OpenMath Content Dictionary: hypergeon2

Canonical URL:

http://www.math.kobe-u.ac.jp/OCD/hypergeon2.tfb

CD File:

hypergeon2.ocd

CD as XML Encoded OpenMath:

hypergeon2.omcd

Defines:

appel_F1, appel_F2, appel_F3, appel_F4, lauricella_FA, lauricella_FC, lauricella_FD, multi_pochhammer

Date:

2003-11-30

Version:

1 (Revision 3)

Review Date:

2017-12-31

Status:

experimental

---

  Author: Nobuki Takayama

This CD defines symbols for classical hypergeometric series of several variables, which include Appell functions and Lauricella functions.

---

multi_pochhammer

Description:

multi_pochhammer is a product of pochhammer symbols.

Commented Mathematical property (CMP):

$ (a)_n = \prod_{i=1}^{m} (a_{i})_{n_{i}} $

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon2" name="multi_pochhammer"/>
      <OMV name="a"/>
      <OMV name="n"/>
    </OMA>
    <OMA><OMS cd="arith1" name="product"/>
      <OMA><OMS cd="interval1" name="integer_interval"/>
        <OMI> 1 </OMI>
        <OMA><OMS cd="linalg4" name="columnsize"/>
          <OMV name="n"/>
        </OMA>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="i"/>
        </OMBVAR>
        <OMA><OMS cd="hypergeo0" name="pochhammer"/>
          <OMA><OMS cd="linalg1" name="vector_selector"/>
            <OMV name="i"/>
            <OMV name="a"/>
          </OMA>
          <OMA><OMS cd="linalg1" name="vector_selector"/>
            <OMV name="i"/>
            <OMV name="n"/>
          </OMA>
        </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="hypergeon2">multi_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">1</cn>
    <apply><csymbol cd="linalg4">columnsize</csymbol><ci>n</ci></apply>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>i</ci></bvar>
    <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
     <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>a</ci></apply>
     <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>n</ci></apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (multi_pochhammer ( a, n) , product (integer_interval ( 1 , columnsize ( n) ) , lambda [ i ] . (pochhammer (vector_selector ( i, a) , vector_selector ( i, n) ) ) ) )

Popcorn

hypergeon2.multi_pochhammer($a, $n) = arith1.product(interval1.integer_interval(1, linalg4.columnsize($n)), fns1.lambda[$i -> hypergeo0.pochhammer(linalg1.vector_selector($i, $a), linalg1.vector_selector($i, $n))])

Rendered Presentation MathML

$$\mathrm{multi\_pochhammer}(a,n)=\prod _{i=1}^{\mathrm{columnsize}\left(n\right)}\mathrm{pochhammer}(a_i,n_i)$$

Signatures:

sts

---

[Next: appel_F1] [Last: lauricella_FD] [Top]

---

appel_F1

Description:

Appell's hypergeometric series F_1 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14

Commented Mathematical property (CMP):

$ F_1(a,b,b',c;x,y) = \sum_{m,n=0}^{\infty} \frac{(a)_{m+n} (b)_m (b')_n}{(c)_{m+n} (1)_{m} (1)_{n}}x^{m}y^{n}$

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon2" name="apple_F1"/>
      <OMV name="a"/>
      <OMV name="b1"/>
      <OMV name="b2"/>
      <OMV name="c"/>
      <OMV name="x"/>
      <OMV name="y"/>
    </OMA>
    <OMA><OMS cd="arith1" name="sum"/>
      <OMA><OMS cd="set1" name="cartesian_product"/>
        <OMS cd="setname1" name="N"/>
        <OMS cd="setname1" name="N"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="m"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="times"/>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMV name="a"/>
                  <OMA><OMS cd="arith1" name="plus"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 1 </OMI>
                      <OMV name="m"/>
                    </OMA>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 2 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMV name="b1"/>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="b2"/>
                <OMA><OMS cd="linalg1" name="vector_selector"/>
                  <OMI> 2 </OMI>
                  <OMV name="m"/>
                </OMA>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="times"/>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMV name="c"/>
                  <OMA><OMS cd="arith1" name="plus"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 1 </OMI>
                      <OMV name="m"/>
                    </OMA>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 2 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMI> 1 </OMI>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMI> 1 </OMI>
                <OMA><OMS cd="linalg1" name="vector_selector"/>
                  <OMI> 2 </OMI>
                  <OMV name="m"/>
                </OMA>
              </OMA>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="x"/>
              <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                <OMI> 1 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="y"/>
              <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                <OMI> 2 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
          </OMA>
        </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="hypergeon2">apple_F1</csymbol>
   <ci>a</ci>
   <ci>b1</ci>
   <ci>b2</ci>
   <ci>c</ci>
   <ci>x</ci>
   <ci>y</ci>
  </apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="set1">cartesian_product</csymbol>
    <csymbol cd="setname1">N</csymbol>
    <csymbol cd="setname1">N</csymbol>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>m</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">times</csymbol>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <ci>a</ci>
         <apply><csymbol cd="arith1">plus</csymbol>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
         </apply>
        </apply>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <ci>b1</ci>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <ci>b2</ci>
        <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
       </apply>
      </apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">times</csymbol>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <ci>c</ci>
         <apply><csymbol cd="arith1">plus</csymbol>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
         </apply>
        </apply>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <cn type="integer">1</cn>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <cn type="integer">1</cn>
        <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
       </apply>
      </apply>
     </apply>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>x</ci>
       <apply><csymbol cd="hypergeon2">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
      </apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>y</ci>
       <apply><csymbol cd="hypergeon2">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
      </apply>
     </apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (apple_F1 ( a, b1, b2, c, x, y) , sum (cartesian_product (N, N) , lambda [ m ] . (times (divide (times (times (pochhammer ( a, plus (vector_selector ( 1 , m) , vector_selector ( 2 , m) ) ) , pochhammer ( b1, vector_selector ( 1 , m) ) ) , pochhammer ( b2, vector_selector ( 2 , m) ) ) , times (times (pochhammer ( c, plus (vector_selector ( 1 , m) , vector_selector ( 2 , m) ) ) , pochhammer ( 1 , vector_selector ( 1 , m) ) ) , pochhammer ( 1 , vector_selector ( 2 , m) ) ) ) , times (power ( x, vector_selector ( 1 , m) ) , power ( y, vector_selector ( 2 , m) ) ) ) ) ) )

