Let Pn(x) be the Legendre polynomial of degree n. Then the generating function for Pn(x) is given by:
∞
1
Pn(x)tn = √
n=0
1 − 2xt + t2
Differentiating both sides with respect to t, we get:
∞
∑nPn(x)tn-1 = -xt(1 − 2xt + t2)-1/2 + (1 − 2xt + t2)-3/2
n=1
Multiplying both sides by (1 − 2xt + t2)1/2, we get:
∞
∑
1. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Legendre’s Function
N. B. Vyas
Department of Mathematics
Atmiya Institute of Technology and Science
Department of Mathematics
N. B. Vyas Legendre’s Function
2. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
The differential equation
N. B. Vyas Legendre’s Function
3. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
The differential equation
2
(1 − x )y − 2xy + n(n + 1)y = 0
N. B. Vyas Legendre’s Function
4. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
The differential equation
2
(1 − x )y − 2xy + n(n + 1)y = 0
is called Legendre’s differential equation,
n is real constant
N. B. Vyas Legendre’s Function
5. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Legendre’s Polynomials:
⇒ P0 (x) = 1
N. B. Vyas Legendre’s Function
6. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Legendre’s Polynomials:
⇒ P0 (x) = 1
⇒ P1 (x) = x
N. B. Vyas Legendre’s Function
7. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Legendre’s Polynomials:
⇒ P0 (x) = 1
⇒ P1 (x) = x
1
⇒ P2 (x) = (3x2 − 1)
2
N. B. Vyas Legendre’s Function
8. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Legendre’s Polynomials:
⇒ P0 (x) = 1
⇒ P1 (x) = x
1
⇒ P2 (x) = (3x2 − 1)
2
1
⇒ P3 (x) = (5x3 − 3x)
2
N. B. Vyas Legendre’s Function
9. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Legendre’s Polynomials:
⇒ P0 (x) = 1
⇒ P1 (x) = x
1
⇒ P2 (x) = (3x2 − 1)
2
1
⇒ P3 (x) = (5x3 − 3x)
2
1
⇒ P4 (x) = (35x3 − 30x2 + 3)
8
N. B. Vyas Legendre’s Function
10. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Legendre’s Polynomials:
⇒ P0 (x) = 1
⇒ P1 (x) = x
1
⇒ P2 (x) = (3x2 − 1)
2
1
⇒ P3 (x) = (5x3 − 3x)
2
1
⇒ P4 (x) = (35x3 − 30x2 + 3)
8
1
⇒ P5 (x) = (63x5 − 70x3 + 15x)
8
N. B. Vyas Legendre’s Function
11. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Ex.1 Express f (x) in terms of
Legendre’s polynomials where
f (x) = x3 + 2x2 − x − 3.
N. B. Vyas Legendre’s Function
12. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
N. B. Vyas Legendre’s Function
13. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
⇒ P0 (x) = 1
∴ 1 = P0 (x)
N. B. Vyas Legendre’s Function
14. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
⇒ P0 (x) = 1
∴ 1 = P0 (x)
⇒ P1 (x) = x
∴ x = P1 (x)
N. B. Vyas Legendre’s Function
15. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
⇒ P0 (x) = 1
∴ 1 = P0 (x)
⇒ P1 (x) = x
∴ x = P1 (x)
1
⇒ P3 (x) = (5x3 − 3x)
2
N. B. Vyas Legendre’s Function
16. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
⇒ P0 (x) = 1
∴ 1 = P0 (x)
⇒ P1 (x) = x
∴ x = P1 (x)
1
⇒ P3 (x) = (5x3 − 3x)
2
∴ 2P3 (x) = (5x3 − 3x)
N. B. Vyas Legendre’s Function
17. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
⇒ P0 (x) = 1
∴ 1 = P0 (x)
⇒ P1 (x) = x
∴ x = P1 (x)
1
⇒ P3 (x) = (5x3 − 3x)
2
∴ 2P3 (x) = (5x3 − 3x)
∴ 2P3 (x) + 3x = 5x3
N. B. Vyas Legendre’s Function
18. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
⇒ P0 (x) = 1
∴ 1 = P0 (x)
⇒ P1 (x) = x
∴ x = P1 (x)
1
⇒ P3 (x) = (5x3 − 3x)
2
∴ 2P3 (x) = (5x3 − 3x)
∴ 2P3 (x) + 3x = 5x3
∴ 2P3 (x) + 3P1 (x) = 5x3 { x = P1 (x)}
N. B. Vyas Legendre’s Function
19. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
⇒ P0 (x) = 1
∴ 1 = P0 (x)
⇒ P1 (x) = x
∴ x = P1 (x)
1
⇒ P3 (x) = (5x3 − 3x)
2
∴ 2P3 (x) = (5x3 − 3x)
∴ 2P3 (x) + 3x = 5x3
∴ 2P3 (x) + 3P1 (x) = 5x3 { x = P1 (x)}
2 3
∴ x3 = P3 (x) + P1 (x)
5 5
N. B. Vyas Legendre’s Function
20. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
1
⇒ P2 (x) = (3x2 − 1)
2
N. B. Vyas Legendre’s Function
21. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
1
⇒ P2 (x) = (3x2 − 1)
2
∴ 2P2 (x) = (3x2 − 1)
N. B. Vyas Legendre’s Function
22. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
1
⇒ P2 (x) = (3x2 − 1)
2
∴ 2P2 (x) = (3x2 − 1)
∴ 2P2 (x) + 1 = 3x2
N. B. Vyas Legendre’s Function
23. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
1
⇒ P2 (x) = (3x2 − 1)
2
∴ 2P2 (x) = (3x2 − 1)
∴ 2P2 (x) + 1 = 3x2
∴ 2P2 (x) + P0 (x) = 3x2 { 1 = P0 (x)}
N. B. Vyas Legendre’s Function
24. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
1
⇒ P2 (x) = (3x2 − 1)
2
∴ 2P2 (x) = (3x2 − 1)
∴ 2P2 (x) + 1 = 3x2
∴ 2P2 (x) + P0 (x) = 3x2 { 1 = P0 (x)}
2 1
∴ x2 = P2 (x) + P0 (x)
3 3
N. B. Vyas Legendre’s Function
25. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
1
⇒ P2 (x) = (3x2 − 1)
2
∴ 2P2 (x) = (3x2 − 1)
∴ 2P2 (x) + 1 = 3x2
∴ 2P2 (x) + P0 (x) = 3x2 { 1 = P0 (x)}
2 1
∴ x2 = P2 (x) + P0 (x)
3 3
Now, f (x) = x3 + 2x2 − x − 3
N. B. Vyas Legendre’s Function
26. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
1
⇒ P2 (x) = (3x2 − 1)
2
∴ 2P2 (x) = (3x2 − 1)
∴ 2P2 (x) + 1 = 3x2
∴ 2P2 (x) + P0 (x) = 3x2 { 1 = P0 (x)}
2 1
∴ x2 = P2 (x) + P0 (x)
3 3
Now, f (x) = x3 + 2x2 − x − 3
f (x) = x3 + 2x2 − x − 3
2 3 4 2
= P3 (x) + P1 (x) + P2 (x) + P0 (x) − P1 (x) − 3P0 (x)
5 5 3 3
N. B. Vyas Legendre’s Function
27. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Ex.2 Express x3 − 5x2 + 6x + 1 in
terms of Legendre’s polynomial.
