TRB PG Assistant - Legendre, Hermite and Laguerre equations - basic properties, Gamma and Beta functions One Line Questions & Answers

1. What is the differential equation for the associated Legendre equation? — (1-x^2)d^2y/dx^2 - 2x dy/dx + [n(n+1) - m^2/(1-x^2)]y = 0

2. 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? — (2n+1)xP_n(x) = (n+1)P_{n+1}(x) + nP_{n-1}(x)

3. 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? — (n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x)

4. What is the value of B(x, y) if x and y are positive integers? — (x-1)!(y-1)! / (x+y-1)!

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

6. For Legendre polynomials P_n(x), what is the value of P_n(-1)? — (-1)^n

7. What is the value of H_n(0) for even n? — (-1)^(n/2) * (n!) / (n/2)!

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

9. What is the value of L_n(0)? — 1

10. 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)? — 0

11. What is the value of B(1, 1)? — 1

12. What is the value of B(2, 3)? — 1/12

13. What is the value of Gamma(3)? — 2

14. Which of the following is a property of the Beta function? — B(x, y) = B(y, x)

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

16. What is the relationship between the Beta function and the Gamma function? — B(x, y) = Gamma(x)Gamma(y) / Gamma(x+y)

17. What is the relation between the Gamma function and the Beta function that is most commonly used? — B(x, y) = Gamma(x)Gamma(y) / Gamma(x+y)

18. What is the differential equation known as Legendre's equation? — (1 - x^2) d^2y/dx^2 - 2x dy/dx + n(n+1)y = 0

19. What is the generating function for Laguerre polynomials L_n(x)? — e^(-xt / (1-t)) / (1-t) = sum(L_n(x) t^n)

20. What is the generating function for Hermite polynomials H_n(x)? — e^(2xt - t^2) = sum(H_n(x) t^n / n!)

21. What is the relationship between the Gamma function and the factorial for non-negative integers? — Gamma(n+1) = n!

22. What is the fundamental recurrence relation for the Gamma function? — Gamma(z+1) = z Gamma(z)

23. Which property of the Gamma function is crucial for relating it to factorials? — Gamma(z+1) = z Gamma(z)

24. 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? — H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)

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

26. What is the Rodrigues' formula for Hermite polynomials H_n(x)? — H_n(x) = (-1)^n e^(x^2) d^n/dx^n (e^(-x^2))

27. What are the first few Hermite polynomials? — H0(x)=1, H1(x)=2x, H2(x)=4x^2-2

28. What are the solutions to Legendre's equation called? — Legendre polynomials

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

30. What is the orthogonality relation for Legendre polynomials P_n(x) over the interval [-1, 1]? — Integral from -1 to 1 of P_m(x)P_n(x) dx = 2/(2n+1) delta_mn

31. What is the orthogonality relation for Hermite polynomials H_n(x) over the interval (-infinity, infinity)? — Integral from -inf to inf of e^(-x^2) H_m(x)H_n(x) dx = 2^n n! sqrt(pi) delta_mn

32. What is the Beta function, denoted by B(x, y)? — Integral from 0 to 1 of t^(x-1) (1-t)^(y-1) dt

33. Which integral representation is valid for the Beta function B(x, y) for x > 0 and y > 0? — Integral from 0 to 1 of t^(x-1) (1-t)^(y-1) dt

34. What is the orthogonality relation for Laguerre polynomials L_n(x) over the interval [0, infinity)? — Integral from 0 to inf of e^(-x) L_m(x)L_n(x) dx = (n+1) delta_mn

35. What is the integral definition of the Gamma function for Re(z) > 0? — Integral from 0 to infinity of x^(z-1) e^(-x) dx

36. What is the Rodrigues' formula for Laguerre polynomials L_n(x)? — L_n(x) = 1/n! e^x d^n/dx^n (x^n e^(-x))

37. What are the first few Laguerre polynomials? — L0(x)=1, L1(x)=1-x, L2(x)=(x^2-4x+2)/2

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

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

40. What are the polynomial solutions to Hermite's differential equation called? — Hermite polynomials

41. What is the value of Gamma(n) for a positive integer n? — (n-1)!

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

43. Which of the following is NOT a property of Legendre polynomials P_n(x)? — P_n(0) = 0 for all n

44. Which of the following is a Rodrigues' formula for Legendre polynomials P_n(x)? — P_n(x) = 1/(2^n n!) d^n/dx^n (x^2 - 1)^n

45. What are the first few Legendre polynomials? — P0(x)=1, P1(x)=x, P2(x)=(3x^2-1)/2

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

47. What is the value of Gamma(1/2)? — sqrt(pi)

48. Which differential equation has solutions that are Laguerre polynomials? — x d^2y/dx^2 + (1-x) dy/dx + ny = 0

Legendre, Hermite and Laguerre equations - basic properties, Gamma and Beta functions - One Line Questions