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# OpenMath Content Dictionary: orthpoly

Canonical URL:
http://www.openxm.org/...
CD File:
orthpoly.ocd
CD as XML Encoded OpenMath:
orthpoly.omcd
Defines:
jacobiG, legendreP, legendreQ
Date:
2002-08-11, 2003-11-30
Version:
0 (Revision 1)
Review Date:
2003-08-11
Status:
experimental
Uses CD:
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

Description:

The first Legendre function. This function is one of the two famous solutions of Legendre differential equation.

Binary
Commented Mathematical property (CMP):
legendreP(v;z) = hypergeo1.hypergeometric2F1(-v,v+1,1;(1-z)/2)
Formal Mathematical property (FMP):
$\mathrm{legendreP}\left(v,z\right)=\mathrm{hypergeometric2F1}\left(-v,v+1,1,\frac{1-z}{2}\right)$
Signatures:
sts

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## legendreQ

Description:

The second Legendre function. This function is the another one of the famous two solutions of Legendre differential equation.

Binary
Commented Mathematical property (CMP):
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)
Formal Mathematical property (FMP):
$\mathrm{legendreQ}\left(v,z\right)=\frac{\frac{\sqrt{\pi }\mathrm{gamma}\left(v+1\right)}{\mathrm{gamma}\left(v+\frac{3}{2}\right)}}{{\left(2z\right)}^{\left(v+1\right)}}\mathrm{hypergeometric2F1}\left(\frac{v+1}{2},\frac{v}{2}+1,v+\frac{3}{2},{\left(\frac{1}{z}\right)}^{2}\right)$
Signatures:
sts

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## jacobiG

Description:

The Jacobi polynomial.

4ary
Commented Mathematical property (CMP):
jacobiG(n,a,c;z) = hypergeometric2F1(-n,a+n,c,z) (c \not\in Z_{<=0})
Formal Mathematical property (FMP):
$-c\notin \mathbb{N}⇒\mathrm{jacobiG}\left(n,a,c,z\right)=\mathrm{hypergeometric2F1}\left(-n,a+n,c,z\right)$
Signatures:
sts

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