Legendre function

where [P.sub.n] and 0 are the Legendre function and the scattering angle, respectively.

After you have entered the two mode numbers, there is a slight delay while the program computes the Legendre function (lines 90 through 220).

proposed a new fractional-order Legendre function with spectral method to solve partial FDEs; based on the operational matrix of these functions, the same authors developed their approach in combination with variational iteration formula to solve a class of FDEs; see [32].

and m = 0,1,..., n, where [P.sup.m.sub.n] and [Q.sup.m.sub.n](x) are the associated Legendre functions of the first and second kinds, respectively.

put forward the orthogonal fractional-order Legendre functions based on shifted Legendre polynomials.

A new modification of variational iteration method using fractional-order Legendre functions is proposed and successfully applied to find the approximate solution of nonlinear fractional differential equations.

Here, the special functions [P.sup.m.sub.n] are the associated Legendre functions, defined in [17] as

A synthesis of unequally spaced antenna arrays using Legendre functions is proposed in [19].

[z.sub.n](x) denotes the spherical Bessel functions: [j.sub.n](xt) = [square root of [pi]/2x][J.sub.n+1/2](x)/x, [h.sup.(1,2).sub.n](x) = [square root of [pi]/2x]H.sup.(1,2).sub.n+1/2](x) /x, [P.sup.m.sub.n](cos[theta]) are the Legendre functions. Superscript [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] means that [z.sub.n] (x) = [j.sub.n](x) (in [15] symbols [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are used for this).

where [Q.sub.0] and [Q.sub.2] are the Legendre functions linearly independent to the Legendre polynomials [P.sub.0] and [P.sub.2] respectively; [B.sub.0] and [B.sub.2] are constants given by