Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m: Where is the Kronecker delta. Also, they satisfy the orthogonality condition for fixed : Negative and/or negative The differential equation is clearly invariant under a change in sign of m. The functions for negative m were shown above to be proportional to those of positive m: (This followed from the Rodrigues' formula definition. This definition also makes the various recurrence formulas work for positive or negative .) The differential equation is also invariant under a change from to , and the functions for negative are defined by Parity From their definition, one can verify that the Associated Legendre functions are either even or odd according to The first few associated Legendre functions The first few associated Legendre functions, including those for negative values of m, are: Recurrence formula These functions have a number of recurrence properties: Helpful identities (initial values for the first recursion): with the double factorial. Gaunt's formula The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. For instance, this turns out to be necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the Coulomb operator are needed. For this we have Gaunt's formula This formula is to be used under the following assumptions: the degrees are non-negative integers all three orders are non-negative integers is the largest of the three orders the orders sum up the degrees obey Other quantities appearing in the formula are defined as The integral is zero unless the sum of degrees is even so that is an integer the triangular condition is satisfied Dong and Lemus (2002) generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials. Generalization via hypergeometric functions These functions may actually be defined for general complex parameters and argument: where is the gamma function and is the hypergeometric function They are called the Legendre functions when defined in this more general way. They satisfy the same differential equation as before: Since this is a second order differential equation, it has a second solution, , defined as: and both obey the various recurrence formulas given previously. Reparameterization in terms of angles These functions are most useful when