TRB PG Assistant - Legendre, Hermite and Laguerre equations - basic properties, Gamma and Beta functions Question Bank with Options

1. What is the limit of Gamma(z) as z approaches infinity along the real axis?

A) Infinity

B) 0

C) 1

D) Euler-Mascheroni constant

2. Which of the following is NOT a property of Legendre polynomials P_n(x)?

A) P_n(1) = 1

B) P_n(-1) = (-1)^n

C) Integral from -1 to 1 of P_n(x) dx = 0 for n > 0

D) P_n(0) = 0 for all n

3. What is the value of Gamma(n+1) for a positive integer n?

A) n!

B) (n+1)!

C) n

D) 1

4. The differential equation d^2y/dx^2 - 2x dy/dx + 2ny = 0 is known as Hermite's equation. What is the value of n for the polynomial solution H_0(x)?

A) 0

B) 1

C) 2

D) n

5. What is the relation between the Gamma function and the Beta function that is most commonly used?

A) B(x, y) = Gamma(x)Gamma(y) / Gamma(x+y)

B) B(x, y) = Gamma(x+y) / (Gamma(x)Gamma(y))

C) B(x, y) = Gamma(x) + Gamma(y)

D) B(x, y) = Gamma(x) - Gamma(y)

6. What is the differential equation for the associated Legendre equation?

A) (1-x^2)d^2y/dx^2 - 2x dy/dx + [n(n+1) - m^2/(1-x^2)]y = 0

B) (1-x^2)d^2y/dx^2 - 2x dy/dx + [n(n+1) + m^2/(1-x^2)]y = 0

C) x d^2y/dx^2 + (1-x) dy/dx + ny = 0

D) d^2y/dx^2 - 2x dy/dx + 2ny = 0

7. The integral form of the Beta function B(x, y) is symmetric in x and y. What does this imply?

A) B(x, y) = B(y, x)

B) B(x, y) = -B(y, x)

C) B(x, y) = 1/B(y, x)

D) B(x, y) = B(x+y, x+y)

8. Which property of the Gamma function is crucial for relating it to factorials?

A) Gamma(z+1) = z Gamma(z)

B) Gamma(z)Gamma(1-z) = pi / sin(pi z)

C) Gamma(z) = Integral from 0 to inf of x^(z-1) e^(-x) dx

D) Gamma(1/2) = sqrt(pi)

9. What are the first few Laguerre polynomials?

A) L0(x)=1, L1(x)=1-x, L2(x)=(x^2-4x+2)/2

B) L0(x)=1, L1(x)=x, L2(x)=x^2

C) L0(x)=1, L1(x)=1-x, L2(x)=x^2-4x+2

D) L0(x)=0, L1(x)=1-x, L2(x)=(x^2-4x+2)/2

10. What are the first few Hermite polynomials?

A) H0(x)=1, H1(x)=2x, H2(x)=4x^2-2

B) H0(x)=1, H1(x)=x, H2(x)=x^2

C) H0(x)=1, H1(x)=2x, H2(x)=4x^2

D) H0(x)=0, H1(x)=2x, H2(x)=4x^2-2

11. What are the first few Legendre polynomials?

A) P0(x)=1, P1(x)=x, P2(x)=(3x^2-1)/2

B) P0(x)=1, P1(x)=1, P2(x)=x^2

C) P0(x)=1, P1(x)=x, P2(x)=x^2

D) P0(x)=0, P1(x)=x, P2(x)=3x^2

12. The differential equation d^2y/dx^2 + (1-x) dy/dx + ny = 0 is related to which special function?

A) Legendre

B) Hermite

C) Laguerre

D) Bessel

13. What is the value of B(2, 3)?

A) 1/12

B) 1/24

C) 1/6

D) 1/10

14. What is the value of Gamma(3)?

A) 2

B) 6

C) 1

D) 24

15. Which integral representation is valid for the Beta function B(x, y) for x > 0 and y > 0?

A) Integral from 0 to 1 of t^(x-1) (1-t)^(y-1) dt

B) Integral from 0 to infinity of t^(x-1) / (1+t)^(x+y) dt

C) Integral from 0 to infinity of x^(t-1) e^(-t) dt

D) Integral from 0 to 1 of x^t dt

16. What is the relationship between the Gamma function and the factorial for non-negative integers?

A) Gamma(n+1) = n!

B) Gamma(n) = n!

C) Gamma(n+1) = (n+1)!

D) Gamma(n) = (n-1)!

17. What is the Rodrigues' formula for Laguerre polynomials L_n(x)?

A) L_n(x) = 1/n! e^x d^n/dx^n (x^n e^(-x))

B) L_n(x) = e^x d^n/dx^n (x^n e^(-x))

C) L_n(x) = 1/n! d^n/dx^n (x^n e^(-x))

D) L_n(x) = (-1)^n/n! e^x d^n/dx^n (x^n e^(-x))

18. The Laguerre polynomials L_n(x) satisfy the recurrence relation: (n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x). What is the correct form?

