OpenMath Content Dictionary: hypergeo1

Canonical URL:

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

CD File:

hypergeo1.ocd

CD as XML Encoded OpenMath:

hypergeo1.omcd

Defines:

hypergeometric0F1, hypergeometric1F1, hypergeometric2F1, hypergeometric_pFq

Date:

2002-11-29

Version:

0 (Revision 1)

Review Date:

2017-12-31

Status:

experimental

---

  Author: Yasushi Tamura

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

---

hypergeometric0F1

Description:

Hypergeometric function {}_0 F_1.

Commented Mathematical property (CMP):

hypergeometric0F1(;a;z) =\sum_{n=0}^{+\infty} \frac{1}{pochhammer(a,n)pochhammer(1,n)} z^n

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo1" name="hypergeometric0F1"/>
      <OMV name="a"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="sum"/>
      <OMA><OMS cd="interval1" name="integer_interval"/>
        <OMI> 0 </OMI>
        <OMS cd="nums1" name="infinity"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="n"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMA><OMS cd="arith1" name="divide"/>
              <OMI> 1 </OMI>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="a"/>
                <OMV name="n"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="hypergeo0" name="pochhammer"/>
              <OMI> 1 </OMI>
              <OMV name="n"/>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="z"/>
            <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="hypergeo1">hypergeometric0F1</csymbol><ci>a</ci><ci>z</ci></apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn type="integer">0</cn>
    <csymbol cd="nums1">infinity</csymbol>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>n</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <cn type="integer">1</cn>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol><ci>a</ci><ci>n</ci></apply>
      </apply>
      <apply><csymbol cd="hypergeo0">pochhammer</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
     </apply>
     <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><ci>n</ci></apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (hypergeometric0F1 ( a, z) , sum (integer_interval ( 0 , infinity) , lambda [ n ] . (times (divide (divide ( 1 , pochhammer ( a, n) ) , pochhammer ( 1 , n) ) , power ( z, n) ) ) ) )

Popcorn

hypergeo1.hypergeometric0F1($a, $z) = arith1.sum(interval1.integer_interval(0, nums1.infinity), fns1.lambda[$n -> 1 / hypergeo0.pochhammer($a, $n) / hypergeo0.pochhammer(1, $n) * $z ^ $n])

Rendered Presentation MathML

$$\mathrm{hypergeometric0F1}(a,z)=\sum _{n=0}^{\infty }\frac{\frac{1}{\mathrm{pochhammer}(a,n)}}{\mathrm{pochhammer}(1,n)}{z}^n$$

Signatures:

sts

---

[Next: hypergeometric1F1] [Last: hypergeometric_pFq] [Top]

---

hypergeometric1F1

Description:

Kummer's confluent hypergeometric function.

Commented Mathematical property (CMP):

hypergeometric1F1(a,b;z) =\sum_{n=0}^{+\infty} \frac{pochhammer(a,n)}{pochhammer(1,n)pochhammer(b,n)} z^n

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo1" name="hypergeometric1F1"/>
      <OMV name="a"/>
      <OMV name="b"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="sum"/>
      <OMA><OMS cd="interval1" name="integer_interval"/>
        <OMI> 0 </OMI>
        <OMS cd="nums1" name="infinity"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="n"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMA><OMS cd="hypergeo0" name="pochhammer"/>
              <OMV name="a"/>
              <OMV name="n"/>
            </OMA>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="b"/>
                <OMV name="n"/>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMI> 1 </OMI>
                <OMV name="n"/>
              </OMA>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="z"/>
            <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="hypergeo1">hypergeometric1F1</csymbol><ci>a</ci><ci>b</ci><ci>z</ci></apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn type="integer">0</cn>
    <csymbol cd="nums1">infinity</csymbol>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>n</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <apply><csymbol cd="hypergeo0">pochhammer</csymbol><ci>a</ci><ci>n</ci></apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol><ci>b</ci><ci>n</ci></apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
      </apply>
     </apply>
     <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><ci>n</ci></apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (hypergeometric1F1 ( a, b, z) , sum (integer_interval ( 0 , infinity) , lambda [ n ] . (times (divide (pochhammer ( a, n) , times (pochhammer ( b, n) , pochhammer ( 1 , n) ) ) , power ( z, n) ) ) ) )

Popcorn

hypergeo1.hypergeometric1F1($a, $b, $z) = arith1.sum(interval1.integer_interval(0, nums1.infinity), fns1.lambda[$n -> hypergeo0.pochhammer($a, $n) / (hypergeo0.pochhammer($b, $n) * hypergeo0.pochhammer(1, $n)) * $z ^ $n])

Rendered Presentation MathML

$$\mathrm{hypergeometric1F1}(a,b,z)=\sum _{n=0}^{\infty }\frac{\mathrm{pochhammer}(a,n)}{\mathrm{pochhammer}(b,n)\mathrm{pochhammer}(1,n)}{z}^n$$

Signatures:

sts

---

[Next: hypergeometric2F1] [Previous: hypergeometric0F1] [Top]

---

hypergeometric2F1

Description:

The Gauss hypergeometric function. This function has a branch cut on [1,+infinity).

