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:

hypergeometric_pFq, hypergeometric0F1, hypergeometric1F1, hypergeometric2F1

Date:

2002-11-29

Version:

0

Review Date:

2003-07-26

Status:

experimental

Uses CD:

alg1, arith1, calculus1, fns1, hypergeo0, interval1, linalg1, linalg4, nums1, relation1

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):

xml prefix mathml

<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>

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

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

Signatures:

sts

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):

xml prefix mathml

<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>

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) ) ) ) )

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

Signatures:

sts

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):

xml prefix mathml

<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>

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) ) ) ) )

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

Example:

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

xml prefix mathml

<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>

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 )

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

Signatures:

sts

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):

xml prefix mathml

<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>

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) ) ) ) )

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

Signatures:

sts