Popcorn

hypergeon2.apple_F1($a, $b1, $b2, $c, $x, $y) = arith1.sum(set1.cartesian_product(setname1.N, setname1.N), fns1.lambda[$m -> (hypergeo0.pochhammer($a, linalg1.vector_selector(1, $m) + linalg1.vector_selector(2, $m)) * hypergeo0.pochhammer($b1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer($b2, linalg1.vector_selector(2, $m))) / (hypergeo0.pochhammer($c, linalg1.vector_selector(1, $m) + linalg1.vector_selector(2, $m)) * hypergeo0.pochhammer(1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer(1, linalg1.vector_selector(2, $m))) * $x ^ hypergeon2.vector_selector(1, $m) * $y ^ hypergeon2.vector_selector(2, $m)])

Rendered Presentation MathML

$$\mathrm{apple\_F1}(a,b_1,b_2,c,x,y)=\sum _{min\mathbb{N}\times \mathbb{N}}\frac{\mathrm{pochhammer}(a,m_1+m_2)\mathrm{pochhammer}(b_1,m_1)\mathrm{pochhammer}(b_2,m_2)}{\mathrm{pochhammer}(c,m_1+m_2)\mathrm{pochhammer}(1,m_1)\mathrm{pochhammer}(1,m_2)}{x}^{\mathrm{vector\_selector}(1,m)}{y}^{\mathrm{vector\_selector}(2,m)}$$

Signatures:

sts

---

[Next: appel_F2] [Previous: multi_pochhammer] [Top]

---

appel_F2

Description:

Appell's hypergeometric series F_2 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14

Commented Mathematical property (CMP):

$ F_2(a,b,b',c,c';x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_{m}(b')_{n}} {(c)_{m} (c')_{n} (1)_{m} (1)_{n}}x^{m}y^{n} $

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon2" name="apple_F2"/>
      <OMV name="a"/>
      <OMV name="b1"/>
      <OMV name="b2"/>
      <OMV name="c1"/>
      <OMV name="c2"/>
      <OMV name="x"/>
      <OMV name="y"/>
    </OMA>
    <OMA><OMS cd="arith1" name="sum"/>
      <OMA><OMS cd="set1" name="cartesian_product"/>
        <OMS cd="setname1" name="N"/>
        <OMS cd="setname1" name="N"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="m"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="times"/>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMV name="a"/>
                  <OMA><OMS cd="arith1" name="plus"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 1 </OMI>
                      <OMV name="m"/>
                    </OMA>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 2 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMV name="b1"/>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="b2"/>
                <OMA><OMS cd="linalg1" name="vector_selector"/>
                  <OMI> 2 </OMI>
                  <OMV name="m"/>
                </OMA>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="times"/>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="c1"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 1 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="c2"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 2 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMI> 1 </OMI>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMI> 1 </OMI>
                <OMA><OMS cd="linalg1" name="vector_selector"/>
                  <OMI> 2 </OMI>
                  <OMV name="m"/>
                </OMA>
              </OMA>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="x"/>
              <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                <OMI> 1 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="y"/>
              <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                <OMI> 2 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
          </OMA>
        </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="hypergeon2">apple_F2</csymbol>
   <ci>a</ci>
   <ci>b1</ci>
   <ci>b2</ci>
   <ci>c1</ci>
   <ci>c2</ci>
   <ci>x</ci>
   <ci>y</ci>
  </apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="set1">cartesian_product</csymbol>
    <csymbol cd="setname1">N</csymbol>
    <csymbol cd="setname1">N</csymbol>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>m</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">times</csymbol>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <ci>a</ci>
         <apply><csymbol cd="arith1">plus</csymbol>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
         </apply>
        </apply>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <ci>b1</ci>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <ci>b2</ci>
        <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
       </apply>
      </apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">times</csymbol>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>c1</ci>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
         </apply>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>c2</ci>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
         </apply>
        </apply>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <cn type="integer">1</cn>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <cn type="integer">1</cn>
        <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
       </apply>
      </apply>
     </apply>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>x</ci>
       <apply><csymbol cd="hypergeon2">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
      </apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>y</ci>
       <apply><csymbol cd="hypergeon2">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
      </apply>
     </apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (apple_F2 ( a, b1, b2, c1, c2, x, y) , sum (cartesian_product (N, N) , lambda [ m ] . (times (divide (times (times (pochhammer ( a, plus (vector_selector ( 1 , m) , vector_selector ( 2 , m) ) ) , pochhammer ( b1, vector_selector ( 1 , m) ) ) , pochhammer ( b2, vector_selector ( 2 , m) ) ) , times (times (times (pochhammer ( c1, vector_selector ( 1 , m) ) , pochhammer ( c2, vector_selector ( 2 , m) ) ) , pochhammer ( 1 , vector_selector ( 1 , m) ) ) , pochhammer ( 1 , vector_selector ( 2 , m) ) ) ) , times (power ( x, vector_selector ( 1 , m) ) , power ( y, vector_selector ( 2 , m) ) ) ) ) ) )

