Generating function and Rodrigue's formula of Hermite polynomial

Generating function of Hermite polynomial:

$$e^{2xt-t^2}=\sum_{n=0}^\infty\frac{H_n(x) +n}{n!}$$

L.H.S

$$ e^{2xt-t^2}$$

$$=e^{2xt}\cdot e^{-t^2}$$

$$=\biggl[1+\frac{(2xt)^1}{1!}+\frac{(2xt)^2}{2!}+\dotsm+\frac{(2xt)^r}{r!}+\dotsm\biggr]\biggl[1+\frac{(-t^2)^1}{1!}+\frac{(-t^2)^2}{2!}+\dotsm+\frac{(-t^2)^5}{5!}+\dotsm\biggr]$$

$$=\sum_{r=0}^\infty \frac{(2xt)^4}{r!}\cdot \sum_{s=0}^\infty \frac{(-t^2)^s}{s!}$$

$$=\sum_{r=0}^\infty\sum_{s=0}^\infty \frac{(-1)^s(2x)^r t^{r+2s}}{r!s!}$$

Put \( r+ 2s= n, r=n-2s\)

$$=\sum_{n=0}^\infty\sum_{s=0}^{\frac n2} \frac{(-1)^s(2x)^{n-2s}+n}{(n-2s)! s!}$$

$$=\sum_{n=0}^\infty\biggl[\sum_{s=0}^{\frac n2} \frac{(-1)^s (2x)^{n-2s}+n}{(n-2s)! s!}$$

$$=\sum_{n=0}^\infty H_n(x) \times \frac {t^n}{n!}$$

$$=\sum_{n=0}^\infty \frac{H_n(x) t_n}{n!}$$

$$=R.H.S$$

Proved.

Again, We have generating function as,

$$e^{2xt-t^2}=\sum_{n=0}^\infty \frac{H_n(x)+n}{n!}$$

Differentiating above equation with respect to t we get,

$$ e^{2xt-t^2}=\sum_{n=0}^\infty \frac{H_n'(x)+n}{n!}$$

$$or,\;\; \sum_{n=0}^\infty \frac{H_n(x) t^n}{n!}(2x-2t)=\sum_{n=0}^\infty \frac{nt^{n-1}H_n(x)}{n!}$$

$$or,\;\; 2x\sum_{n=0}^\infty \frac{H_n(x)t^n}{n!}- 2\sum_{n=0}^\infty \frac{H_n(x) t^{n+1}}{n!}=\sum_{n=0}^\infty\frac{t^{n-1}H_n(x)}{(n-1)!}$$

Now, equating the coefficient of \(t^n\) on both side. We get,

$$or,\;\; 2x\cdot\frac{H_n(x)}{n!}- 2\cdot \frac{H_{n-1}(x)}{(n-1}!=\frac{H_{n+1}(x)}{n!}$$

$$or,\;\; 2x \frac{H_n(x)}{n(n-1)!}-\frac{2 H_{n-1}(x)}{(n-1)!}=\frac{H_{n+1}(x)}{n(n-1)!}$$

$$or,\;\; 2x. H_n(x) – 2nH_{n-1}(x)=H_{n+1}(x)$$

This is the first recurrence relation.

Now, differentiating equation (1) with respect to x, we get,

$$e^{2xt-t^2}\cdot 2t=\sum_{n=0}^\infty \frac{H_n'(x) t^n}{n!}$$

$$ or,\;\; 2t\cdot \sum_{n=0}^\infty \frac{H_n(x)\cdot t^n}{n!}=\sum_{n=0}^\infty\frac{H_n'(x)\cdot t^n}{n!}$$

Now, collecting the coefficient of \(t^n\) on both side.

$$\frac{2\cdot H_{n-1}(x)}{(n-1)!}=\frac{H_n'(x)}{n!}$$

$$or,\;\; \frac{2\cdot H_{n-1}(x)}{(n-1)!}=\frac{H_n'(x)}{n(n-1)!}$$

$$H_n'(x)= 2n H_{n-1}(x)\dotsm(A)$$

This is second recurrence relation.

