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:

2002-08-07, 2003-11-30

Version:

1 (Revision 1)

Review Date:

2003-08-07

Status:

experimental

Uses CD:

hypergeo1, hypergeon0, arith1, fns1, interval1, linalg1, linalg4, relation1, set1

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS 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>

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

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

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS 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 name="vector_selector"/>
                <OMI> 1 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="y"/>
              <OMA><OMS name="vector_selector"/>
                <OMI> 2 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

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) ) ) ) ) ) )

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

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS 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 name="vector_selector"/>
                <OMI> 1 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="y"/>
              <OMA><OMS name="vector_selector"/>
                <OMI> 2 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

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) ) ) ) ) ) )

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

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS 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 name="vector_selector"/>
                <OMI> 1 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="y"/>
              <OMA><OMS name="vector_selector"/>
                <OMI> 2 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

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) ) ) ) ) ) )

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

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS 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 name="vector_selector"/>
                <OMI> 1 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="y"/>
              <OMA><OMS name="vector_selector"/>
                <OMI> 2 </OMI>
                <OMV name="m"/>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

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) ) ) ) ) ) )

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

Signatures:

sts

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):

xml prefix mathml

<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 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 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 name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="b"/>
                        </OMA>
                        <OMA><OMS 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 name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="c"/>
                        </OMA>
                        <OMA><OMS 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 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>

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) ) )

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

Signatures:

sts

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):

xml prefix mathml

<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 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 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 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 name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="c"/>
                        </OMA>
                        <OMA><OMS 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 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>

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) ) )

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

Signatures:

sts

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):

xml prefix mathml

<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 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 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 name="vector_selector"/>
                          <OMV name="i"/>
                          <OMV name="b"/>
                        </OMA>
                        <OMA><OMS 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 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 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>

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) ) )

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

Signatures:

sts