Popcorn

hypergeon2.apple_F2($a, $b1, $b2, $c1, $c2, $x, $y) = arith1.sum(set1.cartesian_product(setname1.N, setname1.N), fns1.lambda[$m -> (hypergeo0.pochhammer($a, linalg1.vector_selector(1, $m) + linalg1.vector_selector(2, $m)) * hypergeo0.pochhammer($b1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer($b2, linalg1.vector_selector(2, $m))) / (hypergeo0.pochhammer($c1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer($c2, linalg1.vector_selector(2, $m)) * hypergeo0.pochhammer(1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer(1, linalg1.vector_selector(2, $m))) * $x ^ hypergeon2.vector_selector(1, $m) * $y ^ hypergeon2.vector_selector(2, $m)])

Rendered Presentation MathML

$$\mathrm{apple\_F2}(a,b_1,b_2,c_1,c_2,x,y)=\sum _{min\mathbb{N}\times \mathbb{N}}\frac{\mathrm{pochhammer}(a,m_1+m_2)\mathrm{pochhammer}(b_1,m_1)\mathrm{pochhammer}(b_2,m_2)}{\mathrm{pochhammer}(c_1,m_1)\mathrm{pochhammer}(c_2,m_2)\mathrm{pochhammer}(1,m_1)\mathrm{pochhammer}(1,m_2)}{x}^{\mathrm{vector\_selector}(1,m)}{y}^{\mathrm{vector\_selector}(2,m)}$$

Signatures:

sts

---

[Next: appel_F3] [Previous: appel_F1] [Top]

---

appel_F3

Description:

Appell's hypergeometric series F_3 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14

Commented Mathematical property (CMP):

$ F_3(a,a',b,b',c;x,y) = \sum_{m,n=0}^{\infty} \frac{(a)_{m}(a')_{n}(b)_{m}(b')_{n}} {(c)_{m+n}(1)_{m}(1)_{n}}x^{m}y^{n}$

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon2" name="apple_F3"/>
      <OMV name="a1"/>
      <OMV name="a2"/>
      <OMV name="b1"/>
      <OMV name="b2"/>
      <OMV name="c"/>
      <OMV name="x"/>
      <OMV name="y"/>
    </OMA>
    <OMA><OMS cd="arith1" name="sum"/>
      <OMA><OMS cd="set1" name="cartesian_product"/>
        <OMS cd="setname1" name="N"/>
        <OMS cd="setname1" name="N"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="m"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="times"/>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="a1"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 1 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="a2"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 1 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMV name="b1"/>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="b2"/>
                <OMA><OMS cd="linalg1" name="vector_selector"/>
                  <OMI> 2 </OMI>
                  <OMV name="m"/>
                </OMA>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="times"/>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMV name="c"/>
                  <OMA><OMS cd="arith1" name="plus"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 1 </OMI>
                      <OMV name="m"/>
                    </OMA>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 2 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMI> 1 </OMI>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMI> 1 </OMI>
                <OMA><OMS cd="linalg1" name="vector_selector"/>
                  <OMI> 2 </OMI>
                  <OMV name="m"/>
                </OMA>
              </OMA>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="x"/>
              <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                <OMI> 1 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="y"/>
              <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                <OMI> 2 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
          </OMA>
        </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="hypergeon2">apple_F3</csymbol>
   <ci>a1</ci>
   <ci>a2</ci>
   <ci>b1</ci>
   <ci>b2</ci>
   <ci>c</ci>
   <ci>x</ci>
   <ci>y</ci>
  </apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="set1">cartesian_product</csymbol>
    <csymbol cd="setname1">N</csymbol>
    <csymbol cd="setname1">N</csymbol>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>m</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">times</csymbol>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>a1</ci>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
         </apply>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>a2</ci>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
         </apply>
        </apply>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <ci>b1</ci>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <ci>b2</ci>
        <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
       </apply>
      </apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">times</csymbol>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <ci>c</ci>
         <apply><csymbol cd="arith1">plus</csymbol>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
         </apply>
        </apply>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <cn type="integer">1</cn>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <cn type="integer">1</cn>
        <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
       </apply>
      </apply>
     </apply>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>x</ci>
       <apply><csymbol cd="hypergeon2">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
      </apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>y</ci>
       <apply><csymbol cd="hypergeon2">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
      </apply>
     </apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (apple_F3 ( a1, a2, b1, b2, c, x, y) , sum (cartesian_product (N, N) , lambda [ m ] . (times (divide (times (times (times (pochhammer ( a1, vector_selector ( 1 , m) ) , pochhammer ( a2, vector_selector ( 1 , m) ) ) , pochhammer ( b1, vector_selector ( 1 , m) ) ) , pochhammer ( b2, vector_selector ( 2 , m) ) ) , times (times (pochhammer ( c, plus (vector_selector ( 1 , m) , vector_selector ( 2 , m) ) ) , pochhammer ( 1 , vector_selector ( 1 , m) ) ) , pochhammer ( 1 , vector_selector ( 2 , m) ) ) ) , times (power ( x, vector_selector ( 1 , m) ) , power ( y, vector_selector ( 2 , m) ) ) ) ) ) )

Popcorn

hypergeon2.apple_F3($a1, $a2, $b1, $b2, $c, $x, $y) = arith1.sum(set1.cartesian_product(setname1.N, setname1.N), fns1.lambda[$m -> (hypergeo0.pochhammer($a1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer($a2, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer($b1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer($b2, linalg1.vector_selector(2, $m))) / (hypergeo0.pochhammer($c, linalg1.vector_selector(1, $m) + linalg1.vector_selector(2, $m)) * hypergeo0.pochhammer(1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer(1, linalg1.vector_selector(2, $m))) * $x ^ hypergeon2.vector_selector(1, $m) * $y ^ hypergeon2.vector_selector(2, $m)])

Rendered Presentation MathML

$$\mathrm{apple\_F3}(a_1,a_2,b_1,b_2,c,x,y)=\sum _{min\mathbb{N}\times \mathbb{N}}\frac{\mathrm{pochhammer}(a_1,m_1)\mathrm{pochhammer}(a_2,m_1)\mathrm{pochhammer}(b_1,m_1)\mathrm{pochhammer}(b_2,m_2)}{\mathrm{pochhammer}(c,m_1+m_2)\mathrm{pochhammer}(1,m_1)\mathrm{pochhammer}(1,m_2)}{x}^{\mathrm{vector\_selector}(1,m)}{y}^{\mathrm{vector\_selector}(2,m)}$$