N. B. Vyas Legendre’s Function
28. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Ex.3 Express 4x3 − 2x2 − 3x + 8 in
terms of Legendre’s polynomial.
N. B. Vyas Legendre’s Function
29. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Generating Function for Pn(x)
∞
1
Pn(x)tn = √
n=0
1 − 2xt + t2
1
= (1 − 2xt + t2)− 2
N. B. Vyas Legendre’s Function
30. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
1
The function (1 − 2xt + t2)− 2 is
called Generating function of
Legendre’s polynomial Pn(x)
N. B. Vyas Legendre’s Function
31. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Ex Show that
N. B. Vyas Legendre’s Function
32. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Ex Show that
(i)Pn(1) = 1
N. B. Vyas Legendre’s Function
33. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Ex Show that
(i)Pn(1) = 1
(ii)Pn(−1) = (−1)n
N. B. Vyas Legendre’s Function
34. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Ex Show that
(i)Pn(1) = 1
(ii)Pn(−1) = (−1)n
(iii)Pn(−x) = (−1)nPn(x)
N. B. Vyas Legendre’s Function
35. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
∞
1
(i) We have Pn (x)tn = (1 − 2xt + t2 )− 2
n=0
Putting x = 1 in eq(1), we get
N. B. Vyas Legendre’s Function
36. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
∞
1
(i) We have Pn (x)tn = (1 − 2xt + t2 )− 2
n=0
Putting x = 1 in eq(1), we get
∞
1
Pn (1)tn = (1 − 2t + t2 )− 2 = (1 − t)−1
n=0
N. B. Vyas Legendre’s Function
37. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
∞
1
(i) We have Pn (x)tn = (1 − 2xt + t2 )− 2
n=0
Putting x = 1 in eq(1), we get
∞
1
Pn (1)tn = (1 − 2t + t2 )− 2 = (1 − t)−1
n=0
∞
1
∴ Pn (1)tn = = 1 + t + t2 + t3 + ...
1−t
n=0
N. B. Vyas Legendre’s Function
38. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
∞
1
(i) We have Pn (x)tn = (1 − 2xt + t2 )− 2
n=0
Putting x = 1 in eq(1), we get
∞
1
Pn (1)tn = (1 − 2t + t2 )− 2 = (1 − t)−1
n=0
∞
1
∴ Pn (1)tn = = 1 + t + t2 + t3 + ...
1−t
n=0
∞ ∞
n
∴ Pn (1)t = tn
n=0 n=0
N. B. Vyas Legendre’s Function
39. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
∞
1
(i) We have Pn (x)tn = (1 − 2xt + t2 )− 2
n=0
Putting x = 1 in eq(1), we get
∞
1
Pn (1)tn = (1 − 2t + t2 )− 2 = (1 − t)−1
n=0
∞
1
∴ Pn (1)tn = = 1 + t + t2 + t3 + ...
1−t
n=0
∞ ∞
n
∴ Pn (1)t = tn
n=0 n=0
Comparing the coefficient of tn both the sides, we get
N. B. Vyas Legendre’s Function
40. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Solution:
∞
1
(i) We have Pn (x)tn = (1 − 2xt + t2 )− 2
n=0
Putting x = 1 in eq(1), we get
∞
1
Pn (1)tn = (1 − 2t + t2 )− 2 = (1 − t)−1
n=0
∞
1
∴ Pn (1)tn = = 1 + t + t2 + t3 + ...
1−t
n=0
∞ ∞
n
∴ Pn (1)t = tn
n=0 n=0
Comparing the coefficient of tn both the sides, we get
Pn (1) = 1
N. B. Vyas Legendre’s Function
41. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(ii) Putting x = −1 in eq(1), we get
N. B. Vyas Legendre’s Function
42. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(ii) Putting x = −1 in eq(1), we get
∞
1
Pn (−1)tn = (1 + 2t + t2 )− 2 = (1 + t)−1
n=0
N. B. Vyas Legendre’s Function
43. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(ii) Putting x = −1 in eq(1), we get
∞
1
Pn (−1)tn = (1 + 2t + t2 )− 2 = (1 + t)−1
n=0
∞
1
∴ Pn (−1)tn = = 1 − t + t2 − t3 + ...
1+t
n=0
N. B. Vyas Legendre’s Function
44. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(ii) Putting x = −1 in eq(1), we get
∞
1
Pn (−1)tn = (1 + 2t + t2 )− 2 = (1 + t)−1
n=0
∞
1
∴ Pn (−1)tn = = 1 − t + t2 − t3 + ...
1+t
n=0
∞ ∞
∴ Pn (−1)tn = (−1)n tn
n=0 n=0
N. B. Vyas Legendre’s Function
45. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(ii) Putting x = −1 in eq(1), we get
∞
1
Pn (−1)tn = (1 + 2t + t2 )− 2 = (1 + t)−1
n=0
∞
1
∴ Pn (−1)tn = = 1 − t + t2 − t3 + ...
1+t
n=0
∞ ∞
∴ Pn (−1)tn = (−1)n tn
n=0 n=0
Comparing coefficients of tn , we get
N. B. Vyas Legendre’s Function
46. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(ii) Putting x = −1 in eq(1), we get
∞
1
Pn (−1)tn = (1 + 2t + t2 )− 2 = (1 + t)−1
n=0
∞
1
∴ Pn (−1)tn = = 1 − t + t2 − t3 + ...