A) (n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x)

B) (n+1)L_{n+1}(x) = (2n+1+x)L_n(x) - nL_{n-1}(x)

C) nL_{n+1}(x) = (2n+1-x)L_n(x) - (n+1)L_{n-1}(x)

D) (n+1)L_{n+1}(x) = (n+1-x)L_n(x) - nL_{n-1}(x)

19. What is the Rodrigues' formula for Hermite polynomials H_n(x)?

A) H_n(x) = (-1)^n e^(x^2) d^n/dx^n (e^(-x^2))

B) H_n(x) = e^(x^2) d^n/dx^n (e^(-x^2))

C) H_n(x) = (-1)^n d^n/dx^n (e^(-x^2))

D) H_n(x) = (-1)^n e^(-x^2) d^n/dx^n (e^(x^2))

20. The Hermite polynomials H_n(x) satisfy the recurrence relation: H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x). What is the correct form?

A) H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)

B) H_{n+1}(x) = 2xH_n(x) + 2nH_{n-1}(x)

C) H_{n+1}(x) = xH_n(x) - nH_{n-1}(x)

D) H_{n+1}(x) = 2xH_n(x) - H_{n-1}(x)

21. Which of the following is a Rodrigues' formula for Legendre polynomials P_n(x)?

A) P_n(x) = 1/(2^n n!) d^n/dx^n (x^2 - 1)^n

B) P_n(x) = 1/(2^n) d^n/dx^n (x^2 - 1)^n

C) P_n(x) = 1/(2^n n!) d^n/dx^n (x^2 + 1)^n

D) P_n(x) = d^n/dx^n (x^2 - 1)^n

22. Legendre polynomials P_n(x) satisfy the recurrence relation: (2n+1)xP_n(x) = (n+1)P_{n+1}(x) + nP_{n-1}(x). What is the correct form of this relation?

A) (2n+1)xP_n(x) = (n+1)P_{n+1}(x) + nP_{n-1}(x)

B) (2n+1)xP_n(x) = nP_{n+1}(x) + (n+1)P_{n-1}(x)

C) (2n-1)xP_n(x) = nP_{n+1}(x) + (n-1)P_{n-1}(x)

D) (2n+1)xP_n(x) = P_{n+1}(x) + P_{n-1}(x)

23. Which of the following is a property of the Beta function?

A) B(x, y) = B(y, x)

B) B(x, y) = -B(y, x)

C) B(x, y) = B(x+1, y+1)

D) B(x, y) = B(x-1, y-1)

24. What is the value of B(x, y) if x and y are positive integers?

A) (x-1)!(y-1)! / (x+y-1)!

B) x!y! / (x+y)!

C) (x-1)!(y-1)!

D) (x+y-1)! / ((x-1)!(y-1)!)

25. What is the value of B(1, 1)?

A) 1

B) 2

C) 1/2

D) 0

26. What is the relationship between the Beta function and the Gamma function?

A) B(x, y) = Gamma(x)Gamma(y) / Gamma(x+y)

B) B(x, y) = Gamma(x+y) / (Gamma(x)Gamma(y))

C) B(x, y) = Gamma(x)Gamma(y)

D) B(x, y) = Gamma(x+y)

27. What is the Beta function, denoted by B(x, y)?

A) Integral from 0 to 1 of t^(x-1) (1-t)^(y-1) dt

B) Integral from 0 to infinity of t^(x-1) (1-t)^(y-1) dt

C) Integral from -1 to 1 of t^(x-1) (1-t)^(y-1) dt

D) Integral from 0 to 1 of t^x (1-t)^y dt

28. What is the value of Gamma(1/2)?

A) sqrt(pi)

B) pi

C) sqrt(pi)/2

D) 1

29. What is the value of Gamma(n) for a positive integer n?

A) n!

B) (n-1)!

C) (n+1)!

D) n

30. What is the fundamental recurrence relation for the Gamma function?

A) Gamma(z+1) = z Gamma(z)

B) Gamma(z+1) = (z+1) Gamma(z)

C) Gamma(z) = z Gamma(z-1)

