OpenMath Content Dictionary: hypergeon1

Canonical URL:

http://www.math.kobe-u.ac.jp/OCD/hypergeon1.tfb

CD File:

hypergeon1.ocd

CD as XML Encoded OpenMath:

hypergeon1.omcd

Defines:

a_hypergeomeric, falling_factorial, falling_multi_factorial, raising_multi_factorial

Date:

2002-07-30, 2003-11-30

Version:

1 (Revision 1)

Review Date:

2003-07-30

Status:

experimental

Uses CD:

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

This CD defines symbols for A-hypergeometric series

falling_factorial

Description:

falling_factorial(n,i) is equal to n*(n-1)* ... *(n-i+1).

Signatures:

sts

raising_multi_factorial

Description:

raising_multi_factorial is a product of pochhammer symbols. 2-ary function. reference: authors: "Saito, Sturmfels, Takayama" title: "Grobner Deformations of Hypergeometric Differential Equations" pages: 127

Commented Mathematical property (CMP):

$ [v]_{u_+} = \prod_{i \in \Z\cap[0,n] :\ u_i > 0} (v_i+1)_{u_i} $

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon1" name="raising_multi_factorial"/>
      <OMV name="v"/>
      <OMV name="u"/>
    </OMA>
    <OMA><OMS cd="arith1" name="product"/>
      <OMA><OMS cd="set1" name="intersect"/>
        <OMA><OMS cd="interval1" name="integer_interval"/>
          <OMI> 1 </OMI>
          <OMV name="n"/>
        </OMA>
        <OMA><OMS cd="set1" name="suchthat"/>
          <OMS cd="setname1" name="Z"/>
          <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="i"/>
            </OMBVAR>
            <OMA><OMS cd="arith1" name="gt"/>
              <OMA><OMS cd="linalg1" name="vector_selector"/>
                <OMV name="i"/>
                <OMV name="u"/>
              </OMA>
              <OMI> 0 </OMI>
            </OMA>
          </OMBIND>
        </OMA>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="i"/>
        </OMBVAR>
        <OMA><OMS cd="hypergeo0" name="pochhammer"/>
          <OMA><OMS cd="arith1" name="plus"/>
            <OMA><OMS cd="linalg1" name="vector_selector"/>
              <OMV name="i"/>
              <OMV name="v"/>
            </OMA>
            <OMI> 1 </OMI>
          </OMA>
          <OMA><OMS cd="linalg1" name="vector_selector"/>
            <OMV name="i"/>
            <OMV name="u"/>
          </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="hypergeon1">raising_multi_factorial</csymbol><ci>v</ci><ci>u</ci></apply>
  <apply><csymbol cd="arith1">product</csymbol>
   <apply><csymbol cd="set1">intersect</csymbol>
    <apply><csymbol cd="interval1">integer_interval</csymbol><cn>1</cn><ci>n</ci></apply>
    <apply><csymbol cd="set1">suchthat</csymbol>
     <csymbol cd="setname1">Z</csymbol>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>i</ci></bvar>
      <apply><csymbol cd="arith1">gt</csymbol>
       <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>u</ci></apply>
       <cn>0</cn>
      </apply>
     </bind>
    </apply>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>i</ci></bvar>
    <apply><csymbol cd="hypergeo0">pochhammer</csymbol>
     <apply><csymbol cd="arith1">plus</csymbol>
      <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>v</ci></apply>
      <cn>1</cn>
     </apply>
     <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>u</ci></apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (raising_multi_factorial ( v, u) , product (intersect (integer_interval ( 1 , n) , suchthat (Z, lambda [ i ] . (gt (vector_selector ( i, u) , 0 ) ) ) ) , lambda [ i ] . (pochhammer (plus (vector_selector ( i, v) , 1 ) , vector_selector ( i, u) ) ) ) )

Popcorn

hypergeon1.raising_multi_factorial($v, $u) = arith1.product(set1.intersect(interval1.integer_interval(1, $n), set1.suchthat(setname1.Z, fns1.lambda[$i -> arith1.gt(linalg1.vector_selector($i, $u), 0)])), fns1.lambda[$i -> hypergeo0.pochhammer(linalg1.vector_selector($i, $v) + 1, linalg1.vector_selector($i, $u))])

Rendered Presentation MathML

$$\mathrm{raising\_multi\_factorial}(v,u)=\prod _{iin[1,n]\cap \left\{i\in \mathbb{Z}\right|\mathrm{gt}(u_i,0)\}}\mathrm{pochhammer}(v_i+1,u_i)$$