1+t
n=0
∞ ∞
∴ Pn (−1)tn = (−1)n tn
n=0 n=0
Comparing coefficients of tn , we get
Pn (−1) = (−1)n
N. B. Vyas Legendre’s Function
47. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(iii) Now replacing x by −x in eq(1), we get
N. B. Vyas Legendre’s Function
48. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(iii) Now replacing x by −x in eq(1), we get
∞
1
Pn (−x)tn = (1 + 2xt + t2 )− 2 —(a)
n=0
N. B. Vyas Legendre’s Function
49. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(iii) Now replacing x by −x in eq(1), we get
∞
1
Pn (−x)tn = (1 + 2xt + t2 )− 2 —(a)
n=0
Now, replacing t by −t in eq(1), we get
N. B. Vyas Legendre’s Function
50. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(iii) Now replacing x by −x in eq(1), we get
∞
1
Pn (−x)tn = (1 + 2xt + t2 )− 2 —(a)
n=0
Now, replacing t by −t in eq(1), we get
∞
1
Pn (x)(−t)n = (1 + 2xt + t2 )− 2 —(b)
n=0
N. B. Vyas Legendre’s Function
51. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(iii) Now replacing x by −x in eq(1), we get
∞
1
Pn (−x)tn = (1 + 2xt + t2 )− 2 —(a)
n=0
Now, replacing t by −t in eq(1), we get
∞
1
Pn (x)(−t)n = (1 + 2xt + t2 )− 2 —(b)
n=0
from equation (a) and (b)
N. B. Vyas Legendre’s Function
52. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(iii) Now replacing x by −x in eq(1), we get
∞
1
Pn (−x)tn = (1 + 2xt + t2 )− 2 —(a)
n=0
Now, replacing t by −t in eq(1), we get
∞
1
Pn (x)(−t)n = (1 + 2xt + t2 )− 2 —(b)
n=0
from equation (a) and (b)
∞ ∞
n
Pn (−x)(t) = Pn (x)(−1)n (t)n
n=0 n=0
N. B. Vyas Legendre’s Function
53. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(iii) Now replacing x by −x in eq(1), we get
∞
1
Pn (−x)tn = (1 + 2xt + t2 )− 2 —(a)
n=0
Now, replacing t by −t in eq(1), we get
∞
1
Pn (x)(−t)n = (1 + 2xt + t2 )− 2 —(b)
n=0
from equation (a) and (b)
∞ ∞
n
Pn (−x)(t) = Pn (x)(−1)n (t)n
n=0 n=0
Comparing the coefficients of tn , both sides, we get
N. B. Vyas Legendre’s Function
54. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(iii) Now replacing x by −x in eq(1), we get
∞
1
Pn (−x)tn = (1 + 2xt + t2 )− 2 —(a)
n=0
Now, replacing t by −t in eq(1), we get
∞
1
Pn (x)(−t)n = (1 + 2xt + t2 )− 2 —(b)
n=0
from equation (a) and (b)
∞ ∞
n
Pn (−x)(t) = Pn (x)(−1)n (t)n
n=0 n=0
Comparing the coefficients of tn , both sides, we get
Pn (−x) = (−1)n Pn (x)
N. B. Vyas Legendre’s Function
55. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Rodrigue’s Formula
1 dn
Pn(x) = n [(x2 − 1)n]
2 n! dxn
N. B. Vyas Legendre’s Function
56. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Proof:
Let y = (x2 − 1)n
N. B. Vyas Legendre’s Function
57. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Proof:
Let y = (x2 − 1)n
Differentiating wit respect to x
N. B. Vyas Legendre’s Function
58. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Proof:
Let y = (x2 − 1)n
Differentiating wit respect to x
∴ y1 = n(x2 − 1)n−1 (2x)
N. B. Vyas Legendre’s Function
59. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Proof:
Let y = (x2 − 1)n
Differentiating wit respect to x
∴ y1 = n(x2 − 1)n−1 (2x)
2nx(x2 − 1)n 2nxy
∴ y1 = = 2
(x2 − 1) x −1
N. B. Vyas Legendre’s Function
60. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Proof:
Let y = (x2 − 1)n
Differentiating wit respect to x
∴ y1 = n(x2 − 1)n−1 (2x)
2nx(x2 − 1)n 2nxy
∴ y1 = = 2
(x2 − 1) x −1
∴ (x2 − 1)y1 = 2nxy
N. B. Vyas Legendre’s Function
61. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Proof:
Let y = (x2 − 1)n
Differentiating wit respect to x
∴ y1 = n(x2 − 1)n−1 (2x)
2nx(x2 − 1)n 2nxy
∴ y1 = = 2
(x2 − 1) x −1
∴ (x2 − 1)y1 = 2nxy
Differentiating with respect to x,
N. B. Vyas Legendre’s Function
62. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Proof:
Let y = (x2 − 1)n
Differentiating wit respect to x
∴ y1 = n(x2 − 1)n−1 (2x)
2nx(x2 − 1)n 2nxy
∴ y1 = = 2
(x2 − 1) x −1
∴ (x2 − 1)y1 = 2nxy
Differentiating with respect to x,
(x2 − 1)y2 +2xy1 = 2nxy1 + 2ny
N. B. Vyas Legendre’s Function
63. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn
(U V ) = nC0 U Vn + nC1 U1 Vn−1 + ... + nCn Un V
dxn
N. B. Vyas Legendre’s Function
64. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn
(U V ) = nC0 U Vn + nC1 U1 Vn−1 + ... + nCn Un V
dxn
dn
→ ((x2 −1)y2 ) = nC0 (x2 −1)yn+2 +nC1 (2x)yn+1 +nC2 (2)yn
dxn
N. B. Vyas Legendre’s Function
65. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn
(U V ) = nC0 U Vn + nC1 U1 Vn−1 + ... + nCn Un V
dxn
dn
→ ((x2 −1)y2 ) = nC0 (x2 −1)yn+2 +nC1 (2x)yn+1 +nC2 (2)yn
dxn
dn
→ (2xy1 ) = nC0 (2x)yn+1 + nC1 (2)yn
dxn
N. B. Vyas Legendre’s Function
66. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn
(U V ) = nC0 U Vn + nC1 U1 Vn−1 + ... + nCn Un V
dxn
dn
→ ((x2 −1)y2 ) = nC0 (x2 −1)yn+2 +nC1 (2x)yn+1 +nC2 (2)yn
dxn
dn
→ (2xy1 ) = nC0 (2x)yn+1 + nC1 (2)yn
dxn
dn
→ (2nxy1 ) = nC0 (2nx)yn+1 + nC1 (2n)yn
dxn
N. B. Vyas Legendre’s Function
67. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn
(U V ) = nC0 U Vn + nC1 U1 Vn−1 + ... + nCn Un V
dxn
dn
→ ((x2 −1)y2 ) = nC0 (x2 −1)yn+2 +nC1 (2x)yn+1 +nC2 (2)yn
dxn
dn
→ (2xy1 ) = nC0 (2x)yn+1 + nC1 (2)yn
dxn
dn
→ (2nxy1 ) = nC0 (2nx)yn+1 + nC1 (2n)yn
dxn
dn
→ (2ny) = 2nyn
dxn
N. B. Vyas Legendre’s Function
68. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn
(U V ) = nC0 U Vn + nC1 U1 Vn−1 + ... + nCn Un V
dxn
dn
→ ((x2 −1)y2 ) = nC0 (x2 −1)yn+2 +nC1 (2x)yn+1 +nC2 (2)yn
dxn
dn
→ (2xy1 ) = nC0 (2x)yn+1 + nC1 (2)yn
dxn
dn
→ (2nxy1 ) = nC0 (2nx)yn+1 + nC1 (2n)yn
dxn
dn
→ (2ny) = 2nyn
dxn
n(n − 1)
Also nC0 = 1, nC1 = n, nC2 =
2!