Signatures:

sts

---

[Next: appel_F4] [Previous: appel_F2] [Top]

---

appel_F4

Description:

Appell's hypergeometric series F_4 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14

Commented Mathematical property (CMP):

$ F_4(a,b,c,c';x,y) = \sum_{m,n=0}^{\infty} \frac{(a)_{m+n} (b)_{m+n}}{(c1)_{m}(c2)_{n}(1)_{m}(1)_{n}}x^{m}y^{n} $

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon2" name="apple_F4"/>
      <OMV name="a"/>
      <OMV name="b"/>
      <OMV name="c1"/>
      <OMV name="c2"/>
      <OMV name="x"/>
      <OMV name="y"/>
    </OMA>
    <OMA><OMS cd="arith1" name="sum"/>
      <OMA><OMS cd="set1" name="cartesian_product"/>
        <OMS cd="setname1" name="N"/>
        <OMS cd="setname1" name="N"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="m"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="a"/>
                <OMA><OMS cd="arith1" name="plus"/>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 2 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="b"/>
                <OMA><OMS cd="arith1" name="plus"/>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 2 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="times"/>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="c1"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 1 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="c2"/>
                    <OMA><OMS cd="linalg1" name="vector_selector"/>
                      <OMI> 2 </OMI>
                      <OMV name="m"/>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                  <OMI> 1 </OMI>
                  <OMA><OMS cd="linalg1" name="vector_selector"/>
                    <OMI> 1 </OMI>
                    <OMV name="m"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMI> 1 </OMI>
                <OMA><OMS cd="linalg1" name="vector_selector"/>
                  <OMI> 2 </OMI>
                  <OMV name="m"/>
                </OMA>
              </OMA>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="x"/>
              <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                <OMI> 1 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="y"/>
              <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                <OMI> 2 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
          </OMA>
        </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="hypergeon2">apple_F4</csymbol>
   <ci>a</ci>
   <ci>b</ci>
   <ci>c1</ci>
   <ci>c2</ci>
   <ci>x</ci>
   <ci>y</ci>
  </apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="set1">cartesian_product</csymbol>
    <csymbol cd="setname1">N</csymbol>
    <csymbol cd="setname1">N</csymbol>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>m</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <ci>a</ci>
        <apply><csymbol cd="arith1">plus</csymbol>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <ci>b</ci>
        <apply><csymbol cd="arith1">plus</csymbol>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
        </apply>
       </apply>
      </apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">times</csymbol>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>c1</ci>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
         </apply>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>c2</ci>
          <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
         </apply>
        </apply>
        <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
         <cn type="integer">1</cn>
         <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
        <cn type="integer">1</cn>
        <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
       </apply>
      </apply>
     </apply>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>x</ci>
       <apply><csymbol cd="hypergeon2">vector_selector</csymbol><cn type="integer">1</cn><ci>m</ci></apply>
      </apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>y</ci>
       <apply><csymbol cd="hypergeon2">vector_selector</csymbol><cn type="integer">2</cn><ci>m</ci></apply>
      </apply>
     </apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (apple_F4 ( a, b, c1, c2, x, y) , sum (cartesian_product (N, N) , lambda [ m ] . (times (divide (times (pochhammer ( a, plus (vector_selector ( 1 , m) , vector_selector ( 2 , m) ) ) , pochhammer ( b, plus (vector_selector ( 1 , m) , vector_selector ( 2 , m) ) ) ) , times (times (times (pochhammer ( c1, vector_selector ( 1 , m) ) , pochhammer ( c2, vector_selector ( 2 , m) ) ) , pochhammer ( 1 , vector_selector ( 1 , m) ) ) , pochhammer ( 1 , vector_selector ( 2 , m) ) ) ) , times (power ( x, vector_selector ( 1 , m) ) , power ( y, vector_selector ( 2 , m) ) ) ) ) ) )

Popcorn

hypergeon2.apple_F4($a, $b, $c1, $c2, $x, $y) = arith1.sum(set1.cartesian_product(setname1.N, setname1.N), fns1.lambda[$m -> (hypergeo0.pochhammer($a, linalg1.vector_selector(1, $m) + linalg1.vector_selector(2, $m)) * hypergeo0.pochhammer($b, linalg1.vector_selector(1, $m) + linalg1.vector_selector(2, $m))) / (hypergeo0.pochhammer($c1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer($c2, linalg1.vector_selector(2, $m)) * hypergeo0.pochhammer(1, linalg1.vector_selector(1, $m)) * hypergeo0.pochhammer(1, linalg1.vector_selector(2, $m))) * $x ^ hypergeon2.vector_selector(1, $m) * $y ^ hypergeon2.vector_selector(2, $m)])

Rendered Presentation MathML

$$\mathrm{apple\_F4}(a,b,c_1,c_2,x,y)=\sum _{min\mathbb{N}\times \mathbb{N}}\frac{\mathrm{pochhammer}(a,m_1+m_2)\mathrm{pochhammer}(b,m_1+m_2)}{\mathrm{pochhammer}(c_1,m_1)\mathrm{pochhammer}(c_2,m_2)\mathrm{pochhammer}(1,m_1)\mathrm{pochhammer}(1,m_2)}{x}^{\mathrm{vector\_selector}(1,m)}{y}^{\mathrm{vector\_selector}(2,m)}$$

Signatures:

sts

---

[Next: lauricella_FA] [Previous: appel_F3] [Top]

---

lauricella_FA

Description:

Lauricella's hypergeometric series F_A of n variables. In case of one variables, it agrees with the Appel function F_2. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages:

Commented Mathematical property (CMP):