Signatures:

sts

falling_multi_factorial

Description:

falling_multi_factorial is a product of falling pochhammer symbols. 2-ary function. reference: authors: "Saito, Sturmfels, Takayama" title: "Grobner Deformations of Hypergeometric Differential Equations" pages: 127

Commented Mathematical property (CMP):

$ [v]_{u_-} = \prod_{i \in \Z\cap[0,n] :\ u_i < 0} v_i (v_i-1) \cdots (v_i + u_i-1) $

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon1" name="falling_multi_factorial"/>
      <OMV name="v"/>
      <OMV name="u"/>
    </OMA>
    <OMA><OMS cd="arith1" name="product"/>
      <OMA><OMS cd="set1" name="intersect"/>
        <OMA><OMS cd="interval1" name="integer_interval"/>
          <OMI> 1 </OMI>
          <OMV name="n"/>
        </OMA>
        <OMA><OMS cd="set1" name="suchthat"/>
          <OMS cd="setname1" name="Z"/>
          <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="i"/>
            </OMBVAR>
            <OMA><OMS cd="arith1" name="lt"/>
              <OMA><OMS cd="linalg1" name="vector_selector"/>
                <OMV name="i"/>
                <OMV name="u"/>
              </OMA>
              <OMI> 0 </OMI>
            </OMA>
          </OMBIND>
        </OMA>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="i"/>
        </OMBVAR>
        <OMA><OMS cd="hypergeon1" name="falling_factorial"/>
          <OMA><OMS cd="linalg1" name="vector_selector"/>
            <OMV name="i"/>
            <OMV name="v"/>
          </OMA>
          <OMA><OMS cd="arith1" name="unary_minus"/>
            <OMA><OMS cd="linalg1" name="vector_selector"/>
              <OMV name="i"/>
              <OMV name="u"/>
            </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="hypergeon1">falling_multi_factorial</csymbol><ci>v</ci><ci>u</ci></apply>
  <apply><csymbol cd="arith1">product</csymbol>
   <apply><csymbol cd="set1">intersect</csymbol>
    <apply><csymbol cd="interval1">integer_interval</csymbol><cn>1</cn><ci>n</ci></apply>
    <apply><csymbol cd="set1">suchthat</csymbol>
     <csymbol cd="setname1">Z</csymbol>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>i</ci></bvar>
      <apply><csymbol cd="arith1">lt</csymbol>
       <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>u</ci></apply>
       <cn>0</cn>
      </apply>
     </bind>
    </apply>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>i</ci></bvar>
    <apply><csymbol cd="hypergeon1">falling_factorial</csymbol>
     <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>v</ci></apply>
     <apply><csymbol cd="arith1">unary_minus</csymbol>
      <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>u</ci></apply>
     </apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (falling_multi_factorial ( v, u) , product (intersect (integer_interval ( 1 , n) , suchthat (Z, lambda [ i ] . (lt (vector_selector ( i, u) , 0 ) ) ) ) , lambda [ i ] . (falling_factorial (vector_selector ( i, v) , unary_minus (vector_selector ( i, u) ) ) ) ) )

Popcorn

hypergeon1.falling_multi_factorial($v, $u) = arith1.product(set1.intersect(interval1.integer_interval(1, $n), set1.suchthat(setname1.Z, fns1.lambda[$i -> arith1.lt(linalg1.vector_selector($i, $u), 0)])), fns1.lambda[$i -> hypergeon1.falling_factorial(linalg1.vector_selector($i, $v), -(linalg1.vector_selector($i, $u)))])

Rendered Presentation MathML

$$\mathrm{falling\_multi\_factorial}(v,u)=\prod _{iin[1,n]\cap \left\{i\in \mathbb{Z}\right|\mathrm{lt}(u_i,0)\}}\mathrm{falling\_factorial}(v_i,-u_i)$$

Signatures:

sts

a_hypergeomeric

Description:

A-hypergeometric series reference: authors: "Saito, Sturmfels, Takayama" title: "Grobner Deformations of Hypergeometric Differential Equations" pages: 127

Commented Mathematical property (CMP):

$ \phi(A,v,x) = \sum_{u \in \kernel{\Z^n \stackrel \Z^d}} \frac{[v]_{u_-}}{[v+u]_{u_+}} x^{v+u} $