N. B. Vyas Legendre’s Function
69. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
N. B. Vyas Legendre’s Function
70. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
N. B. Vyas Legendre’s Function
71. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
∴ (x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0
N. B. Vyas Legendre’s Function
72. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
∴ (x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0
∴ (1 − x2 )yn+2 − 2xyn+1 + n(n + 1)yn = 0
N. B. Vyas Legendre’s Function
73. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
∴ (x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0
∴ (1 − x2 )yn+2 − 2xyn+1 + n(n + 1)yn = 0
dn y
Let v = yn = n
dx
N. B. Vyas Legendre’s Function
74. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
∴ (x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0
∴ (1 − x2 )yn+2 − 2xyn+1 + n(n + 1)yn = 0
dn y
Let v = yn = n
dx
∴ (1 − x2 )v2 − 2xv1 + n(n + 1)v = 0 ——(2)
N. B. Vyas Legendre’s Function
75. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
∴ (x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0
∴ (1 − x2 )yn+2 − 2xyn+1 + n(n + 1)yn = 0
dn y
Let v = yn = n
dx
∴ (1 − x2 )v2 − 2xv1 + n(n + 1)v = 0 ——(2)
Equation (2) is a Legendre’s equation in variables v and x
N. B. Vyas Legendre’s Function
76. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
∴ (x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0
∴ (1 − x2 )yn+2 − 2xyn+1 + n(n + 1)yn = 0
dn y
Let v = yn = n
dx
∴ (1 − x2 )v2 − 2xv1 + n(n + 1)v = 0 ——(2)
Equation (2) is a Legendre’s equation in variables v and x
⇒ Pn (x) is a solution of equation (2)
N. B. Vyas Legendre’s Function
77. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
∴ (x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0
∴ (1 − x2 )yn+2 − 2xyn+1 + n(n + 1)yn = 0
dn y
Let v = yn = n
dx
∴ (1 − x2 )v2 − 2xv1 + n(n + 1)v = 0 ——(2)
Equation (2) is a Legendre’s equation in variables v and x
⇒ Pn (x) is a solution of equation (2)
Also, v = f (x) is a solution of equation (2)
N. B. Vyas Legendre’s Function
78. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (x2 − 1)yn+2 + 2nxyn+1 + n(n − 1)yn
+2xyn+1 + 2nyn = 2nxyn+1 + n(2n)yn + 2nyn
∴ (x2 − 1)yn+2 + 2xyn+1 + (n2 − n + 2n − 2n2 − 2n)yn = 0
∴ (x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0
∴ (1 − x2 )yn+2 − 2xyn+1 + n(n + 1)yn = 0
dn y
Let v = yn = n
dx
∴ (1 − x2 )v2 − 2xv1 + n(n + 1)v = 0 ——(2)
Equation (2) is a Legendre’s equation in variables v and x
⇒ Pn (x) is a solution of equation (2)
Also, v = f (x) is a solution of equation (2)
Pn = cv where c is constant
N. B. Vyas Legendre’s Function
79. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn y dn
∴ Pn (x) = c = c n (x2 − 1)n ——(3)
dxn dx
N. B. Vyas Legendre’s Function
80. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn y dn
∴ Pn (x) = c = c n (x2 − 1)n ——(3)
dxn dx
Now y = (x2 − 1)n
N. B. Vyas Legendre’s Function
81. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn y dn
∴ Pn (x) = c = c n (x2 − 1)n ——(3)
dxn dx
Now y = (x2 − 1)n
= (x + 1)n (x − 1)n
N. B. Vyas Legendre’s Function
82. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn y dn
∴ Pn (x) = c = c n (x2 − 1)n ——(3)
dxn dx
Now y = (x2 − 1)n
= (x + 1)n (x − 1)n
dn y dn
∴ = (x + 1)n n ((x − 1)n )
dxn dx
N. B. Vyas Legendre’s Function
83. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn y dn
∴ Pn (x) = c = c n (x2 − 1)n ——(3)
dxn dx
Now y = (x2 − 1)n
= (x + 1)n (x − 1)n
dn y dn
∴ = (x + 1)n n ((x − 1)n )
dxn dx
dn−1
+n(x + 1)n−1 n−1 ((x − 1)n )
dx
N. B. Vyas Legendre’s Function
84. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn y dn
∴ Pn (x) = c = c n (x2 − 1)n ——(3)
dxn dx
Now y = (x2 − 1)n
= (x + 1)n (x − 1)n
dn y dn
∴ = (x + 1)n n ((x − 1)n )
dxn dx
dn−1
+n(x + 1)n−1 n−1 ((x − 1)n )
dx
+...
