Special Functions

The following are the assignments on special functions you need to submit. The questions are given below:

Assignment Problems

Due Date: 26th Nov.

Q1: Prove that the generating function for Hermite polynomial is $e^{2tx-t^2}$. That is, if

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

then find $H_n(x)$.

Q2: Find the values of H_2n(0) and H_2n+1(0).

Q3: Show that $H_n(x)$ $=(-1)^ne^{x^2} \dfrac{d^n}{dx^n}\left( e^{-x^2}\right)$.

Q4: For n≥1, establish the following recurrence relations of H_n(x):

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

Q5: Using the concept of beta gamma functions, compute the following integrals:

1. $\int_{0}^{\pi/2} \cos^4 x \, dx$
2. $\int_{0}^{\pi/2} \sin^5 x \cos^6 x \, dx$.

Also Study the topics listed below:

• Tchebichef polynomial of the first kind and its orthogonality condition.
• Charlier polynomial and its orthogonality relation.