orthpoly http://www.openxm.org/... 2003-08-11 2002-08-11, 2003-11-30 0 1 experimental arith1 relation1 calculus1 alg1 interval1 nums1 hypergeo0 hypergeo1 This CD defines orthogonal polynomials which are hypergeometric polynomials. These functions are described in the following books. (1) Handbook of Mathematical Functions, Abramowitz, Stegun (2) Higher transcendental functions. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. legendreP The first Legendre function. This function is one of the two famous solutions of Legendre differential equation. Binary legendreP(v;z) = hypergeo1.hypergeometric2F1(-v,v+1,1;(1-z)/2) orthpoly1.legendreP(v,z) ~relation1.eq~ hypergeo1.hypergeometric2F1(arith1.unary_minus(v), v ~arith1.plus~ 1, 1, 1 ~arith1.minus~ z ~arith1.divide~ 2); legendreQ The second Legendre function. This function is the another one of the famous two solutions of Legendre differential equation. Binary legendreQ(v;z) = \frac{\sqrt{\pi}\Gamma(v+1)}{\Gamma(v+3/2)} /(2z)^{v+1} hypergeo1.hypergeometric2F1((v+1)/2,v/2+1,v+3/2;1/z^2) orthpoly1.legendreQ(v,z) ~relation1.eq~ ( arith1.root(nums1.pi,2) ~arith1.times~ hypergeo0.gamma(v ~arith1.plus~ 1) ~arith1.divide~ hypergeo0.gamma(v ~arith1.plus~ (3 ~arith1.divide~ 2)) ~arith1.divide~ (2 ~arith1.times~ z ~arith1.power~ (v ~arith1.plus~ 1)) ~arith1.times~ hypergeo1.hypergeometric2F1(v ~arith1.plus~ 1 ~arith1.divide~ 2, v ~arith1.divide~ 2 ~arith1.plus~ 1, v ~arith1.plus~ (3 ~arith1.divide~ 2), 1 ~arith1.divide~ z ~arith1.power~ 2)); jacobiG The Jacobi polynomial. 4ary jacobiG(n,a,c;z) = hypergeometric2F1(-n,a+n,c,z) (c \not\in Z_{<=0}) arith1.unary_minus(c) ~set1.notin~ setname1.N ~logic1.implies~ (orthpoly1.jacobiG(n,a,c,z) ~relation1.eq~ hypergeo1.hypergeometric2F1(arith1.unary_minus(n), a ~arith1.plus~ n, c, z));