Commented Mathematical property (CMP):

hypergeometric2F1(a,b,c;z) =\sum_{n=0}^{+\infty} \frac{pochhammer(a,n)pochhammer(b,n)}{pochhammer(c,n)pochhammer(1,n)} z^n

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo1" name="hypergeometric2F1"/>
      <OMV name="a"/>
      <OMV name="b"/>
      <OMV name="c"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="sum"/>
      <OMA><OMS cd="interval1" name="integer_interval"/>
        <OMI> 0 </OMI>
        <OMS cd="nums1" name="infinity"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="n"/>
        </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"/>
                <OMV name="n"/>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="b"/>
                <OMV name="n"/>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMV name="c"/>
                <OMV name="n"/>
              </OMA>
              <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                <OMI> 1 </OMI>
                <OMV name="n"/>
              </OMA>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="z"/>
            <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="hypergeo1">hypergeometric2F1</csymbol>
   <ci>a</ci>
   <ci>b</ci>
   <ci>c</ci>
   <ci>z</ci>
  </apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn type="integer">0</cn>
    <csymbol cd="nums1">infinity</csymbol>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>n</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><ci>n</ci></apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol><ci>b</ci><ci>n</ci></apply>
      </apply>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol><ci>c</ci><ci>n</ci></apply>
       <apply><csymbol cd="hypergeo0">pochhammer</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
      </apply>
     </apply>
     <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><ci>n</ci></apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (hypergeometric2F1 ( a, b, c, z) , sum (integer_interval ( 0 , infinity) , lambda [ n ] . (times (divide (times (pochhammer ( a, n) , pochhammer ( b, n) ) , times (pochhammer ( c, n) , pochhammer ( 1 , n) ) ) , power ( z, n) ) ) ) )

Popcorn

hypergeo1.hypergeometric2F1($a, $b, $c, $z) = arith1.sum(interval1.integer_interval(0, nums1.infinity), fns1.lambda[$n -> (hypergeo0.pochhammer($a, $n) * hypergeo0.pochhammer($b, $n)) / (hypergeo0.pochhammer($c, $n) * hypergeo0.pochhammer(1, $n)) * $z ^ $n])

Rendered Presentation MathML

$$\mathrm{hypergeometric2F1}(a,b,c,z)=\sum _{n=0}^{\infty }\frac{\mathrm{pochhammer}(a,n)\mathrm{pochhammer}(b,n)}{\mathrm{pochhammer}(c,n)\mathrm{pochhammer}(1,n)}{z}^n$$

Example:

z (1-z) d^2 F/dz^2 + (c - (a+b+1) z) d F/dz - a b F = 0

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="arith1" name="minus"/>
      <OMA><OMS cd="arith1" name="plus"/>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="times"/>
            <OMV name="z"/>
            <OMA><OMS cd="arith1" name="minus"/>
              <OMI> 1 </OMI>
              <OMV name="z"/>
            </OMA>
          </OMA>
          <OMA><OMS cd="calculus1" name="diff"/>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="z"/>
              </OMBVAR>
              <OMA><OMS cd="calculus1" name="diff"/>
                <OMBIND>
                  <OMS cd="fns1" name="lambda"/>
                  <OMBVAR>
                    <OMV name="z"/>
                  </OMBVAR>
                  <OMA><OMS cd="hypergeo1" name="hypergeometric2F1"/>
                    <OMV name="a"/>
                    <OMV name="b"/>
                    <OMV name="c"/>
                    <OMV name="z"/>
                  </OMA>
                </OMBIND>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="minus"/>
              <OMV name="c"/>
              <OMA><OMS cd="arith1" name="plus"/>
                <OMA><OMS cd="arith1" name="plus"/>
                  <OMV name="a"/>
                  <OMV name="b"/>
                </OMA>
                <OMI> 1 </OMI>
              </OMA>
            </OMA>
            <OMV name="z"/>
          </OMA>
          <OMA><OMS cd="calculus1" name="diff"/>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="z"/>
              </OMBVAR>
              <OMA><OMS cd="hypergeo1" name="hypergeometric2F1"/>
                <OMV name="a"/>
                <OMV name="b"/>
                <OMV name="c"/>
                <OMV name="z"/>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
        <OMA><OMS cd="arith1" name="times"/>
          <OMV name="a"/>
          <OMV name="b"/>
        </OMA>
        <OMA><OMS cd="hypergeo1" name="hypergeometric2F1"/>
          <OMV name="a"/>
          <OMV name="b"/>
          <OMV name="c"/>
          <OMV name="z"/>
        </OMA>
      </OMA>
    </OMA>
    <OMI> 0 </OMI>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">minus</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <ci>z</ci>
      <apply><csymbol cd="arith1">minus</csymbol><cn type="integer">1</cn><ci>z</ci></apply>
     </apply>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><csymbol cd="calculus1">diff</csymbol>
        <bind><csymbol cd="fns1">lambda</csymbol>
         <bvar><ci>z</ci></bvar>
         <apply><csymbol cd="hypergeo1">hypergeometric2F1</csymbol>
          <ci>a</ci>
          <ci>b</ci>
          <ci>c</ci>
          <ci>z</ci>
         </apply>
        </bind>
       </apply>
      </bind>
     </apply>
    </apply>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">minus</csymbol>
       <ci>c</ci>
       <apply><csymbol cd="arith1">plus</csymbol>
        <apply><csymbol cd="arith1">plus</csymbol><ci>a</ci><ci>b</ci></apply>
        <cn type="integer">1</cn>
       </apply>
      </apply>
      <ci>z</ci>
     </apply>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><csymbol cd="hypergeo1">hypergeometric2F1</csymbol>
        <ci>a</ci>
        <ci>b</ci>
        <ci>c</ci>
        <ci>z</ci>
       </apply>
      </bind>
     </apply>
    </apply>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="arith1">times</csymbol><ci>a</ci><ci>b</ci></apply>
    <apply><csymbol cd="hypergeo1">hypergeometric2F1</csymbol>
     <ci>a</ci>
     <ci>b</ci>
     <ci>c</ci>
     <ci>z</ci>
    </apply>
   </apply>
  </apply>
  <cn type="integer">0</cn>
 </apply>
</math>

Prefix

eq (minus (plus (times (times ( z, minus ( 1 , z) ) , diff (lambda [ z ] . (diff (lambda [ z ] . (hypergeometric2F1 ( a, b, c, z) ) ) ) ) ) , times (times (minus ( c, plus (plus ( a, b) , 1 ) ) , z) , diff (lambda [ z ] . (hypergeometric2F1 ( a, b, c, z) ) ) ) ) , times (times ( a, b) , hypergeometric2F1 ( a, b, c, z) ) ) , 0 )

Popcorn

($z * (1 - $z) * calculus1.diff(fns1.lambda[$z -> calculus1.diff(fns1.lambda[$z -> hypergeo1.hypergeometric2F1($a, $b, $c, $z)])]) + ($c - ($a + $b + 1)) * $z * calculus1.diff(fns1.lambda[$z -> hypergeo1.hypergeometric2F1($a, $b, $c, $z)])) - $a * $b * hypergeo1.hypergeometric2F1($a, $b, $c, $z) = 0

Rendered Presentation MathML

$$z(1-z)\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{hypergeometric2F1}(a,b,c,z)\right)\right)+(c-a+b+1)z\frac{d}{dz}\left(\mathrm{hypergeometric2F1}(a,b,c,z)\right)-ab\mathrm{hypergeometric2F1}(a,b,c,z)=0$$

Signatures:

sts

---

[Next: hypergeometric_pFq] [Previous: hypergeometric1F1] [Top]

---

hypergeometric_pFq

Description:

Generalized hypergeometric function. This function has a branch cut on [1,+infinity).