$ F_A(a,b,c;x) = \sum_{k \in \N^n}^{\infty} \frac{(a)_{\sum k_i} \prod (b_i)_{k_i}} {\prod (c_i)_{k_i} \prod (1)_{k_i}} x^{k} $

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="hypergeon0" name="where"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="n"/>
      </OMBVAR>
      <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="hypergeon2" name="lauricella_FA"/>
          <OMV name="a"/>
          <OMV name="b"/>
          <OMV name="c"/>
          <OMV name="x"/>
        </OMA>
        <OMA><OMS cd="arith1" name="sum"/>
          <OMA><OMS cd="hypergeon0" name="cartesian_product_n"/>
            <OMS cd="setname1" name="N"/>
            <OMV name="n"/>
          </OMA>
          <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="k"/>
            </OMBVAR>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="divide"/>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="a"/>
                    <OMA><OMS cd="arith1" name="sum"/>
                      <OMA><OMS cd="interval1" name="integer_interval"/>
                        <OMI> 1 </OMI>
                        <OMV name="n"/>
                      </OMA>
                      <OMBIND>
                        <OMS cd="fns1" name="lambda"/>
                        <OMBVAR>
                          <OMV name="i"/>
                        </OMBVAR>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMBIND>
                    </OMA>
                  </OMA>
                  <OMA><OMS cd="arith1" name="prod"/>
                    <OMA><OMS cd="interval1" name="integer_interval"/>
                      <OMI> 1 </OMI>
                      <OMV name="n"/>
                    </OMA>
                    <OMBIND>
                      <OMS cd="fns1" name="lambda"/>
                      <OMBVAR>
                        <OMV name="i"/>
                      </OMBVAR>
                      <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="b"/>
                        </OMA>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMA>
                    </OMBIND>
                  </OMA>
                </OMA>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="arith1" name="prod"/>
                    <OMA><OMS cd="interval1" name="integer_interval"/>
                      <OMI> 1 </OMI>
                      <OMV name="n"/>
                    </OMA>
                    <OMBIND>
                      <OMS cd="fns1" name="lambda"/>
                      <OMBVAR>
                        <OMV name="i"/>
                      </OMBVAR>
                      <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="c"/>
                        </OMA>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMA>
                    </OMBIND>
                  </OMA>
                  <OMA><OMS cd="arith1" name="prod"/>
                    <OMA><OMS cd="interval1" name="integer_interval"/>
                      <OMI> 1 </OMI>
                      <OMV name="n"/>
                    </OMA>
                    <OMBIND>
                      <OMS cd="fns1" name="lambda"/>
                      <OMBVAR>
                        <OMV name="i"/>
                      </OMBVAR>
                      <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                        <OMI> 1 </OMI>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMA>
                    </OMBIND>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeon0" name="multi_power"/>
                <OMV name="x"/>
                <OMV name="k"/>
              </OMA>
            </OMA>
          </OMBIND>
        </OMA>
      </OMA>
    </OMBIND>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMV name="n"/>
      <OMA><OMS cd="linalg4" name="rowcount"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="hypergeon0">where</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>n</ci></bvar>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="hypergeon2">lauricella_FA</csymbol>
     <ci>a</ci>
     <ci>b</ci>
     <ci>c</ci>
     <ci>x</ci>
    </apply>
    <apply><csymbol cd="arith1">sum</csymbol>
     <apply><csymbol cd="hypergeon0">cartesian_product_n</csymbol><csymbol cd="setname1">N</csymbol><ci>n</ci></apply>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>k</ci></bvar>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">divide</csymbol>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>a</ci>
          <apply><csymbol cd="arith1">sum</csymbol>
           <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
           <bind><csymbol cd="fns1">lambda</csymbol>
            <bvar><ci>i</ci></bvar>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </bind>
          </apply>
         </apply>
         <apply><csymbol cd="arith1">prod</csymbol>
          <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
          <bind><csymbol cd="fns1">lambda</csymbol>
           <bvar><ci>i</ci></bvar>
           <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>b</ci></apply>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </apply>
          </bind>
         </apply>
        </apply>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="arith1">prod</csymbol>
          <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
          <bind><csymbol cd="fns1">lambda</csymbol>
           <bvar><ci>i</ci></bvar>
           <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>c</ci></apply>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </apply>
          </bind>
         </apply>
         <apply><csymbol cd="arith1">prod</csymbol>
          <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
          <bind><csymbol cd="fns1">lambda</csymbol>
           <bvar><ci>i</ci></bvar>
           <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
            <cn type="integer">1</cn>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </apply>
          </bind>
         </apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeon0">multi_power</csymbol><ci>x</ci><ci>k</ci></apply>
      </apply>
     </bind>
    </apply>
   </apply>
  </bind>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>n</ci>
   <apply><csymbol cd="linalg4">rowcount</csymbol><ci>b</ci></apply>
  </apply>
 </apply>
</math>

Prefix

where (lambda [ n ] . (eq (lauricella_FA ( a, b, c, x) , sum (cartesian_product_n (N, n) , lambda [ k ] . (times (divide (times (pochhammer ( a, sum (integer_interval ( 1 , n) , lambda [ i ] . (vector_selector ( i, k) ) ) ) , prod (integer_interval ( 1 , n) , lambda [ i ] . (pochhammer (vector_selector ( i, b) , vector_selector ( i, k) ) ) ) ) , times (prod (integer_interval ( 1 , n) , lambda [ i ] . (pochhammer (vector_selector ( i, c) , vector_selector ( i, k) ) ) ) , prod (integer_interval ( 1 , n) , lambda [ i ] . (pochhammer ( 1 , vector_selector ( i, k) ) ) ) ) ) , multi_power ( x, k) ) ) ) ) ) , eq ( n, rowcount ( b) ) )

