a polynomial of degree m, show that f(x) may be expressed in the form $$f(x)=\sum_{r=0}^m c_rL_r(x)$$ with $c_r=\int_{0}^\infty e^{-x}L_r(x)f(x)dx$.

Deduce that $\int_{0}^\infty e^{-x}x^kL_n(t)dt=0$ if $k<n$

and $\int_{0}^\infty e^{-x}x^kL_n(t)dt=0$ if $k=n$

Aksel'sRose
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  • Have you tried working from the definitions? Laguerre Polynomials have some nice well-known properties. Perhaps it helps to see that the integrand can be written as a sum of terms of the form $\int_{0}^{\infty}\alpha_n e^{-x}x^n$. Integrating by parts... – Nobody Jul 27 '16 at 14:54
  • I think my confusion right now is is this a 3 part problem? If so I don't understand what i am being asked to do on the f(x) with $c_r$ part. – Aksel'sRose Jul 27 '16 at 16:34
  • Ah, I'd say this is a two part problem since the deductions arise naturally by arriving at the first part (the part with $c_r$'s). I'm assuming, however, that you meant $\int_{0}^{\infty}e^{-x}x^k L_{n}(x),dx=0$ if $k<n$ and, moreover, that you meant $\int_{0}^{\infty}e^{-x}x^n L_n (x),dx\ne 0$. Use the generating function of the $L_n$'s if you know it to avoid a hassle with sums. – Nobody Jul 27 '16 at 20:23

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