To derive: The Legendre polynomial of degree 6 and 7. The Legendre polynomial of degree 6 is P 6 ( x ) = 1 16 ( 231 x 6 − 315 x 4 + 105 x 2 − 5 ) _ and the Legendre polynomial of degree 7 is P 7 ( x ) = 1 16 ( 429 x 7 − 693 x 5 + 315 x 3 − 35 x ) _ . Result used: “The equation ( 11 ) is, P n ( x ) = ∑ m = 0 M ( − 1 ) m ( 2 n − 2 m ) ! 2 n m ! ( n − m ) ! ( n − 2 m ) ! x n − 2 m where M = n 2 or n − 1 2 whichever is an integer.” Calculation: Substitute n = 6 in equation (11) to obtain the Legendre polynomial of degree zero as shown below. P 6 ( x ) = { ( − 1 ) 0 ( 2 ( 6 ) − 2 ( 0 ) ) ! 2 6 ( 0 ) ! ( 6 − ( 0 ) ) ! ( 6 − 2 ( 0 ) ) ! x ( 6 ) − 2 ( 0 ) + ( − 1 ) 1 ( 2 ( 6 ) − 2 ( 1 ) ) ! 2 6 ( 1 ) ! ( 6 − ( 1 ) ) ! ( 6 − 2 ( 1 ) ) ! x ( 6 ) − 2 ( 1 ) + ( − 1 ) 2 ( 2 ( 6 ) − 2 ( 2 ) ) ! 2 6 ( 2 ) ! ( 6 − ( 2 ) ) ! ( 6 − 2 ( 2 ) ) ! x 6 − 2 ( 2 ) + ( − 1 ) 3 ( 2 ( 6 ) − 2 ( 3 ) ) ! 2 6 ( 3 ) ! ( 6 − ( 3 ) ) ! ( 6 − 2 ( 3 ) ) ! x 6 − 2 ( 3 ) } = { 12 ! 64 ( 6 ) ! ( 6 ) ! x 6 − 10 ! 64 ( 1 ) ! ( 5 ) ! ( 4 ) ! x 4 + 8 ! 64 ( 2 ) ! ( 4 ) ! ( 2 ) ! x 2 − 6 ! 64 ( 3 ) ! ( 3 ) ! ( 0 ) ! x 0 } = 231 16 x 6 − 315 16 x 4 + 105 16 x 2 − 5 16 = 1 16 ( 231 x 6 − 315 x 4 + 105 x 2 − 5 ) Thus, the Legendre polynomial of degree 6 is P 6 ( x ) = 1 16 ( 231 x 6 − 315 x 4 + 105 x 2 − 5 ) _ . Substitute n = 7 in equation (11) to obtain the Legendre polynomial of degree zero as shown below. P 7 ( x ) = { ( − 1 ) 0 ( 2 ( 7 ) − 2 ( 0 ) ) ! 2 7 ( 0 ) ! ( 7 − ( 0 ) ) ! ( 7 − 2 ( 0 ) ) ! x 7 − 2 ( 0 ) + ( − 1 ) 1 ( 2 ( 7 ) − 2 ( 1 ) ) ! 2 7 ( 1 ) ! ( 7 − ( 1 ) ) ! ( 7 − 2 ( 1 ) ) ! x ( 7 ) − 2 ( 1 ) + ( − 1 ) 2 ( 2 ( 7 ) − 2 ( 2 ) ) ! 2 7 ( 2 ) ! ( 7 − ( 2 ) ) ! ( 7 − 2 ( 2 ) ) ! x 7 − 2 ( 2 ) + ( − 1 ) 3 ( 2 ( 7 ) − 2 ( 3 ) ) ! 2 7 ( 3 ) ! ( 7 − ( 3 ) ) ! ( 7 − 2 ( 3 ) ) ! x 7 − 2 ( 3 ) } = { 14 ! 128 ( 7 ) ! ( 7 ) ! x 7 − 12 ! 128 ( 1 ) ! ( 6 ) ! ( 5 ) ! x 5 + 10 ! 128 ( 2 ) ! ( 5 ) ! ( 3 ) ! x 3 − 8 ! 128 ( 3 ) ! ( 4 ) ! ( 1 ) ! x 1 } = 429 16 x 7 − 693 16 x 5 + 315 16 x 3 − 35 16 x = 1 16 ( 429 x 7 − 693 x 5 + 315 x 3 − 35 x ) Thus, the Legendre polynomial of degree 7 is P 7 ( x ) = 1 16 ( 429 x 7 − 693 x 5 + 315 x 3 − 35 x ) _ .

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