Popcorn

hypergeon0.where(fns1.lambda[$n -> hypergeon2.lauricella_FA($a, $b, $c, $x) = arith1.sum(hypergeon0.cartesian_product_n(setname1.N, $n), fns1.lambda[$k -> (hypergeo0.pochhammer($a, arith1.sum(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeon2.vector_selector($i, $k)])) * arith1.prod(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeo0.pochhammer(hypergeon2.vector_selector($i, $b), hypergeon2.vector_selector($i, $k))])) / (arith1.prod(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeo0.pochhammer(hypergeon2.vector_selector($i, $c), hypergeon2.vector_selector($i, $k))]) * arith1.prod(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeo0.pochhammer(1, hypergeon2.vector_selector($i, $k))])) * hypergeon0.multi_power($x, $k)])], $n = linalg4.rowcount($b))

Rendered Presentation MathML

$$\mathrm{where}(\lambda \hspace{0.17em}n.\mathrm{lauricella\_FA}(a,b,c,x)=\sum _{kin\mathrm{cartesian\_product\_n}(\mathbb{N},n)}\frac{\mathrm{pochhammer}(a,\sum _{i=1}^n\mathrm{vector\_selector}(i,k))\mathrm{prod}([1,n],\lambda \hspace{0.17em}i.\mathrm{pochhammer}(\mathrm{vector\_selector}(i,b),\mathrm{vector\_selector}(i,k)))}{\mathrm{prod}([1,n],\lambda \hspace{0.17em}i.\mathrm{pochhammer}(\mathrm{vector\_selector}(i,c),\mathrm{vector\_selector}(i,k)))\mathrm{prod}([1,n],\lambda \hspace{0.17em}i.\mathrm{pochhammer}(1,\mathrm{vector\_selector}(i,k)))}\mathrm{multi\_power}(x,k),n=\mathrm{rowcount}\left(b\right))$$

Signatures:

sts

---

[Next: lauricella_FC] [Previous: appel_F4] [Top]

---

lauricella_FC

Description:

Lauricella's hypergeometric series F_C of n variables. In case of two variable, it agree with the Appel function F_4. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages:

Commented Mathematical property (CMP):

$ F_C(a,b,c;x) = \sum_{k \in {\bf N}^n}^{\infty} \frac{(a)_{\sum k_i} (b)_{\sum k_i} } {\prod (c_i)_{k_i} \prod (1)_{k_i}} x^{k} $

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="hypergeon0" name="where"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="n"/>
      </OMBVAR>
      <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="hypergeon2" name="lauricella_FC"/>
          <OMV name="a"/>
          <OMV name="b"/>
          <OMV name="c"/>
          <OMV name="x"/>
        </OMA>
        <OMA><OMS cd="arith1" name="sum"/>
          <OMA><OMS cd="hypergeon0" name="cartesian_product_n"/>
            <OMS cd="setname1" name="N"/>
            <OMV name="n"/>
          </OMA>
          <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="k"/>
            </OMBVAR>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="divide"/>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="a"/>
                    <OMA><OMS cd="arith1" name="sum"/>
                      <OMA><OMS cd="interval1" name="integer_interval"/>
                        <OMI> 1 </OMI>
                        <OMV name="n"/>
                      </OMA>
                      <OMBIND>
                        <OMS cd="fns1" name="lambda"/>
                        <OMBVAR>
                          <OMV name="i"/>
                        </OMBVAR>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMBIND>
                    </OMA>
                  </OMA>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="b"/>
                    <OMA><OMS cd="arith1" name="sum"/>
                      <OMA><OMS cd="interval1" name="integer_interval"/>
                        <OMI> 1 </OMI>
                        <OMV name="n"/>
                      </OMA>
                      <OMBIND>
                        <OMS cd="fns1" name="lambda"/>
                        <OMBVAR>
                          <OMV name="i"/>
                        </OMBVAR>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMBIND>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="arith1" name="prod"/>
                    <OMA><OMS cd="interval1" name="integer_interval"/>
                      <OMI> 1 </OMI>
                      <OMV name="n"/>
                    </OMA>
                    <OMBIND>
                      <OMS cd="fns1" name="lambda"/>
                      <OMBVAR>
                        <OMV name="i"/>
                      </OMBVAR>
                      <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="c"/>
                        </OMA>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMA>
                    </OMBIND>
                  </OMA>
                  <OMA><OMS cd="arith1" name="prod"/>
                    <OMA><OMS cd="interval1" name="integer_interval"/>
                      <OMI> 1 </OMI>
                      <OMV name="n"/>
                    </OMA>
                    <OMBIND>
                      <OMS cd="fns1" name="lambda"/>
                      <OMBVAR>
                        <OMV name="i"/>
                      </OMBVAR>
                      <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                        <OMI> 1 </OMI>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMA>
                    </OMBIND>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeon0" name="multi_power"/>
                <OMV name="x"/>
                <OMV name="k"/>
              </OMA>
            </OMA>
          </OMBIND>
        </OMA>
      </OMA>
    </OMBIND>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMV name="n"/>
      <OMA><OMS cd="linalg4" name="rowcount"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="hypergeon0">where</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>n</ci></bvar>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="hypergeon2">lauricella_FC</csymbol>
     <ci>a</ci>
     <ci>b</ci>
     <ci>c</ci>
     <ci>x</ci>
    </apply>
    <apply><csymbol cd="arith1">sum</csymbol>
     <apply><csymbol cd="hypergeon0">cartesian_product_n</csymbol><csymbol cd="setname1">N</csymbol><ci>n</ci></apply>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>k</ci></bvar>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">divide</csymbol>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>a</ci>
          <apply><csymbol cd="arith1">sum</csymbol>
           <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
           <bind><csymbol cd="fns1">lambda</csymbol>
            <bvar><ci>i</ci></bvar>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </bind>
          </apply>
         </apply>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>b</ci>
          <apply><csymbol cd="arith1">sum</csymbol>
           <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
           <bind><csymbol cd="fns1">lambda</csymbol>
            <bvar><ci>i</ci></bvar>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </bind>
          </apply>
         </apply>
        </apply>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="arith1">prod</csymbol>
          <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
          <bind><csymbol cd="fns1">lambda</csymbol>
           <bvar><ci>i</ci></bvar>
           <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>c</ci></apply>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </apply>
          </bind>
         </apply>
         <apply><csymbol cd="arith1">prod</csymbol>
          <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
          <bind><csymbol cd="fns1">lambda</csymbol>
           <bvar><ci>i</ci></bvar>
           <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
            <cn type="integer">1</cn>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </apply>
          </bind>
         </apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeon0">multi_power</csymbol><ci>x</ci><ci>k</ci></apply>
      </apply>
     </bind>
    </apply>
   </apply>
  </bind>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>n</ci>
   <apply><csymbol cd="linalg4">rowcount</csymbol><ci>b</ci></apply>
  </apply>
 </apply>
</math>