N. B. Vyas Legendre’s Function
85. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
dn y dn
∴ Pn (x) = c = c n (x2 − 1)n ——(3)
dxn dx
Now y = (x2 − 1)n
= (x + 1)n (x − 1)n
dn y dn
∴ = (x + 1)n n ((x − 1)n )
dxn dx
dn−1
+n(x + 1)n−1 n−1 ((x − 1)n )
dx
+...
dn ((x + 1)n )
+ (x − 1)n
dxn
N. B. Vyas Legendre’s Function
86. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
Recurrence Relations for Pn(x) : −
N. B. Vyas Legendre’s Function
87. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
N. B. Vyas Legendre’s Function
88. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
Proof: We have
N. B. Vyas Legendre’s Function
89. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
Proof: We have
∞ 1
−
n 2 ) 2 ——–(i)
Pn (x)t = (1 − 2xt + t
n=0
N. B. Vyas Legendre’s Function
90. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
Proof: We have
∞ 1
−
n 2 ) 2 ——–(i)
Pn (x)t = (1 − 2xt + t
n=0
Differentiating equation (i) partially with respect to t, we
get
N. B. Vyas Legendre’s Function
91. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
Proof: We have
∞ 1
−
n 2 ) 2 ——–(i)
Pn (x)t = (1 − 2xt + t
n=0
Differentiating equation (i) partially with respect to t, we
get
∞
1 3
nPn (x)tn−1 = − (1 − 2xt + t2 )− 2 (−2x + 2t)
2
n=1
N. B. Vyas Legendre’s Function
92. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
Proof: We have
∞ 1
−
n 2 ) 2 ——–(i)
Pn (x)t = (1 − 2xt + t
n=0
Differentiating equation (i) partially with respect to t, we
get
∞
1 3
nPn (x)tn−1 = − (1 − 2xt + t2 )− 2 (−2x + 2t)
2
n=1
1
= (1 − 2xt + t2 )−1 (1 − 2xt + t2 )− 2 (x − t)
N. B. Vyas Legendre’s Function
93. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
Proof: We have
∞ 1
−
n 2 ) 2 ——–(i)
Pn (x)t = (1 − 2xt + t
n=0
Differentiating equation (i) partially with respect to t, we
get
∞
1 3
nPn (x)tn−1 = − (1 − 2xt + t2 )− 2 (−2x + 2t)
2
n=1
1
= (1 − 2xt + t2 )−1 (1 − 2xt + t2 )− 2 (x − t)
1
(1 − 2xt + t2 )− 2
= (x − t)
(1 − 2xt + t2 )
N. B. Vyas Legendre’s Function
94. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
Proof: We have
∞ 1
−
n 2 ) 2 ——–(i)
Pn (x)t = (1 − 2xt + t
n=0
Differentiating equation (i) partially with respect to t, we
get
∞
1 3
nPn (x)tn−1 = − (1 − 2xt + t2 )− 2 (−2x + 2t)
2
n=1
1
= (1 − 2xt + t2 )−1 (1 − 2xt + t2 )− 2 (x − t)
1
(1 − 2xt + t2 )− 2
= (x − t)
(1 − 2xt + t2 )
from (i)
N. B. Vyas Legendre’s Function
95. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(1) (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
Proof: We have
∞ 1
−
n 2 ) 2 ——–(i)
Pn (x)t = (1 − 2xt + t
n=0
Differentiating equation (i) partially with respect to t, we
get
∞
1 3
nPn (x)tn−1 = − (1 − 2xt + t2 )− 2 (−2x + 2t)
2
n=1
1
= (1 − 2xt + t2 )−1 (1 − 2xt + t2 )− 2 (x − t)
1
(1 − 2xt + t2 )− 2
= (x − t)
(1 − 2xt + t2 )
from (i)
∞ ∞
n−1
(1 − 2xt + t2 ) nPn (x)t = (x − t) Pn (x)tn
n=1 n=0
N. B. Vyas Legendre’s Function
96. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞ ∞
∴ nPn (x)tn−1 − 2x nPn (x)tn + nPn (x)tn+1 =
n=1 n=1 n=1
∞ ∞
x Pn (x)tn − Pn (x)tn+1
n=0 n=0
N. B. Vyas Legendre’s Function
97. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞ ∞
∴ nPn (x)tn−1 − 2x nPn (x)tn + nPn (x)tn+1 =
n=1 n=1 n=1
∞ ∞
x Pn (x)tn − Pn (x)tn+1
n=0 n=0
replacing n by n+1 in 1st term, n by n-1 in 3rd term in
L.H.S.
N. B. Vyas Legendre’s Function
98. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞ ∞
∴ nPn (x)tn−1 − 2x nPn (x)tn + nPn (x)tn+1 =
n=1 n=1 n=1
∞ ∞
x Pn (x)tn − Pn (x)tn+1
n=0 n=0
replacing n by n+1 in 1st term, n by n-1 in 3rd term in
L.H.S.
replacing n by n-1 in 2nd term in R.H.S
N. B. Vyas Legendre’s Function
99. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞ ∞
∴ nPn (x)tn−1 − 2x nPn (x)tn + nPn (x)tn+1 =
n=1 n=1 n=1
∞ ∞
x Pn (x)tn − Pn (x)tn+1
n=0 n=0
replacing n by n+1 in 1st term, n by n-1 in 3rd term in
L.H.S.
replacing n by n-1 in 2nd term in R.H.S
∞ ∞ ∞
(n + 1)Pn+1 (x)tn − 2x nPn (x)tn + (n −
n=0 n=1 n=2
∞ ∞
1)Pn−1 (x)tn = x Pn (x)tn − Pn−1 (x)tn
n=0 n=1
N. B. Vyas Legendre’s Function
100. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞ ∞
∴ nPn (x)tn−1 − 2x nPn (x)tn + nPn (x)tn+1 =
n=1 n=1 n=1
∞ ∞
x Pn (x)tn − Pn (x)tn+1
n=0 n=0
replacing n by n+1 in 1st term, n by n-1 in 3rd term in
L.H.S.
