Write an expression for generating function for Hermite polynomials. Using this expression evaluate the integral
### Generating Function for Hermite Polynomials: The generating function for Hermite polynomials \( H_n(x) \) is given by: \[ G(x,t) = e^{2xt - t^2} = \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!} \] where \( H_n(x) \) are the Hermite polynomials. ### Evaluating the Integral: We need to evaluate _____ _______ _______ ______ ______ ____ _____ _____ _________ _____ _____.
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