OpenMath Content Dictionary: hypergeo0

Canonical URL:

http://www.math.kobe-u.ac.jp/OCD/

CD File:

hypergeo0.ocd

CD as XML Encoded OpenMath:

hypergeo0.omcd

Defines:

beta, gamma, pochhammer

Date:

2002-11-29

Version:

0 (Revision 1)

Review Date:

2017-12-31

Status:

experimental

---

  Author: Yasushi Tamura

This CD defines some basic hypergeometric integrals and symbols necessary to define hypergeometric functions. These functions are described in the following books. (1) Handbook of Mathematical Functions, Abramowitz, Stegun (2) Higher transcendental functions. Vol. III. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (3) From Gauss to Painleve, Vieweg, Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, Masaaki Yoshida

---

gamma

Description:

Euler's gamma function

Commented Mathematical property (CMP):

gamma(z)=\int_0^{+\infty} t^{z-1} e^{-z} dt (Re(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="relation1" name="gt"/>
      <OMA><OMS cd="complex1" name="real"/>
        <OMV name="z"/>
      </OMA>
      <OMI> 0 </OMI>
    </OMA>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="hypergeo0" name="gamma"/>
        <OMV name="z"/>
      </OMA>
      <OMA><OMS cd="calculus1" name="defint"/>
        <OMA><OMS cd="interval1" name="interval"/>
          <OMI> 0 </OMI>
          <OMS cd="nums1" name="infinity"/>
        </OMA>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="t"/>
          </OMBVAR>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="t"/>
              <OMA><OMS cd="arith1" name="minus"/>
                <OMV name="z"/>
                <OMI> 1 </OMI>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMS cd="nums1" name="e"/>
              <OMA><OMS cd="arith1" name="unary_minus"/>
                <OMV name="z"/>
              </OMA>
            </OMA>
          </OMA>
        </OMBIND>
      </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="relation1">gt</csymbol>
   <apply><csymbol cd="complex1">real</csymbol><ci>z</ci></apply>
   <cn type="integer">0</cn>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="hypergeo0">gamma</csymbol><ci>z</ci></apply>
   <apply><csymbol cd="calculus1">defint</csymbol>
    <apply><csymbol cd="interval1">interval</csymbol>
     <cn type="integer">0</cn>
     <csymbol cd="nums1">infinity</csymbol>
    </apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>t</ci></bvar>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>t</ci>
       <apply><csymbol cd="arith1">minus</csymbol><ci>z</ci><cn type="integer">1</cn></apply>
      </apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <csymbol cd="nums1">e</csymbol>
       <apply><csymbol cd="arith1">unary_minus</csymbol><ci>z</ci></apply>
      </apply>
     </apply>
    </bind>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (gt (real ( z) , 0 ) , eq (gamma ( z) , defint (interval ( 0 , infinity) , lambda [ t ] . (times (power ( t, minus ( z, 1 ) ) , power (e, unary_minus ( z) ) ) ) ) ) )

Popcorn

complex1.real($z) > 0 ==> hypergeo0.gamma($z) = calculus1.defint(interval1.interval(0, nums1.infinity), fns1.lambda[$t -> $t ^ ($z - 1) * nums1.e ^ -($z)])

Rendered Presentation MathML

$$\mathrm{real}\left(z\right)>0\Rightarrow \mathrm{gamma}\left(z\right)=\underset{0}{\overset{\infty }{\int }}{t}^{(z-1)}{e}^{\left(-z\right)}dt$$

Example:

gamma(n) = (n-1)! (n \in N)

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="logic1" name="implies"/>
    <OMA><OMS cd="set1" name="in"/>
      <OMV name="n"/>
      <OMS cd="setname1" name="N"/>
    </OMA>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="hypergeo0" name="gamma"/>
        <OMV name="n"/>
      </OMA>
      <OMA><OMS cd="integer1" name="factorial"/>
        <OMA><OMS cd="arith1" name="minus"/>
          <OMV name="n"/>
          <OMI> 1 </OMI>
        </OMA>
      </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">in</csymbol><ci>n</ci><csymbol cd="setname1">N</csymbol></apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="hypergeo0">gamma</csymbol><ci>n</ci></apply>
   <apply><csymbol cd="integer1">factorial</csymbol>
    <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><cn type="integer">1</cn></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (in ( n, N) , eq (gamma ( n) , factorial (minus ( n, 1 ) ) ) )

Popcorn

set1.in($n, setname1.N) ==> hypergeo0.gamma($n) = integer1.factorial($n - 1)

Rendered Presentation MathML

$$n\in \mathbb{N}\Rightarrow \mathrm{gamma}\left(n\right)=(n-1)!$$

Signatures:

sts

---

[Next: beta] [Last: pochhammer] [Top]

---

beta

Description:

Euler's beta function

Commented Mathematical property (CMP):

beta(p,q)=\frac{gamma(p)gamma(q)}{gamma(p+q)}(p,q \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="logic1" name="and"/>
      <OMA><OMS cd="set1" name="notin"/>
        <OMA><OMS cd="arith1" name="unary_minus"/>
          <OMV name="p"/>
        </OMA>
        <OMS cd="setname1" name="N"/>
      </OMA>
      <OMA><OMS cd="set1" name="notin"/>
        <OMA><OMS cd="arith1" name="unary_minus"/>
          <OMV name="q"/>
        </OMA>
        <OMS cd="setname1" name="N"/>
      </OMA>
    </OMA>
    <OMA><OMS cd="arith1" name="divide"/>
      <OMA><OMS cd="arith1" name="times"/>
        <OMA><OMS cd="hypergeo0" name="gamma"/>
          <OMV name="p"/>
        </OMA>
        <OMA><OMS cd="hypergeo0" name="gamma"/>
          <OMV name="q"/>
        </OMA>
      </OMA>
      <OMA><OMS cd="hypergeo0" name="gamma"/>
        <OMA><OMS cd="arith1" name="plus"/>
          <OMV name="p"/>
          <OMV name="q"/>
        </OMA>
      </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="logic1">and</csymbol>
   <apply><csymbol cd="set1">notin</csymbol>
    <apply><csymbol cd="arith1">unary_minus</csymbol><ci>p</ci></apply>
    <csymbol cd="setname1">N</csymbol>
   </apply>
   <apply><csymbol cd="set1">notin</csymbol>
    <apply><csymbol cd="arith1">unary_minus</csymbol><ci>q</ci></apply>
    <csymbol cd="setname1">N</csymbol>
   </apply>
  </apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="hypergeo0">gamma</csymbol><ci>p</ci></apply>
    <apply><csymbol cd="hypergeo0">gamma</csymbol><ci>q</ci></apply>
   </apply>
   <apply><csymbol cd="hypergeo0">gamma</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol><ci>p</ci><ci>q</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (and (notin (unary_minus ( p) , N) , notin (unary_minus ( q) , N) ) , divide (times (gamma ( p) , gamma ( q) ) , gamma (plus ( p, q) ) ) )

Popcorn

set1.notin( -($p), setname1.N) and set1.notin( -($q), setname1.N) ==> (hypergeo0.gamma($p) * hypergeo0.gamma($q)) / hypergeo0.gamma($p + $q)

Rendered Presentation MathML

$$-p\notin \mathbb{N}\wedge -q\notin \mathbb{N}\Rightarrow \frac{\mathrm{gamma}\left(p\right)\mathrm{gamma}\left(q\right)}{\mathrm{gamma}\left(p+q\right)}$$

Example:

beta(p,q)=\int_0^1 t^{p-1} (1-t)^{q-1} dt (Re(p),Re(q)>0)

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="logic1" name="implies"/>
    <OMA><OMS cd="logic1" name="and"/>
      <OMA><OMS cd="relation1" name="gt"/>
        <OMA><OMS cd="complex1" name="real"/>
          <OMV name="p"/>
        </OMA>
        <OMI> 0 </OMI>
      </OMA>
      <OMA><OMS cd="relation1" name="gt"/>
        <OMA><OMS cd="complex1" name="real"/>
          <OMV name="q"/>
        </OMA>
        <OMI> 0 </OMI>
      </OMA>
    </OMA>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="hypergeo0" name="beta"/>
        <OMV name="p"/>
        <OMV name="q"/>
      </OMA>
      <OMA><OMS cd="calculus1" name="defint"/>
        <OMA><OMS cd="interval1" name="interval"/>
          <OMI> 0 </OMI>
          <OMI> 1 </OMI>
        </OMA>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="t"/>
          </OMBVAR>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="t"/>
              <OMA><OMS cd="arith1" name="minus"/>
                <OMV name="p"/>
                <OMI> 1 </OMI>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMA><OMS cd="arith1" name="minus"/>
                <OMI> 1 </OMI>
                <OMV name="t"/>
              </OMA>
              <OMA><OMS cd="arith1" name="minus"/>
                <OMV name="q"/>
                <OMI> 1 </OMI>
              </OMA>
            </OMA>
          </OMA>
        </OMBIND>
      </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="logic1">and</csymbol>
   <apply><csymbol cd="relation1">gt</csymbol>
    <apply><csymbol cd="complex1">real</csymbol><ci>p</ci></apply>
    <cn type="integer">0</cn>
   </apply>
   <apply><csymbol cd="relation1">gt</csymbol>
    <apply><csymbol cd="complex1">real</csymbol><ci>q</ci></apply>
    <cn type="integer">0</cn>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="hypergeo0">beta</csymbol><ci>p</ci><ci>q</ci></apply>
   <apply><csymbol cd="calculus1">defint</csymbol>
    <apply><csymbol cd="interval1">interval</csymbol>
     <cn type="integer">0</cn>
     <cn type="integer">1</cn>
    </apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>t</ci></bvar>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>t</ci>
       <apply><csymbol cd="arith1">minus</csymbol><ci>p</ci><cn type="integer">1</cn></apply>
      </apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <apply><csymbol cd="arith1">minus</csymbol><cn type="integer">1</cn><ci>t</ci></apply>
       <apply><csymbol cd="arith1">minus</csymbol><ci>q</ci><cn type="integer">1</cn></apply>
      </apply>
     </apply>
    </bind>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (and (gt (real ( p) , 0 ) , gt (real ( q) , 0 ) ) , eq (beta ( p, q) , defint (interval ( 0 , 1 ) , lambda [ t ] . (times (power ( t, minus ( p, 1 ) ) , power (minus ( 1 , t) , minus ( q, 1 ) ) ) ) ) ) )