replacing n by n-1 in 2nd term in R.H.S
∞ ∞ ∞
(n + 1)Pn+1 (x)tn − 2x nPn (x)tn + (n −
n=0 n=1 n=2
∞ ∞
1)Pn−1 (x)tn = x Pn (x)tn − Pn−1 (x)tn
n=0 n=1
comparing the coefficients of tn on both the sides
N. B. Vyas Legendre’s Function
101. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (n + 1)Pn+1 (x) − 2xnPn (x) + (n − 1)Pn−1 (x) =
xPn (x) − Pn−1 (x)
N. B. Vyas Legendre’s Function
102. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (n + 1)Pn+1 (x) − 2xnPn (x) + (n − 1)Pn−1 (x) =
xPn (x) − Pn−1 (x)
∴ (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − (n − 1 + 1)Pn−1 (x)
N. B. Vyas Legendre’s Function
103. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (n + 1)Pn+1 (x) − 2xnPn (x) + (n − 1)Pn−1 (x) =
xPn (x) − Pn−1 (x)
∴ (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − (n − 1 + 1)Pn−1 (x)
∴ (n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
N. B. Vyas Legendre’s Function
104. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
N. B. Vyas Legendre’s Function
105. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
Proof: We have
N. B. Vyas Legendre’s Function
106. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
Proof: We have
∞ 1
−
n
Pn (x)t = (1 − 2xt + t2 ) 2 ——–(i)
n=0
N. B. Vyas Legendre’s Function
107. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
Proof: We have
∞ 1
−
n
Pn (x)t = (1 − 2xt + t2 ) 2 ——–(i)
n=0
Differentiating equation (i) partially with respect to x, we
get
N. B. Vyas Legendre’s Function
108. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
Proof: We have
∞ 1
−
n
Pn (x)t = (1 − 2xt + t2 ) 2 ——–(i)
n=0
Differentiating equation (i) partially with respect to x, we
get
∞
1 3
Pn (x)tn = − (1 − 2xt + t2 )− 2 (−2t)
2
n=0
N. B. Vyas Legendre’s Function
109. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
Proof: We have
∞ 1
−
n
Pn (x)t = (1 − 2xt + t2 ) 2 ——–(i)
n=0
Differentiating equation (i) partially with respect to x, we
get
∞
1 3
Pn (x)tn = − (1 − 2xt + t2 )− 2 (−2t)
2
n=0
∞ 1
n t(1 − 2xt + t2 )− 2
∴ Pn (x)t = ————-(ii)
(1 − 2xt + t2 )
n=0
N. B. Vyas Legendre’s Function
110. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
Proof: We have
∞ 1
−
n
Pn (x)t = (1 − 2xt + t2 ) 2 ——–(i)
n=0
Differentiating equation (i) partially with respect to x, we
get
∞
1 3
Pn (x)tn = − (1 − 2xt + t2 )− 2 (−2t)
2
n=0
∞ 1
t(1 − 2xt + t2 )− 2
n
∴ Pn (x)t = ————-(ii)
(1 − 2xt + t2 )
n=0
⇒ Differentiating equation (i) partially with respect to t, we
get
N. B. Vyas Legendre’s Function
111. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
Proof: We have
∞ 1
−
n
Pn (x)t = (1 − 2xt + t2 ) 2 ——–(i)
n=0
Differentiating equation (i) partially with respect to x, we
get
∞
1 3
Pn (x)tn = − (1 − 2xt + t2 )− 2 (−2t)
2
n=0
∞ 1
t(1 − 2xt + t2 )− 2
n
∴ Pn (x)t = ————-(ii)
(1 − 2xt + t2 )
n=0
⇒ Differentiating equation (i) partially with respect to t, we
get
∞
1 3
nPn (x)tn−1 = − (1 − 2xt + t2 )− 2 (−2x + 2t)
2
n=1
N. B. Vyas Legendre’s Function
112. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(2) nPn (x) = xPn (x) − Pn−1 (x)
Proof: We have
∞ 1
−
n
Pn (x)t = (1 − 2xt + t2 ) 2 ——–(i)
n=0
Differentiating equation (i) partially with respect to x, we
get
∞
1 3
Pn (x)tn = − (1 − 2xt + t2 )− 2 (−2t)
2
n=0
∞ 1
t(1 − 2xt + t2 )− 2
n
∴ Pn (x)t = ————-(ii)
(1 − 2xt + t2 )
n=0
⇒ Differentiating equation (i) partially with respect to t, we
get
∞
1 3
nPn (x)tn−1 = − (1 − 2xt + t2 )− 2 (−2x + 2t)
2
n=1
∞ 1
n−1 (x − t)(1 −Legendre’s )− 2
N. B. Vyas
2xt + t2 Function
113. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞
n−1 (x − t)
nPn (x)t = Pn (x)tn {by eq. (ii)
t
n=1 n=0
N. B. Vyas Legendre’s Function
114. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞
n−1 (x − t)
nPn (x)t = Pn (x)tn {by eq. (ii)
t
n=1 n=0
∞ ∞
∴ t nPn (x)tn−1 = (x − t) Pn (x)tn
n=1 n=0
N. B. Vyas Legendre’s Function
115. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞
n−1 (x − t)
nPn (x)t = Pn (x)tn {by eq. (ii)
t
n=1 n=0
∞ ∞
∴ t nPn (x)tn−1 = (x − t) Pn (x)tn
n=1 n=0
∞ ∞ ∞
∴ nPn (x)tn = x Pn (x)tn − Pn (x)tn+1
n=1 n=0 n=0
N. B. Vyas Legendre’s Function
116. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞
n−1 (x − t)
nPn (x)t = Pn (x)tn {by eq. (ii)
t
n=1 n=0
∞ ∞
∴ t nPn (x)tn−1 = (x − t) Pn (x)tn
n=1 n=0
∞ ∞ ∞
∴ nPn (x)tn = x Pn (x)tn − Pn (x)tn+1
n=1 n=0 n=0
Replacing n by n-1 in 2nd term in R.H.S.
N. B. Vyas Legendre’s Function
117. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞
n−1 (x − t)
nPn (x)t = Pn (x)tn {by eq. (ii)
t
n=1 n=0
∞ ∞
∴ t nPn (x)tn−1 = (x − t) Pn (x)tn
n=1 n=0
∞ ∞ ∞
∴ nPn (x)tn = x Pn (x)tn − Pn (x)tn+1
n=1 n=0 n=0
Replacing n by n-1 in 2nd term in R.H.S.
