IGNOU MPH 1 2026 - Obtain orthogonality relation for Hermite polynomials using the genera

Question

Obtain orthogonality relation for Hermite polynomials using the generating function:

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

19 Jan 2026

Answer :

Word Count : 173

Consider the generating function of Hermite polynomials [ e^{2xt-t^2}=\sum_{n=0}^{\infty}H_n(x)\frac{t^n}{n!}. ] Write a similar generating function with another parameter (s): [ e^{2xs-s^2}=\sum_{m=0}^{\infty}H_m(x)\frac{s^m}{m!}. ] Multiply the two generating functions: [ e^{2xt-t^2},e^{2xs-s^2}=e^{2x(t+s)-(t^2+s^2)}. ] Now multiply both sides by (e^{-x^2}) and integrate with respect to (x) over ((-\infty,\infty)): [ \int_{-\infty}^{\infty} e^{-x^2}

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