Popcorn

complex1.real($p) > 0 and complex1.real($q) > 0 ==> hypergeo0.beta($p, $q) = calculus1.defint(interval1.interval(0, 1), fns1.lambda[$t -> $t ^ ($p - 1) * (1 - $t) ^ ($q - 1)])

Rendered Presentation MathML

$$\mathrm{real}\left(p\right)>0\wedge \mathrm{real}\left(q\right)>0\Rightarrow \mathrm{beta}(p,q)=\underset{0}{\overset{1}{\int }}{t}^{(p-1)}{(1-t)}^{(q-1)}dt$$

Signatures:

sts

---

[Next: pochhammer] [Previous: gamma] [Top]

---

pochhammer

Description:

Pochhammer symbol

Commented Mathematical property (CMP):

pochhammer(a,n) = gamma(a+n)/gamma(a)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo0" name="pochhammer"/>
      <OMV name="alpha"/>
      <OMV name="n"/>
    </OMA>
    <OMA><OMS cd="arith1" name="divide"/>
      <OMA><OMS cd="hypergeo0" name="gamma"/>
        <OMA><OMS cd="arith1" name="plus"/>
          <OMV name="alpha"/>
          <OMV name="n"/>
        </OMA>
      </OMA>
      <OMA><OMS cd="hypergeo0" name="gamma"/>
        <OMV name="alpha"/>
      </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="hypergeo0">pochhammer</csymbol><ci>alpha</ci><ci>n</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="hypergeo0">gamma</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol><ci>alpha</ci><ci>n</ci></apply>
   </apply>
   <apply><csymbol cd="hypergeo0">gamma</csymbol><ci>alpha</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (pochhammer ( alpha, n) , divide (gamma (plus ( alpha, n) ) , gamma ( alpha) ) )

Popcorn

hypergeo0.pochhammer($alpha, $n) = hypergeo0.gamma($alpha + $n) / hypergeo0.gamma($alpha)

Rendered Presentation MathML

$$\mathrm{pochhammer}(\mathrm{alpha},n)=\frac{\mathrm{gamma}\left(\mathrm{alpha}+n\right)}{\mathrm{gamma}\left(\mathrm{alpha}\right)}$$

Example:

pochhammer(a,n) = \prod_0^{n-1} (a+i)

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo0" name="pochhammer"/>
      <OMV name="a"/>
      <OMV name="n"/>
    </OMA>
    <OMA><OMS cd="arith1" name="product"/>
      <OMA><OMS cd="interval1" name="integer_interval"/>
        <OMI> 0 </OMI>
        <OMA><OMS cd="arith1" name="minus"/>
          <OMV name="n"/>
          <OMI> 1 </OMI>
        </OMA>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="i"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="plus"/>
          <OMV name="a"/>
          <OMV name="i"/>
        </OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="hypergeo0">pochhammer</csymbol><ci>a</ci><ci>n</ci></apply>
  <apply><csymbol cd="arith1">product</csymbol>
   <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn type="integer">0</cn>
    <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><cn type="integer">1</cn></apply>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>i</ci></bvar>
    <apply><csymbol cd="arith1">plus</csymbol><ci>a</ci><ci>i</ci></apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (pochhammer ( a, n) , product (integer_interval ( 0 , minus ( n, 1 ) ) , lambda [ i ] . (plus ( a, i) ) ) )

Popcorn

hypergeo0.pochhammer($a, $n) = arith1.product(interval1.integer_interval(0, $n - 1), fns1.lambda[$i -> $a + $i])

Rendered Presentation MathML

$$\mathrm{pochhammer}(a,n)=\prod _{i=0}^{n-1}a+i$$

Signatures:

sts

---

[First: gamma] [Previous: beta] [Top]