∞ ∞ ∞
n n
∴ nPn (x)t = x Pn (x)t − Pn−1 (x)tn
n=1 n=0 n=1
N. B. Vyas Legendre’s Function
118. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞
n−1 (x − t)
nPn (x)t = Pn (x)tn {by eq. (ii)
t
n=1 n=0
∞ ∞
∴ t nPn (x)tn−1 = (x − t) Pn (x)tn
n=1 n=0
∞ ∞ ∞
∴ nPn (x)tn = x Pn (x)tn − Pn (x)tn+1
n=1 n=0 n=0
Replacing n by n-1 in 2nd term in R.H.S.
∞ ∞ ∞
n n
∴ nPn (x)t = x Pn (x)t − Pn−1 (x)tn
n=1 n=0 n=1
comparing the coefficients of tn on both sides, we get
N. B. Vyas Legendre’s Function
119. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∞ ∞
n−1 (x − t)
nPn (x)t = Pn (x)tn {by eq. (ii)
t
n=1 n=0
∞ ∞
∴ t nPn (x)tn−1 = (x − t) Pn (x)tn
n=1 n=0
∞ ∞ ∞
∴ nPn (x)tn = x Pn (x)tn − Pn (x)tn+1
n=1 n=0 n=0
Replacing n by n-1 in 2nd term in R.H.S.
∞ ∞ ∞
n n
∴ nPn (x)t = x Pn (x)t − Pn−1 (x)tn
n=1 n=0 n=1
comparing the coefficients of tn on both sides, we get
nPn (x) = xPn (x) − Pn−1 (x)
N. B. Vyas Legendre’s Function
120. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
N. B. Vyas Legendre’s Function
121. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
N. B. Vyas Legendre’s Function
122. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
∴ (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x) — (a)
N. B. Vyas Legendre’s Function
123. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
∴ (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x) — (a)
differentiating equation (a) partially with respect to x, We
get
N. B. Vyas Legendre’s Function
124. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
∴ (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x) — (a)
differentiating equation (a) partially with respect to x, We
get
∴ (2n + 1)Pn (x) + (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x)
—(b)
N. B. Vyas Legendre’s Function
125. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
∴ (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x) — (a)
differentiating equation (a) partially with respect to x, We
get
∴ (2n + 1)Pn (x) + (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x)
—(b)
Also from relation (2)
N. B. Vyas Legendre’s Function
126. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
∴ (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x) — (a)
differentiating equation (a) partially with respect to x, We
get
∴ (2n + 1)Pn (x) + (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x)
—(b)
Also from relation (2)
nPn (x) = xPn (x) − Pn−1 (x)
N. B. Vyas Legendre’s Function
127. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
∴ (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x) — (a)
differentiating equation (a) partially with respect to x, We
get
∴ (2n + 1)Pn (x) + (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x)
—(b)
Also from relation (2)
nPn (x) = xPn (x) − Pn−1 (x)
∴ xPn (x) = nPn (x) + Pn−1 (x)—– (c)
N. B. Vyas Legendre’s Function
128. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
∴ (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x) — (a)
differentiating equation (a) partially with respect to x, We
get
∴ (2n + 1)Pn (x) + (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x)
—(b)
Also from relation (2)
nPn (x) = xPn (x) − Pn−1 (x)
∴ xPn (x) = nPn (x) + Pn−1 (x)—– (c)
Substituting the value of (c) in equation (b), we get
N. B. Vyas Legendre’s Function
129. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(3) (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)n
Proof: We have ( from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
∴ (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x) — (a)
differentiating equation (a) partially with respect to x, We
get
∴ (2n + 1)Pn (x) + (2n + 1)xPn (x) = (n + 1)Pn+1 (x) + nPn−1 (x)
—(b)
Also from relation (2)
nPn (x) = xPn (x) − Pn−1 (x)
∴ xPn (x) = nPn (x) + Pn−1 (x)—– (c)
Substituting the value of (c) in equation (b), we get
∴ (2n + 1)Pn (x) + (2n + 1)[nPn (x) + Pn−1 (x)] =
(n + 1)Pn+1 (x) + nPn−1 (x)
N. B. Vyas Legendre’s Function
130. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (2n + 1)(n + 1)Pn (x) + (2n + 1)Pn−1 (x) =
(n + 1)Pn+1 (x) + nPn−1 (x)
N. B. Vyas Legendre’s Function
131. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (2n + 1)(n + 1)Pn (x) + (2n + 1)Pn−1 (x) =
(n + 1)Pn+1 (x) + nPn−1 (x)
∴ (2n + 1)(n + 1)Pn (x) = (n + 1)Pn+1 (x) − (n + 1)Pn−1 (x)
N. B. Vyas Legendre’s Function
132. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (2n + 1)(n + 1)Pn (x) + (2n + 1)Pn−1 (x) =
(n + 1)Pn+1 (x) + nPn−1 (x)
∴ (2n + 1)(n + 1)Pn (x) = (n + 1)Pn+1 (x) − (n + 1)Pn−1 (x)
∴ dividing by (n + 1), we get
N. B. Vyas Legendre’s Function
133. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
∴ (2n + 1)(n + 1)Pn (x) + (2n + 1)Pn−1 (x) =
(n + 1)Pn+1 (x) + nPn−1 (x)
∴ (2n + 1)(n + 1)Pn (x) = (n + 1)Pn+1 (x) − (n + 1)Pn−1 (x)
∴ dividing by (n + 1), we get
∴ (2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x)
N. B. Vyas Legendre’s Function
134. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
N. B. Vyas Legendre’s Function
135. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
N. B. Vyas Legendre’s Function
136. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
(2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x) —–(a)
N. B. Vyas Legendre’s Function
137. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
(2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x) —–(a)
Also we have (from relation (2) ),
N. B. Vyas Legendre’s Function
138. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
(2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x) —–(a)
Also we have (from relation (2) ),
∴ nPn (x) = xPn (x) − Pn−1 (x) ——(b)
N. B. Vyas Legendre’s Function
139. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
(2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x) —–(a)
Also we have (from relation (2) ),
∴ nPn (x) = xPn (x) − Pn−1 (x) ——(b)
Taking (a) - (b), we get
N. B. Vyas Legendre’s Function
140. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
(2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x) —–(a)
Also we have (from relation (2) ),
∴ nPn (x) = xPn (x) − Pn−1 (x) ——(b)
