The recurrence relations of the Legendre polynomials describe how each polynomial in the sequence can be obtained from its predecessors.

Some useful recurrence relations of the Legendre polynomials include

where .

To derive these relations, we differentiate eq345 with respect to and to give

and

respectively.

Substituting eq345 in eq346f yields

Equating the coefficents of gives

Substituting eq346e in eq346f results in . When we equate the coefficients of in this equation, we get eq346b. Combining eq346b and eq346g gives eq346a. Substracting eq346b from eq346a yields eq346c.

The last recurrence relation is obtained by substituting eq345 in eq346e to give .Â Expanding and equating coefficients of , we have eq346d.