Prefix

where (lambda [ n ] . (eq (lauricella_FC ( a, b, c, x) , sum (cartesian_product_n (N, n) , lambda [ k ] . (times (divide (times (pochhammer ( a, sum (integer_interval ( 1 , n) , lambda [ i ] . (vector_selector ( i, k) ) ) ) , pochhammer ( b, sum (integer_interval ( 1 , n) , lambda [ i ] . (vector_selector ( i, k) ) ) ) ) , times (prod (integer_interval ( 1 , n) , lambda [ i ] . (pochhammer (vector_selector ( i, c) , vector_selector ( i, k) ) ) ) , prod (integer_interval ( 1 , n) , lambda [ i ] . (pochhammer ( 1 , vector_selector ( i, k) ) ) ) ) ) , multi_power ( x, k) ) ) ) ) ) , eq ( n, rowcount ( b) ) )

Popcorn

hypergeon0.where(fns1.lambda[$n -> hypergeon2.lauricella_FC($a, $b, $c, $x) = arith1.sum(hypergeon0.cartesian_product_n(setname1.N, $n), fns1.lambda[$k -> (hypergeo0.pochhammer($a, arith1.sum(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeon2.vector_selector($i, $k)])) * hypergeo0.pochhammer($b, arith1.sum(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeon2.vector_selector($i, $k)]))) / (arith1.prod(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeo0.pochhammer(hypergeon2.vector_selector($i, $c), hypergeon2.vector_selector($i, $k))]) * arith1.prod(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeo0.pochhammer(1, hypergeon2.vector_selector($i, $k))])) * hypergeon0.multi_power($x, $k)])], $n = linalg4.rowcount($b))

Rendered Presentation MathML

$$\mathrm{where}(\lambda \hspace{0.17em}n.\mathrm{lauricella\_FC}(a,b,c,x)=\sum _{kin\mathrm{cartesian\_product\_n}(\mathbb{N},n)}\frac{\mathrm{pochhammer}(a,\sum _{i=1}^n\mathrm{vector\_selector}(i,k))\mathrm{pochhammer}(b,\sum _{i=1}^n\mathrm{vector\_selector}(i,k))}{\mathrm{prod}([1,n],\lambda \hspace{0.17em}i.\mathrm{pochhammer}(\mathrm{vector\_selector}(i,c),\mathrm{vector\_selector}(i,k)))\mathrm{prod}([1,n],\lambda \hspace{0.17em}i.\mathrm{pochhammer}(1,\mathrm{vector\_selector}(i,k)))}\mathrm{multi\_power}(x,k),n=\mathrm{rowcount}\left(b\right))$$

Signatures:

sts

---

[Next: lauricella_FD] [Previous: lauricella_FA] [Top]

---

lauricella_FD

Description:

Lauricella's hypergeometric series F_D of n variables. In case of two variables, it agree with the Appell function F_1. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages:

Commented Mathematical property (CMP):

$ F_D(a,b,c;x) = \sum_{k \in {\bf N}^n}^{\infty} \frac{(a)_{\sum k_i} \prod (b_i)_{k_i}} {(c)_{\sum k_i} \prod (1)_{k_i}} x^{k} $

