Advanced Special Function-I - BU 2021 Question Paper

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EF-461

M.A./M.Sc. III^rd Semester (Reg./Pvt./ATKT)

Examination, 2021-22

Maths

Paper - IV

Advanced Special Function-I

Time : 3 Hours]

[Maximum Marks : Reg. 85

Pvt. 100

Note :- Attempt all the questions.

SECTION - 'A'

Objective Type Questions

1. Choose the correct answer :

5×3=15

(i) $(\alpha)_{2n} = $

$$2^n (\alpha/2)_n (\alpha/2+1/2)_n$$
$$2^{2n} (\alpha/2)_n (\alpha/2+1/2)_n$$
$$2^{2n} (\alpha/2)_n$$
None of these

(ii) $n-k! = $

$$\frac{(-1)^k n!}{(-n)_k}$$
$$\frac{(-1)^k (-n)_k}{n!}$$
$$\frac{(-1)^k}{(-n)_k}$$
None of these

(iii) $ (1-z)^{-a} = $

$$_1F_1 (a; a; z)$$
$$_1F_0 (a; -; z)$$
$$_0F_1 (a; -; z)$$
None of these

(iv) $_2F_1 \begin{bmatrix} -k, \alpha + n \\ \alpha \end{bmatrix}; 1 = $

$$\frac{1}{(\alpha)_k}$$
$$\frac{(-n)_k}{(\alpha)_k}$$
$$\frac{(\alpha)_k}{(-n)_k}$$
None of these

(v) $\sum_{n=0}^\infty \sum_{k=0}^n A(k, n) = $

$$\sum_{n=0}^\infty \sum_{k=0}^\infty A(k, n-k)$$
$$\sum_{k=0}^\infty \sum_{n=0}^k A(k, n-k)$$
$$\sum_{n=0}^\infty \sum_{k=0}^n A(n, n-k)$$
None of these

SECTION - 'B'

Short Answer Type Questions

5×5=25

2. Show that $ \Gamma(z+1) = z\Gamma(z) $

To show that

$$\beta(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$$

3. Show that

$$\frac{d}{dx} {}_2F_1 \begin{bmatrix} a, b \\ c \end{bmatrix}; x = \frac{ab}{c} {}_2F_1 \begin{bmatrix} a+1, b+1 \\ c+1 \end{bmatrix}; x$$

Show that

$ (a-b) F(a) = aF(a+) - bF(b+) $

4. Prove that

$$_2F_1 \begin{bmatrix} -n, a, b \\ c \end{bmatrix}; 1 = \frac{(c-a)_n (c-b)_n}{(c)_n (c-a-b)_n}$$

Show that

$$\int_0^1 x^{t-1} (1-x^2)^{t-1/2} [1-t^2x^2]^{-1/2} dx = \frac{\pi}{2} \frac{\Gamma(t+1/2)}{\Gamma(t)} {}_2F_1 \begin{bmatrix} 1/4, 3/4 \\ 1 \end{bmatrix}; \frac{t^4}{16}$$

SECTION - 'C'

Long Answer Type Questions

9×5=45

5. Prove that

$$b {}_2F_1 (a, b+1; b; z) - (a-b) {}_2F_1 (a+1, b; b; z) - (a-b) {}_2F_1 (a, b+1; b+1; z) = ab {}_2F_1 (a+1, b+1; b; z)$$

Show that

$$_1F_1 (a; b; z) = e^z {}_1F_1 (b-a; b; -z)$$

6. Show that

$$\sum_{n=0}^\infty \sum_{k=0}^n A(k, n) = \sum_{k=0}^\infty \sum_{n=k}^\infty A(k, n-k)$$

To show that

$$\sum_{n=0}^\infty \sum_{k=0}^{[n/2]} A(k, n) = \sum_{k=0}^\infty \sum_{n=2k}^\infty A(k, n-k)$$

7. To show that for $0 \le m \le n$,

$$(\alpha)_{n-m} = \frac{(-1)^m (\alpha)_n}{(1-\alpha-n)_m}$$

To show that

$$\Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin \pi z}$$

8. To show that if $2b$ is neither zero nor a negative integer and if $|y| < \frac{1}{2}$ and $|\frac{y}{1-y}| < 1$ then

$$(1-y)^{-a} {}_2F_1 \begin{bmatrix} a/2, a/2+1/2 \\ b+1/2 \end{bmatrix}; \frac{y^2}{(1-y)^2} = {}_2F_1 \begin{bmatrix} a, b \\ 2b \end{bmatrix}; 2y$$

If $|z| < 1$

Show that

$$_2F_1 \begin{bmatrix} a, b \\ c \end{bmatrix}; z = (1-z)^{c-a-b} {}_2F_1 \begin{bmatrix} c-a, c-b \\ c \end{bmatrix}; z$$

9. If $n$ is a non negative integer and If $a$ and $b$ are independent of $n$ then show that

$$_3F_2 \begin{bmatrix} -n, a+n+1/2, b \\ 1+a+b-n, 1/2 \end{bmatrix}; 1 = \frac{(b)_n}{(1+a-b)_n} \frac{(1+a-b)_n}{(1+a-b)_n}$$

If neither $a-b$ nor $a-c$ nor $a$ is a negative integer then.

$$(1-x)^{-a} {}_3F_2 \begin{bmatrix} a, b, c \\ 1+a-b, 1+a-c \end{bmatrix}; \frac{4x}{(1-x)^2} = {}_3F_2 \begin{bmatrix} a/2, a/2+1/2, 1+a-b-c \\ 1+a-b, 1+a-c \end{bmatrix}; 4x$$

10. Show that if $2a$ is not an odd integer $< 0$ then.

$$e^{-z} {}_2F_1 (a, 2a; 2z) = {}_0F_1 (-; a+1/2; z^2/4)$$

Show that

$$\frac{d^k}{dz^k} [e^{-z} {}_1F_1 (a; b; z)] = (-1)^k (b-a)_k e^{-z} {}_1F_1 (a; b+k; z)$$

11. If $|z| < 1$ and $| \frac{z}{1-z} | < 1$ then

$$_2F_1 \begin{bmatrix} a, b \\ c \end{bmatrix}; z = (1-z)^{-a} {}_2F_1 \begin{bmatrix} a, c-b \\ c \end{bmatrix}; \frac{-z}{1-z}$$

To show that

$$_0F_1 (-; a; x) {}_0F_1 (-; b; x) = {}_4F_3 \begin{bmatrix} a/2, a/2+1/2, b/2, b/2+1/2 \\ a, b, a+b-1 \end{bmatrix}; 4x$$