2
$\begingroup$

From wikipedia, the Hermite polynomial has a combinatorial description as $$ H_r (x)= r!\sum^{[r/2]}_{s=0}\frac{(-1)^s}{s! (r-2s)!}(2x)^{r-2s} $$ I am trying to express the following equation in terms of the Hermite polynomials $$ f_r (x)= r!\sum^{[r/2]}_{s=0}\frac{(v)^s}{s! (r-2s)!}(x)^{r-2s} $$ but am unable to figure out how to do this, as it is slightly different.

$\endgroup$
2
$\begingroup$

From $$ H_r (x)= r!\sum^{[r/2]}_{s=0}\frac{(-1)^s}{s! (r-2s)!}(2x)^{r-2s} $$ we have \begin{align*} H_r (ix/(2\sqrt{v})) &= r!\sum^{[r/2]}_{s=0}\frac{(-1)^s}{s! (r-2s)!}(ix/\sqrt{v})^{r-2s}\\ & = r!\sum^{[r/2]}_{s=0}\frac{(-1)^s (\sqrt{v}/i)^{2s - r} }{s! (r-2s)!}x^{r-2s}\\ & = r!\sum^{[r/2]}_{s=0}\frac{(-1)^s (-i\sqrt{v})^{2s - r} }{s! (r-2s)!}x^{r-2s}\\ & = r!\sum^{[r/2]}_{s=0}\frac{(-1)^s (-i)^{2s} \sqrt{v}^{2s}(-i\sqrt{v})^{-r} }{s! (r-2s)!}x^{r-2s}\\ & = r!\sum^{[r/2]}_{s=0}\frac{v^{s}(-i\sqrt{v})^{-r} }{s! (r-2s)!}x^{r-2s}\\ &= (-i\sqrt{v})^{-r} f_r(x) \end{align*} if I haven't made any mistakes.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.