Taking (a) - (b), we get
∴ (n + 1)Pn (x) = Pn+1 (x) − xPn (x)
N. B. Vyas Legendre’s Function
141. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
(2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x) —–(a)
Also we have (from relation (2) ),
∴ nPn (x) = xPn (x) − Pn−1 (x) ——(b)
Taking (a) - (b), we get
∴ (n + 1)Pn (x) = Pn+1 (x) − xPn (x)
replacing n by n − 1, we get
N. B. Vyas Legendre’s Function
142. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
(2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x) —–(a)
Also we have (from relation (2) ),
∴ nPn (x) = xPn (x) − Pn−1 (x) ——(b)
Taking (a) - (b), we get
∴ (n + 1)Pn (x) = Pn+1 (x) − xPn (x)
replacing n by n − 1, we get
∴ nPn−1 (x) = Pn (x) − xPn−1 (x)
N. B. Vyas Legendre’s Function
143. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(4) Pn (x) = xPn−1 (x) + nPn−1 (x)
Proof: We have (from relation (3) ),
(2n + 1)Pn (x) = Pn+1 (x) − Pn−1 (x) —–(a)
Also we have (from relation (2) ),
∴ nPn (x) = xPn (x) − Pn−1 (x) ——(b)
Taking (a) - (b), we get
∴ (n + 1)Pn (x) = Pn+1 (x) − xPn (x)
replacing n by n − 1, we get
∴ nPn−1 (x) = Pn (x) − xPn−1 (x)
∴ Pn (x) = xPn−1 (x) + nPn−1 (x)
N. B. Vyas Legendre’s Function
144. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
N. B. Vyas Legendre’s Function
145. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
N. B. Vyas Legendre’s Function
146. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
Pn (x) = xPn−1 (x) + nPn−1 (x) ——- (a)
N. B. Vyas Legendre’s Function
147. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
Pn (x) = xPn−1 (x) + nPn−1 (x) ——- (a)
also we have (from relation (2) )
N. B. Vyas Legendre’s Function
148. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
Pn (x) = xPn−1 (x) + nPn−1 (x) ——- (a)
also we have (from relation (2) )
nPn (x) = xPn (x) − Pn−1 (x)
N. B. Vyas Legendre’s Function
149. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
Pn (x) = xPn−1 (x) + nPn−1 (x) ——- (a)
also we have (from relation (2) )
nPn (x) = xPn (x) − Pn−1 (x)
xPn (x) = nPn (x) + Pn−1 (x) ——– (b)
N. B. Vyas Legendre’s Function
150. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
Pn (x) = xPn−1 (x) + nPn−1 (x) ——- (a)
also we have (from relation (2) )
nPn (x) = xPn (x) − Pn−1 (x)
xPn (x) = nPn (x) + Pn−1 (x) ——– (b)
taking (a) - x X (b), we get
N. B. Vyas Legendre’s Function
151. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
Pn (x) = xPn−1 (x) + nPn−1 (x) ——- (a)
also we have (from relation (2) )
nPn (x) = xPn (x) − Pn−1 (x)
xPn (x) = nPn (x) + Pn−1 (x) ——– (b)
taking (a) - x X (b), we get
(1 − x2 )Pn (x) = xPn−1 (x) + nPn−1 (x) − nxPn (x) − xPn−1 (x)
N. B. Vyas Legendre’s Function
152. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
Pn (x) = xPn−1 (x) + nPn−1 (x) ——- (a)
also we have (from relation (2) )
nPn (x) = xPn (x) − Pn−1 (x)
xPn (x) = nPn (x) + Pn−1 (x) ——– (b)
taking (a) - x X (b), we get
(1 − x2 )Pn (x) = xPn−1 (x) + nPn−1 (x) − nxPn (x) − xPn−1 (x)
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
N. B. Vyas Legendre’s Function
153. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(5) (1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
Proof: We have (from relation (4) )
Pn (x) = xPn−1 (x) + nPn−1 (x) ——- (a)
also we have (from relation (2) )
nPn (x) = xPn (x) − Pn−1 (x)
xPn (x) = nPn (x) + Pn−1 (x) ——– (b)
taking (a) - x X (b), we get
(1 − x2 )Pn (x) = xPn−1 (x) + nPn−1 (x) − nxPn (x) − xPn−1 (x)
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)]
N. B. Vyas Legendre’s Function
154. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
N. B. Vyas Legendre’s Function
155. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
N. B. Vyas Legendre’s Function
156. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
N. B. Vyas Legendre’s Function
157. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
also we have (from relation (1) )
N. B. Vyas Legendre’s Function
158. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
also we have (from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
N. B. Vyas Legendre’s Function
159. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
also we have (from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + nxPn (x) − nPn−1 (x)
N. B. Vyas Legendre’s Function
160. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
also we have (from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + nxPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + n[xPn (x) − Pn−1 (x)]
N. B. Vyas Legendre’s Function
161. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
also we have (from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + nxPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + n[xPn (x) − Pn−1 (x)]
n(Pn−1 (x) − xPn (x)) = (n + 1)(xPn (x) − Pn+1 (x))
N. B. Vyas Legendre’s Function
162. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
also we have (from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + nxPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + n[xPn (x) − Pn−1 (x)]
n(Pn−1 (x) − xPn (x)) = (n + 1)(xPn (x) − Pn+1 (x))
from equation (a),
N. B. Vyas Legendre’s Function
163. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
also we have (from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + nxPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + n[xPn (x) − Pn−1 (x)]
n(Pn−1 (x) − xPn (x)) = (n + 1)(xPn (x) − Pn+1 (x))
from equation (a),
(1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
N. B. Vyas Legendre’s Function
164. Legendre’s Polynomials
Examples of Legendre’s Polynomials
Generating Function for Pn (x)
Rodrigue’s Formula
Recurrence Relations for Pn (x)
(6) (1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
Proof: We have (from relation (5) )
(1 − x2 )Pn (x) = n[Pn−1 (x) − xPn (x)] ——(a)
also we have (from relation (1) )
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + nxPn (x) − nPn−1 (x)
(n + 1)Pn+1 (x) = (n + 1)xPn (x) + n[xPn (x) − Pn−1 (x)]
n(Pn−1 (x) − xPn (x)) = (n + 1)(xPn (x) − Pn+1 (x))
from equation (a),
(1 − x2 )Pn (x) = (n + 1)[xPn (x) − Pn+1 (x)]
N. B. Vyas Legendre’s Function