OpenMath Content Dictionary: hypergeo0

Canonical URL:

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

CD File:

hypergeo0.ocd

CD as XML Encoded OpenMath:

hypergeo0.omcd

Defines:

beta, gamma, pochhammer

Date:

2002-11-29

Version:

0

Review Date:

2003-04-01

Status:

experimental

Uses CD:

alg1, arith1, calculus1, complex1, fns1, integer1, interval1, logic1, nums1, relation1

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

gamma

Description:

Euler's gamma function

Commented Mathematical property (CMP):

gamma(z)=\int_0^{+\infty} t^{z-1} e^{-z} dt (Re(z)>0)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="logic1" name="implies"/>
    <OMA><OMS cd="relation1" name="gt"/>
      <OMA><OMS cd="complex1" name="real"/>
        <OMV name="z"/>
      </OMA>
      <OMI> 0 </OMI>
    </OMA>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMA><OMS cd="hypergeo0" name="gamma"/>
        <OMV name="z"/>
      </OMA>
      <OMA><OMS cd="calculus1" name="defint"/>
        <OMA><OMS cd="interval1" name="interval"/>
          <OMI> 0 </OMI>
          <OMS cd="nums1" name="infinity"/>
        </OMA>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="t"/>
          </OMBVAR>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="t"/>
              <OMA><OMS cd="arith1" name="minus"/>
                <OMV name="z"/>
                <OMI> 1 </OMI>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMS cd="nums1" name="e"/>
              <OMA><OMS cd="arith1" name="unary_minus"/>
                <OMV name="z"/>
              </OMA>
            </OMA>
          </OMA>
        </OMBIND>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

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

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

Example:

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

xml prefix mathml

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

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

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

Signatures:

sts

beta

Description:

Euler's beta function

Commented Mathematical property (CMP):

beta(p,q)=\frac{gamma(p)gamma(q)}{gamma(p+q)}(p,q \not\in Z_{<=0})

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="logic1" name="implies"/>
    <OMA><OMS cd="logic1" name="and"/>
      <OMA><OMS cd="set1" name="notin"/>
        <OMA><OMS cd="arith1" name="unary_minus"/>
          <OMV name="p"/>
        </OMA>
        <OMS cd="setname1" name="N"/>
      </OMA>
      <OMA><OMS cd="set1" name="notin"/>
        <OMA><OMS cd="arith1" name="unary_minus"/>
          <OMV name="q"/>
        </OMA>
        <OMS cd="setname1" name="N"/>
      </OMA>
    </OMA>
    <OMA><OMS cd="arith1" name="divide"/>
      <OMA><OMS cd="arith1" name="times"/>
        <OMA><OMS cd="hypergeo0" name="gamma"/>
          <OMV name="p"/>
        </OMA>
        <OMA><OMS cd="hypergeo0" name="gamma"/>
          <OMV name="q"/>
        </OMA>
      </OMA>
      <OMA><OMS cd="hypergeo0" name="gamma"/>
        <OMA><OMS cd="arith1" name="plus"/>
          <OMV name="p"/>
          <OMV name="q"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

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

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

Example:

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

xml prefix mathml

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

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

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

Signatures:

sts

pochhammer

Description:

Pochhammer symbol

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

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

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

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

Example:

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

xml prefix mathml

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

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

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

Signatures:

sts