Differentiating equation (A) with respect to x, we get

$$H_n''(x)=2nH_{n-1}'(x)$$

$$=2n\cdot 2(n-1) H_{n-2}(x)$$

$$= 2^2 n(n-1) H_{n-2}(x)\dotsm(B)$$

Again, Differentiating equation (B) with respect to x, we get

$$H_n'''(x)= 2^2 n(n-1) H_{n-2}'(x)$$

$$=2^2n(n-1)\cdot 2(n-2) H_{n-3}(x)$$

$$=2^3 n(n-1)(n-2)H_{n-3} (x)$$

Now for m times,

$$\frac{d^mH_n(x)}{dx^m}=2^m n(n-1) (n-2)\dotsm (n-m+1)H_{n-m(x)}\times \frac{(n-m)(n-m-1)\dotsm 3.2.1}{(n-m)(n-m-1)\dotsm 3.2.1}$$

$$\frac{d^mH_n(x)}{dx^m}=\frac{2^mn!H_{n-m}(x)}{(n-m)!}$$

Rodrigue's Formula:

$$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}(e-x)^2$$

Proof:

We know the generating function of Hermit polynomial is,

$$\sum_{n=0}^\infty \frac{H_n(x)t^n}{n!}= e^{2xt-t^2}\dotsm(1)$$

Now, differentiating equation(1) with respect to 't' for n times and take limit \(t\to0\).

$$\lim\limits_{t\to 0}\frac{d^n}{dt^n}\sum_{n=0}^\infty\frac{H_n(x) t^n}{n!}=\lim\limits_{t\to0}\frac{d^n}{dt^n} e^{2xt-t^2}$$

$$or,\;\; 0+\frac{n! H_n(x)}{n!}+0=\lim\limits_{t\to0} \frac{d^n}{dt^n}e^{x^2-(t-x)^2}$$

$$or,\;\; H_n(x)= e^{x^2} \lim\limits_{t\to0}\frac{d^n}{dt^n}e^{-(t-x)^2}$$

Put t – x = p

When \(t\to 0, p\to –x\)

$$=e^{x^2} \lim\limits_{p\to –x}\frac{d^n}{dp^n} e^{-p^2}$$

$$or,\;\; e^{x^2}\cdot\frac{d^n e^{-x^2}}{d(-x)^n}=e^{x^2} \frac{d^n}{(-1)^n d(x)^n}e^{-x^2}$$

$$=(-1)^n e^{x^2}\frac{d^n}{d(x)^n}e^{-x^2}$$

Hermite polynomial:

$$ H_n(-x)= (+1)^n e^{x^2}\frac{d^n}{d(x)^n}e^{-x^2}$$

Put n=0,

$$ H_0(x)=(-1)^0\cdot e^{x^2}\cdot e^{x^2}\cdot e^{-x^2}=1$$

Put n=1,

$$H_1(x) = (-1)! e^{x^2} \frac{d}{dx} (e^{-x^2})$$

$$= -e^{x^2}[e^{-x^2}\times (-2x)]$$

$$= 2x$$

Put n=2,

$$H_2(x)=(-1)^2 e^{x^2} \frac{d^2}{dx^2}(e^{-x^2})$$

$$= ex^2\frac{d}{dx}[-2x e^{-x^2}]$$

$$=-2 e^{x^2}\cdot [ 1\cdot e^{-x^2} + xe^{-x^2}\times(-2x)]$$

$$= -2+4x^2$$

$$= 4x^2-2$$

Reference:

1. Dass, H. K.Mathematical Physics. New Delhi: S. Chand, 2005.
2. Vaughn, Michael T.Introduction to Mathematical Physics. Weinheim: Wiley-VCH, 2007
3. Butkov, Eugene.Mathematical Physics. Reading, MA: Addison-Wesley Pub., 1968.
4. Carroll, Robert W.Mathematical Physics. Amsterdam: North-Holland, 1988.
5. Adhikari, Pitri Bhakta, and Dya Nidhi Chhatkuli.A Textbook of Physics. Third Revised Edition ed. Vol. III. Kathmandu: SUKUNDA PUSTAK BHAWAN, 2072.