Proof that these polynomials form a base

Question

Prove that the polynomials

$$f'_n=e^{-z}\frac{d^n}{dz^n}(z^{2n}e^z), n=0,1,2,...$$

form a basis in the vector space of all polynomials. Find the expansion coefficients of the $f'_n$ in terms of the basis functions $f_n=z^n$.

I have tried for some time to prove this but I start to believe there is an error with what is being asked. I know that these really look like the Laguerre polynomials, except that the $z^{2n}$ term messes everything up.

Any ideas?

$f^0'=1$ and $f_n'=z^{2n}+2n z^{2n-1}$ for $n\geq 1$. So $z$ is not in the span of the $f_n '$. — Jens Schwaiger, Sep 03 '22 at 15:05
I misread the definition. I only saw the first derivative. I'm sorry. — Jens Schwaiger, Sep 03 '22 at 15:13

score 3 · Answer 1 · answered Sep 03 '22 at 16:11

3

The degree of $f_n'$ equals $2n$. So any non trivial linear combination has even degree. This implies that for example $z$ is not a linear combination of the $f_n'$.

answered Sep 03 '22 at 16:11

Jens Schwaiger

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Proof that these polynomials form a base

1 Answers1