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:

2003-11-30

Version:

0 (Revision 3)

Review Date:

2017-12-31

Status:

experimental

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  Author: Yasushi Tamura

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.

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legendreP

Description:

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

Commented Mathematical property (CMP):

legendreP(v;z) = hypergeo1.hypergeometric2F1(-v,v+1,1;(1-z)/2)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="orthpoly1" name="legendreP"/>
      <OMV name="v"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="hypergeo1" name="hypergeometric2F1"/>
      <OMA><OMS cd="arith1" name="unary_minus"/>
        <OMV name="v"/>
      </OMA>
      <OMA><OMS cd="arith1" name="plus"/>
        <OMV name="v"/>
        <OMI> 1 </OMI>
      </OMA>
      <OMI> 1 </OMI>
      <OMA><OMS cd="arith1" name="divide"/>
        <OMA><OMS cd="arith1" name="minus"/>
          <OMI> 1 </OMI>
          <OMV name="z"/>
        </OMA>
        <OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="orthpoly1">legendreP</csymbol><ci>v</ci><ci>z</ci></apply>
  <apply><csymbol cd="hypergeo1">hypergeometric2F1</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><ci>v</ci></apply>
   <apply><csymbol cd="arith1">plus</csymbol><ci>v</ci><cn type="integer">1</cn></apply>
   <cn type="integer">1</cn>
   <apply><csymbol cd="arith1">divide</csymbol>
    <apply><csymbol cd="arith1">minus</csymbol><cn type="integer">1</cn><ci>z</ci></apply>
    <cn type="integer">2</cn>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (legendreP ( v, z) , hypergeometric2F1 (unary_minus ( v) , plus ( v, 1 ) , 1 , divide (minus ( 1 , z) , 2 ) ) )

Popcorn

orthpoly1.legendreP($v, $z) = hypergeo1.hypergeometric2F1( -($v), $v + 1, 1, (1 - $z) / 2)

Rendered Presentation MathML

$$\mathrm{legendreP}(v,z)=\mathrm{hypergeometric2F1}(-v,v+1,1,\frac{1-z}{2})$$

Signatures:

sts

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[Next: legendreQ] [Last: jacobiG] [Top]

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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.

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):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="orthpoly1" name="legendreQ"/>
      <OMV name="v"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="times"/>
      <OMA><OMS cd="arith1" name="divide"/>
        <OMA><OMS cd="arith1" name="divide"/>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="root"/>
              <OMS cd="nums1" name="pi"/>
              <OMI> 2 </OMI>
            </OMA>
            <OMA><OMS cd="hypergeo0" name="gamma"/>
              <OMA><OMS cd="arith1" name="plus"/>
                <OMV name="v"/>
                <OMI> 1 </OMI>
              </OMA>
            </OMA>
          </OMA>
          <OMA><OMS cd="hypergeo0" name="gamma"/>
            <OMA><OMS cd="arith1" name="plus"/>
              <OMV name="v"/>
              <OMA><OMS cd="arith1" name="divide"/>
                <OMI> 3 </OMI>
                <OMI> 2 </OMI>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
        <OMA><OMS cd="arith1" name="power"/>
          <OMA><OMS cd="arith1" name="times"/>
            <OMI> 2 </OMI>
            <OMV name="z"/>
          </OMA>
          <OMA><OMS cd="arith1" name="plus"/>
            <OMV name="v"/>
            <OMI> 1 </OMI>
          </OMA>
        </OMA>
      </OMA>
      <OMA><OMS cd="hypergeo1" name="hypergeometric2F1"/>
        <OMA><OMS cd="arith1" name="divide"/>
          <OMA><OMS cd="arith1" name="plus"/>
            <OMV name="v"/>
            <OMI> 1 </OMI>
          </OMA>
          <OMI> 2 </OMI>
        </OMA>
        <OMA><OMS cd="arith1" name="plus"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMV name="v"/>
            <OMI> 2 </OMI>
          </OMA>
          <OMI> 1 </OMI>
        </OMA>
        <OMA><OMS cd="arith1" name="plus"/>
          <OMV name="v"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMI> 3 </OMI>
            <OMI> 2 </OMI>
          </OMA>
        </OMA>
        <OMA><OMS cd="arith1" name="power"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMI> 1 </OMI>
            <OMV name="z"/>
          </OMA>
          <OMI> 2 </OMI>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="orthpoly1">legendreQ</csymbol><ci>v</ci><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">divide</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">root</csymbol>
       <csymbol cd="nums1">pi</csymbol>
       <cn type="integer">2</cn>
      </apply>
      <apply><csymbol cd="hypergeo0">gamma</csymbol>
       <apply><csymbol cd="arith1">plus</csymbol><ci>v</ci><cn type="integer">1</cn></apply>
      </apply>
     </apply>
     <apply><csymbol cd="hypergeo0">gamma</csymbol>
      <apply><csymbol cd="arith1">plus</csymbol>
       <ci>v</ci>
       <apply><csymbol cd="arith1">divide</csymbol>
        <cn type="integer">3</cn>
        <cn type="integer">2</cn>
       </apply>
      </apply>
     </apply>
    </apply>
    <apply><csymbol cd="arith1">power</csymbol>
     <apply><csymbol cd="arith1">times</csymbol><cn type="integer">2</cn><ci>z</ci></apply>
     <apply><csymbol cd="arith1">plus</csymbol><ci>v</ci><cn type="integer">1</cn></apply>
    </apply>
   </apply>
   <apply><csymbol cd="hypergeo1">hypergeometric2F1</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol>
     <apply><csymbol cd="arith1">plus</csymbol><ci>v</ci><cn type="integer">1</cn></apply>
     <cn type="integer">2</cn>
    </apply>
    <apply><csymbol cd="arith1">plus</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol><ci>v</ci><cn type="integer">2</cn></apply>
     <cn type="integer">1</cn>
    </apply>
    <apply><csymbol cd="arith1">plus</csymbol>
     <ci>v</ci>
     <apply><csymbol cd="arith1">divide</csymbol>
      <cn type="integer">3</cn>
      <cn type="integer">2</cn>
     </apply>
    </apply>
    <apply><csymbol cd="arith1">power</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol><cn type="integer">1</cn><ci>z</ci></apply>
     <cn type="integer">2</cn>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (legendreQ ( v, z) , times (divide (divide (times (root (pi, 2 ) , gamma (plus ( v, 1 ) ) ) , gamma (plus ( v, divide ( 3 , 2 ) ) ) ) , power (times ( 2 , z) , plus ( v, 1 ) ) ) , hypergeometric2F1 (divide (plus ( v, 1 ) , 2 ) , plus (divide ( v, 2 ) , 1 ) , plus ( v, divide ( 3 , 2 ) ) , power (divide ( 1 , z) , 2 ) ) ) )