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="hypergeon0" name="where"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="n"/>
      </OMBVAR>
      <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="hypergeon2" name="lauricella_FD"/>
          <OMV name="a"/>
          <OMV name="b"/>
          <OMV name="c"/>
          <OMV name="x"/>
        </OMA>
        <OMA><OMS cd="arith1" name="sum"/>
          <OMA><OMS cd="hypergeon0" name="cartesian_product_n"/>
            <OMS cd="setname1" name="N"/>
            <OMV name="n"/>
          </OMA>
          <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="k"/>
            </OMBVAR>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="divide"/>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="a"/>
                    <OMA><OMS cd="arith1" name="sum"/>
                      <OMA><OMS cd="interval1" name="integer_interval"/>
                        <OMI> 1 </OMI>
                        <OMV name="n"/>
                      </OMA>
                      <OMBIND>
                        <OMS cd="fns1" name="lambda"/>
                        <OMBVAR>
                          <OMV name="i"/>
                        </OMBVAR>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMBIND>
                    </OMA>
                  </OMA>
                  <OMA><OMS cd="arith1" name="prod"/>
                    <OMA><OMS cd="interval1" name="integer_interval"/>
                      <OMI> 1 </OMI>
                      <OMV name="n"/>
                    </OMA>
                    <OMBIND>
                      <OMS cd="fns1" name="lambda"/>
                      <OMBVAR>
                        <OMV name="i"/>
                      </OMBVAR>
                      <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="b"/>
                        </OMA>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMA>
                    </OMBIND>
                  </OMA>
                </OMA>
                <OMA><OMS cd="arith1" name="times"/>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMV name="c"/>
                    <OMA><OMS cd="arith1" name="sum"/>
                      <OMA><OMS cd="interval1" name="integer_interval"/>
                        <OMI> 1 </OMI>
                        <OMV name="n"/>
                      </OMA>
                      <OMBIND>
                        <OMS cd="fns1" name="lambda"/>
                        <OMBVAR>
                          <OMV name="i"/>
                        </OMBVAR>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMBIND>
                    </OMA>
                  </OMA>
                  <OMA><OMS cd="arith1" name="prod"/>
                    <OMA><OMS cd="interval1" name="integer_interval"/>
                      <OMI> 1 </OMI>
                      <OMV name="n"/>
                    </OMA>
                    <OMBIND>
                      <OMS cd="fns1" name="lambda"/>
                      <OMBVAR>
                        <OMV name="i"/>
                      </OMBVAR>
                      <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                        <OMI> 1 </OMI>
                        <OMA><OMS cd="hypergeon2" name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="k"/>
                        </OMA>
                      </OMA>
                    </OMBIND>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeon0" name="multi_power"/>
                <OMV name="x"/>
                <OMV name="k"/>
              </OMA>
            </OMA>
          </OMBIND>
        </OMA>
      </OMA>
    </OMBIND>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMV name="n"/>
      <OMA><OMS cd="linalg4" name="rowcount"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="hypergeon0">where</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>n</ci></bvar>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="hypergeon2">lauricella_FD</csymbol>
     <ci>a</ci>
     <ci>b</ci>
     <ci>c</ci>
     <ci>x</ci>
    </apply>
    <apply><csymbol cd="arith1">sum</csymbol>
     <apply><csymbol cd="hypergeon0">cartesian_product_n</csymbol><csymbol cd="setname1">N</csymbol><ci>n</ci></apply>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>k</ci></bvar>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">divide</csymbol>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>a</ci>
          <apply><csymbol cd="arith1">sum</csymbol>
           <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
           <bind><csymbol cd="fns1">lambda</csymbol>
            <bvar><ci>i</ci></bvar>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </bind>
          </apply>
         </apply>
         <apply><csymbol cd="arith1">prod</csymbol>
          <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
          <bind><csymbol cd="fns1">lambda</csymbol>
           <bvar><ci>i</ci></bvar>
           <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>b</ci></apply>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </apply>
          </bind>
         </apply>
        </apply>
        <apply><csymbol cd="arith1">times</csymbol>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <ci>c</ci>
          <apply><csymbol cd="arith1">sum</csymbol>
           <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
           <bind><csymbol cd="fns1">lambda</csymbol>
            <bvar><ci>i</ci></bvar>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </bind>
          </apply>
         </apply>
         <apply><csymbol cd="arith1">prod</csymbol>
          <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
          <bind><csymbol cd="fns1">lambda</csymbol>
           <bvar><ci>i</ci></bvar>
           <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
            <cn type="integer">1</cn>
            <apply><csymbol cd="hypergeon2">vector_selector</csymbol><ci>i</ci><ci>k</ci></apply>
           </apply>
          </bind>
         </apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeon0">multi_power</csymbol><ci>x</ci><ci>k</ci></apply>
      </apply>
     </bind>
    </apply>
   </apply>
  </bind>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>n</ci>
   <apply><csymbol cd="linalg4">rowcount</csymbol><ci>b</ci></apply>
  </apply>
 </apply>
</math>

Prefix

where (lambda [ n ] . (eq (lauricella_FD ( a, b, c, x) , sum (cartesian_product_n (N, n) , lambda [ k ] . (times (divide (times (pochhammer ( a, sum (integer_interval ( 1 , n) , lambda [ i ] . (vector_selector ( i, k) ) ) ) , prod (integer_interval ( 1 , n) , lambda [ i ] . (pochhammer (vector_selector ( i, b) , vector_selector ( i, k) ) ) ) ) , times (pochhammer ( c, sum (integer_interval ( 1 , n) , lambda [ i ] . (vector_selector ( i, k) ) ) ) , prod (integer_interval ( 1 , n) , lambda [ i ] . (pochhammer ( 1 , vector_selector ( i, k) ) ) ) ) ) , multi_power ( x, k) ) ) ) ) ) , eq ( n, rowcount ( b) ) )

Popcorn

hypergeon0.where(fns1.lambda[$n -> hypergeon2.lauricella_FD($a, $b, $c, $x) = arith1.sum(hypergeon0.cartesian_product_n(setname1.N, $n), fns1.lambda[$k -> (hypergeo0.pochhammer($a, arith1.sum(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeon2.vector_selector($i, $k)])) * arith1.prod(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeo0.pochhammer(hypergeon2.vector_selector($i, $b), hypergeon2.vector_selector($i, $k))])) / (hypergeo0.pochhammer($c, arith1.sum(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeon2.vector_selector($i, $k)])) * arith1.prod(interval1.integer_interval(1, $n), fns1.lambda[$i -> hypergeo0.pochhammer(1, hypergeon2.vector_selector($i, $k))])) * hypergeon0.multi_power($x, $k)])], $n = linalg4.rowcount($b))

Rendered Presentation MathML

$$\mathrm{where}(\lambda \hspace{0.17em}n.\mathrm{lauricella\_FD}(a,b,c,x)=\sum _{kin\mathrm{cartesian\_product\_n}(\mathbb{N},n)}\frac{\mathrm{pochhammer}(a,\sum _{i=1}^n\mathrm{vector\_selector}(i,k))\mathrm{prod}([1,n],\lambda \hspace{0.17em}i.\mathrm{pochhammer}(\mathrm{vector\_selector}(i,b),\mathrm{vector\_selector}(i,k)))}{\mathrm{pochhammer}(c,\sum _{i=1}^n\mathrm{vector\_selector}(i,k))\mathrm{prod}([1,n],\lambda \hspace{0.17em}i.\mathrm{pochhammer}(1,\mathrm{vector\_selector}(i,k)))}\mathrm{multi\_power}(x,k),n=\mathrm{rowcount}\left(b\right))$$

Signatures:

sts

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