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 name="a_hypergeometric"/>
          <OMV name="a"/>
          <OMV name="v"/>
          <OMV name="x"/>
        </OMA>
        <OMA><OMS cd="arith1" name="sum"/>
          <OMA><OMS cd="hypergeon0" name="kernel"/>
            <OMA><OMS cd="hypergeon0" name="cartesian_product_n"/>
              <OMS cd="setname1" name="Z"/>
              <OMV name="n"/>
            </OMA>
            <OMV name="a"/>
          </OMA>
          <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="u"/>
            </OMBVAR>
            <OMA><OMS cd="arith1" name="times"/>
              <OMA><OMS cd="arith1" name="divide"/>
                <OMA><OMS cd="hypergeon1" name="falling_multi_factorial"/>
                  <OMV name="v"/>
                  <OMA><OMS cd="hyergeon0" name="minus_part"/>
                    <OMV name="u"/>
                  </OMA>
                </OMA>
                <OMA><OMS cd="hypergeon1" name="raising_multi_factorial"/>
                  <OMA><OMS cd="arith1" name="plus"/>
                    <OMV name="v"/>
                    <OMV name="u"/>
                  </OMA>
                  <OMA><OMS cd="hypergeon0" name="plus_part"/>
                    <OMV name="u"/>
                  </OMA>
                </OMA>
              </OMA>
              <OMA><OMS cd="hypergeon0" name="multi_power"/>
                <OMV name="x"/>
                <OMA><OMS cd="arith1" name="plus"/>
                  <OMV name="v"/>
                  <OMV name="u"/>
                </OMA>
              </OMA>
            </OMA>
          </OMBIND>
        </OMA>
      </OMA>
    </OMBIND>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMV name="n"/>
      <OMA><OMS cd="linalg4" name="columncount"/>
        <OMV name="a"/>
      </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="">a_hypergeometric</csymbol><ci>a</ci><ci>v</ci><ci>x</ci></apply>
    <apply><csymbol cd="arith1">sum</csymbol>
     <apply><csymbol cd="hypergeon0">kernel</csymbol>
      <apply><csymbol cd="hypergeon0">cartesian_product_n</csymbol><csymbol cd="setname1">Z</csymbol><ci>n</ci></apply>
      <ci>a</ci>
     </apply>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>u</ci></bvar>
      <apply><csymbol cd="arith1">times</csymbol>
       <apply><csymbol cd="arith1">divide</csymbol>
        <apply><csymbol cd="hypergeon1">falling_multi_factorial</csymbol>
         <ci>v</ci>
         <apply><csymbol cd="hyergeon0">minus_part</csymbol><ci>u</ci></apply>
        </apply>
        <apply><csymbol cd="hypergeon1">raising_multi_factorial</csymbol>
         <apply><csymbol cd="arith1">plus</csymbol><ci>v</ci><ci>u</ci></apply>
         <apply><csymbol cd="hypergeon0">plus_part</csymbol><ci>u</ci></apply>
        </apply>
       </apply>
       <apply><csymbol cd="hypergeon0">multi_power</csymbol>
        <ci>x</ci>
        <apply><csymbol cd="arith1">plus</csymbol><ci>v</ci><ci>u</ci></apply>
       </apply>
      </apply>
     </bind>
    </apply>
   </apply>
  </bind>
  <apply><csymbol cd="relation1">eq</csymbol>
   <ci>n</ci>
   <apply><csymbol cd="linalg4">columncount</csymbol><ci>a</ci></apply>
  </apply>
 </apply>
</math>

Prefix

where (lambda [ n ] . (eq (a_hypergeometric ( a, v, x) , sum (kernel (cartesian_product_n (Z, n) , a) , lambda [ u ] . (times (divide (falling_multi_factorial ( v, minus_part ( u) ) , raising_multi_factorial (plus ( v, u) , plus_part ( u) ) ) , multi_power ( x, plus ( v, u) ) ) ) ) ) ) , eq ( n, columncount ( a) ) )

Popcorn

hypergeon0.where(fns1.lambda[$n -> .a_hypergeometric($a, $v, $x) = arith1.sum(hypergeon0.kernel(hypergeon0.cartesian_product_n(setname1.Z, $n), $a), fns1.lambda[$u -> hypergeon1.falling_multi_factorial($v, hyergeon0.minus_part($u)) / hypergeon1.raising_multi_factorial($v + $u, hypergeon0.plus_part($u)) * hypergeon0.multi_power($x, $v + $u)])], $n = linalg4.columncount($a))

Rendered Presentation MathML

$$\mathrm{where}(\lambda \hspace{0.17em}n.\mathrm{a\_hypergeometric}(a,v,x)=\sum _{uin\mathrm{kernel}(\mathrm{cartesian\_product\_n}(\mathbb{Z},n),a)}\frac{\mathrm{falling\_multi\_factorial}(v,\mathrm{minus\_part}\left(u\right))}{\mathrm{raising\_multi\_factorial}(v+u,\mathrm{plus\_part}\left(u\right))}\mathrm{multi\_power}(x,v+u),n=\mathrm{columncount}\left(a\right))$$

Signatures:

sts