Let's define
$$\begin{align*} \frac{dx}{du}&=e^{x^2}\\ \frac{d^2x}{du^2}&=2xe^{2x^2}\\ \frac{d^3x}{du^3}&=(2+8x^2)e^{3x^2}\\ \vdots\\ \frac{d^nx}{du^n}&=P_n(x)e^{nx^2}\\ \frac{d^{n+1}x}{du^{n+1}}&=(P'_n(x)+2nxP_n(x))e^{(n+1)x^2}\\ \frac{d^{n+1}x}{du^{n+1}}&=P_{n+1}(x)e^{(n+1)x^2}\\ \end{align*}$$
$$P_{n+1}(x)=P'_n(x)+2nxP_n(x)$$
I would like to find the differential equation that satisfies $P_{n}(x)$.
And then I would like to find an orthogonal relation such as if possible
if $m \neq n$ $$\int_a^{b} P_n(x)P_m(x)W(x)\;dx=0$$
if $m = n$ $$\int_a^{b} P^2_n(x)W(x)\;dx=a_n$$
Could you please help me about steps?
Thanks
From the Rodrigues formula for the Hermite polynomials, making a few variable substitutions gives the relation
$$\exp(-z^2)\frac{\mathrm d^n}{\mathrm dz^n}\exp(z^2)=i^{-n}H_n(iz)$$
Making the appropriate substitutions into the Hermite DE yields
$$w^{\prime\prime}+2zw^\prime-2nw=0$$
I am not too sure about how one could derive an orthogonality relation from this, though; the substitutions don't really work for the usual orthogonality relation for the Hermite polynomials.