The Maxima on-line user's manual

Algebra Calculator

Hgfred

Function: hgfred (<a>, <b>, <t>) Simplify the generalized hypergeometric function in terms of other, simpler, forms. <a> is a list of numerator parameters and <b> is a list of the denominator parameters.

If `hgfred` cannot simplify the hypergeometric function, it returns an expression of the form `%f[p,q]([a], [b], x)` where <p> is the number of elements in <a>, and <q> is the number of elements in <b>. This is the usual `pFq` generalized hypergeometric function.

```          (%i1) assume(not(equal(z,0)));
(%o1)                          [notequal(z, 0)]
(%i2) hgfred([v+1/2],[2*v+1],2*%i*z);```

v/2 %i z 4 bessel_j(v, z) gamma(v + 1) %e

`          (%o2)               ---------------------------------------`
`                                                 v`
`                                                z`
`          (%i3) hgfred([1,1],[2],z);`

```                                             log(1 - z)
(%o3)                            - ----------
z
(%i4) hgfred([a,a+1/2],[3/2],z^2);```

1 - 2 a 1 - 2 a (z + 1) - (1 - z)

`          (%o4)                   -------------------------------`
`                                           2 (1 - 2 a) z`

It can be beneficial to load orthopoly too as the following example shows. Note that <L> is the generalized Laguerre polynomial.

```          (%i5) load(orthopoly)\$
(%i6) hgfred([-2],[a],z);```

(a - 1) 2 L (z) 2

`          (%o6)                            -------------`
`                                             a (a + 1)`
`          (%i7) ev(%);`

2 z 2 z

`          (%o7)                         --------- - --- + 1`
`                                        a (a + 1)    a`

```(%o1)                                true
(%i2) ```

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