Commented Mathematical property (CMP):

hypergeometric_pFq(a,b;z) =\sum_{n=0}^{+\infty} \frac{\Pi_i pochhammer(a_i,n)}{\Pi_i pochhammer(b_i,n)pochhammer(1,n)} z^n

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo1" name="hypergeometric_pFq"/>
      <OMV name="a"/>
      <OMV name="b"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="sum"/>
      <OMA><OMS cd="interval1" name="integer_interval"/>
        <OMI> 0 </OMI>
        <OMS cd="nums1" name="infinity"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="n"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="divide"/>
            <OMA><OMS cd="arith1" name="divide"/>
              <OMA><OMS cd="arith1" name="product"/>
                <OMA><OMS cd="interval1" name="integer_interval"/>
                  <OMI> 1 </OMI>
                  <OMA><OMS cd="linalg4" name="size"/>
                    <OMV name="a"/>
                  </OMA>
                </OMA>
                <OMBIND>
                  <OMS cd="fns1" name="lambda"/>
                  <OMBVAR>
                    <OMV name="i"/>
                  </OMBVAR>
                  <OMA><OMS cd="hypergeo0" name="pochhammer"/>
                    <OMA><OMS cd="lialg1" name="vector_selector"/>
                      <OMV name="i"/>
                      <OMV name="a"/>
                    </OMA>
                    <OMV name="n"/>
                  </OMA>
                </OMBIND>
              </OMA>
              <OMA><OMS cd="arith1" name="product"/>
                <OMA><OMS cd="interval1" name="integer_interval"/>
                  <OMI> 1 </OMI>
                  <OMA><OMS cd="linalg4" name="size"/>
                    <OMV name="b"/>
                  </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="b"/>
                    </OMA>
                    <OMV name="n"/>
                  </OMA>
                </OMBIND>
              </OMA>
            </OMA>
            <OMA><OMS cd="hypergeo0" name="pochhammer"/>
              <OMI> 1 </OMI>
              <OMV name="n"/>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="z"/>
            <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="hypergeo1">hypergeometric_pFq</csymbol><ci>a</ci><ci>b</ci><ci>z</ci></apply>
  <apply><csymbol cd="arith1">sum</csymbol>
   <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn type="integer">0</cn>
    <csymbol cd="nums1">infinity</csymbol>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>n</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">divide</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <apply><csymbol cd="arith1">product</csymbol>
        <apply><csymbol cd="interval1">integer_interval</csymbol>
         <cn type="integer">1</cn>
         <apply><csymbol cd="linalg4">size</csymbol><ci>a</ci></apply>
        </apply>
        <bind><csymbol cd="fns1">lambda</csymbol>
         <bvar><ci>i</ci></bvar>
         <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
          <apply><csymbol cd="lialg1">vector_selector</csymbol><ci>i</ci><ci>a</ci></apply>
          <ci>n</ci>
         </apply>
        </bind>
       </apply>
       <apply><csymbol cd="arith1">product</csymbol>
        <apply><csymbol cd="interval1">integer_interval</csymbol>
         <cn type="integer">1</cn>
         <apply><csymbol cd="linalg4">size</csymbol><ci>b</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>b</ci></apply>
          <ci>n</ci>
         </apply>
        </bind>
       </apply>
      </apply>
      <apply><csymbol cd="hypergeo0">pochhammer</csymbol><cn type="integer">1</cn><ci>n</ci></apply>
     </apply>
     <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><ci>n</ci></apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (hypergeometric_pFq ( a, b, z) , sum (integer_interval ( 0 , infinity) , lambda [ n ] . (times (divide (divide (product (integer_interval ( 1 , size ( a) ) , lambda [ i ] . (pochhammer (vector_selector ( i, a) , n) ) ) , product (integer_interval ( 1 , size ( b) ) , lambda [ i ] . (pochhammer (vector_selector ( i, b) , n) ) ) ) , pochhammer ( 1 , n) ) , power ( z, n) ) ) ) )

Popcorn

hypergeo1.hypergeometric_pFq($a, $b, $z) = arith1.sum(interval1.integer_interval(0, nums1.infinity), fns1.lambda[$n -> arith1.product(interval1.integer_interval(1, linalg4.size($a)), fns1.lambda[$i -> hypergeo0.pochhammer(lialg1.vector_selector($i, $a), $n)]) / arith1.product(interval1.integer_interval(1, linalg4.size($b)), fns1.lambda[$i -> hypergeo0.pochhammer(linalg1.vector_selector($i, $b), $n)]) / hypergeo0.pochhammer(1, $n) * $z ^ $n])

Rendered Presentation MathML

$$\mathrm{hypergeometric\_pFq}(a,b,z)=\sum _{n=0}^{\infty }\frac{\frac{\prod _{i=1}^{\mathrm{size}\left(a\right)}}{\mathrm{pochhammer}(\mathrm{vector\_selector}(i,a),n)}}{\mathrm{pochhammer}(1,n)}{z}^n$$

Signatures:

sts

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