Popcorn

orthpoly1.legendreQ($v, $z) = (arith1.root(nums1.pi, 2) * hypergeo0.gamma($v + 1)) / hypergeo0.gamma($v + 3 / 2) / (2 * $z) ^ ($v + 1) * hypergeo1.hypergeometric2F1(($v + 1) / 2, $v / 2 + 1, $v + 3 / 2, (1 / $z) ^ 2)

Rendered Presentation MathML

$$\mathrm{legendreQ}(v,z)=\frac{\frac{\sqrt{\pi }\mathrm{gamma}\left(v+1\right)}{\mathrm{gamma}\left(v+\frac{3}{2}\right)}}{{(2z)}^{(v+1)}}\mathrm{hypergeometric2F1}(\frac{v+1}{2},\frac{v}{2}+1,v+\frac{3}{2},{\left(\frac{1}{z}\right)}^2)$$

Signatures:

sts

---

[Next: jacobiG] [Previous: legendreP] [Top]

---

jacobiG

Description:

The Jacobi polynomial.

Commented Mathematical property (CMP):

jacobiG(n,a,c;z) = hypergeometric2F1(-n,a+n,c,z) (c \not\in Z_{<=0})

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="logic1" name="implies"/>
    <OMA><OMS cd="set1" name="notin"/>
      <OMA><OMS cd="arith1" name="unary_minus"/>
        <OMV name="c"/>
      </OMA>
      <OMS cd="setname1" name="N"/>
    </OMA>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="orthpoly1" name="jacobiG"/>
        <OMV name="n"/>
        <OMV name="a"/>
        <OMV name="c"/>
        <OMV name="z"/>
      </OMA>
      <OMA><OMS cd="hypergeo1" name="hypergeometric2F1"/>
        <OMA><OMS cd="arith1" name="unary_minus"/>
          <OMV name="n"/>
        </OMA>
        <OMA><OMS cd="arith1" name="plus"/>
          <OMV name="a"/>
          <OMV name="n"/>
        </OMA>
        <OMV name="c"/>
        <OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">notin</csymbol>
   <apply><csymbol cd="arith1">unary_minus</csymbol><ci>c</ci></apply>
   <csymbol cd="setname1">N</csymbol>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="orthpoly1">jacobiG</csymbol>
    <ci>n</ci>
    <ci>a</ci>
    <ci>c</ci>
    <ci>z</ci>
   </apply>
   <apply><csymbol cd="hypergeo1">hypergeometric2F1</csymbol>
    <apply><csymbol cd="arith1">unary_minus</csymbol><ci>n</ci></apply>
    <apply><csymbol cd="arith1">plus</csymbol><ci>a</ci><ci>n</ci></apply>
    <ci>c</ci>
    <ci>z</ci>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (notin (unary_minus ( c) , N) , eq (jacobiG ( n, a, c, z) , hypergeometric2F1 (unary_minus ( n) , plus ( a, n) , c, z) ) )

Popcorn

set1.notin( -($c), setname1.N) ==> orthpoly1.jacobiG($n, $a, $c, $z) = hypergeo1.hypergeometric2F1( -($n), $a + $n, $c, $z)

Rendered Presentation MathML

$$-c\notin \mathbb{N}\Rightarrow \mathrm{jacobiG}(n,a,c,z)=\mathrm{hypergeometric2F1}(-n,a+n,c,z)$$

Signatures:

sts

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