D) Gamma(z+1) = Gamma(z) / z

31. What is the integral definition of the Gamma function for Re(z) > 0?

A) Integral from 0 to infinity of x^(z-1) e^(-x) dx

B) Integral from 0 to infinity of x^z e^(-x) dx

C) Integral from 0 to 1 of x^(z-1) e^(-x) dx

D) Integral from -infinity to infinity of x^(z-1) e^(-x) dx

32. The Gamma function, denoted by Gamma(z), is a generalization of which mathematical function?

A) Sine function

B) Cosine function

C) Exponential function

D) Factorial function

33. What is the orthogonality relation for Laguerre polynomials L_n(x) over the interval [0, infinity)?

A) Integral from 0 to inf of L_m(x)L_n(x) dx = 0 for m != n

B) Integral from 0 to inf of e^(-x) L_m(x)L_n(x) dx = delta_mn

C) Integral from 0 to inf of e^(-x) L_m(x)L_n(x) dx = (n+1) delta_mn

D) Integral from 0 to inf of e^(-x) L_m(x)L_n(x) dx = 2n+1 delta_mn

34. What is the value of L_n(0)?

A) 0

B) 1

C) n

D) (-1)^n

35. What is the generating function for Laguerre polynomials L_n(x)?

A) e^(-xt / (1-t)) / (1-t) = sum(L_n(x) t^n)

B) e^(xt / (1-t)) / (1-t) = sum(L_n(x) t^n)

C) e^(xt) = sum(L_n(x) t^n / n!)

D) ln(1-t) = -sum(t^n / n)

36. The generalized Laguerre polynomials are denoted by L_n^alpha(x). What is the value of L_n^0(x)?

A) H_n(x)

B) P_n(x)

C) L_n(x)

D) J_n(x)

37. Which differential equation has solutions that are Laguerre polynomials?

A) x d^2y/dx^2 + (1-x) dy/dx + ny = 0

B) d^2y/dx^2 - 2x dy/dx + 2ny = 0

C) (1 - x^2) d^2y/dx^2 - 2x dy/dx + n(n+1)y = 0

D) x^2 d^2y/dx^2 + x dy/dx + (x^2 - n^2)y = 0

38. What is the orthogonality relation for Hermite polynomials H_n(x) over the interval (-infinity, infinity)?

A) Integral from -inf to inf of H_m(x)H_n(x) dx = 0 for m != n

B) Integral from -inf to inf of e^(-x^2) H_m(x)H_n(x) dx = 2^n n! sqrt(pi) delta_mn

C) Integral from -inf to inf of e^(-x^2) H_m(x)H_n(x) dx = 0 for m != n

D) Integral from -inf to inf of H_m(x)H_n(x) dx = sqrt(pi) delta_mn

39. What is the value of H_n(0) for odd n?

A) 0

B) (-1)^((n-1)/2) * (n!) / ((n-1)/2)!

C) (-1)^((n-1)/2) * (n!)

D) 1

40. What is the value of H_n(0) for even n?

A) 0

B) (-1)^(n/2) * (n!) / (n/2)!

C) (-1)^(n/2) * (n!)

D) 1

41. What is the generating function for Hermite polynomials H_n(x)?

A) e^(2xt - t^2) = sum(H_n(x) t^n / n!)

B) e^(-t^2) = sum(H_n(x) t^n / n!)

C) (1-t)^(-1/2) = sum(P_n(x) t^n)

D) e^x = sum(x^n / n!)

42. What are the polynomial solutions to Hermite's differential equation called?

A) Legendre polynomials

B) Hermite polynomials

C) Laguerre polynomials

D) Chebyshev polynomials

43. Which special function is a solution to the differential equation d^2y/dx^2 - 2x dy/dx + 2ny = 0?

A) Legendre equation

B) Hermite equation

C) Laguerre equation

D) Bessel equation

44. What is the orthogonality relation for Legendre polynomials P_n(x) over the interval [-1, 1]?

A) Integral from -1 to 1 of P_m(x)P_n(x) dx = 0 for m != n

B) Integral from -1 to 1 of x^m P_n(x) dx = 0 for m != n

C) Integral from -1 to 1 of P_m(x)P_n(x) dx = 2/(2n+1) delta_mn

D) Integral from -1 to 1 of P_m(x)P_n(x) dx = 0 for all m, n

45. For Legendre polynomials P_n(x), what is the value of P_n(-1)?

A) 0

B) 1

C) n

D) (-1)^n

46. For Legendre polynomials P_n(x), what is the value of P_n(1)?

A) 0

B) 1

C) n

D) (-1)^n

47. What are the solutions to Legendre's equation called?

A) Hermite polynomials

B) Laguerre polynomials

C) Legendre polynomials

D) Bessel functions

48. What is the differential equation known as Legendre's equation?

A) d^2y/dx^2 + x dy/dx + (n^2 - x^2)y = 0

B) (1 - x^2) d^2y/dx^2 - 2x dy/dx + n(n+1)y = 0

C) d^2y/dx^2 - x dy/dx + ny = 0

D) d^2y/dx^2 + (n+1)x dy/dx + (n^2 - x^2)y = 0

Legendre, Hermite and Laguerre equations - basic properties, Gamma and Beta functions - Question Bank