# Question:How to expand or simplify more forms of the generalized hypergeometric functions?

## Question:How to expand or simplify more forms of the generalized hypergeometric functions?

Maple 2024

In many special cases, generalized hypergeometric functions can be transformed into other (possibly less general) functions, but sometimes Maple fails to convert some of them to other functions when possible.
For instance,

```4*hypergeom([1, 1], [2, 3, 7/2], -(x/2)**2) -
35*hypergeom([1, 1], [2, 5/2, 3], -(x/2)**2) -
2*hypergeom([1, 1], [3, 3, 7/2], -(x/2)**2) +
10*hypergeom([1, 1], [5/2, 3, 3], -(x/2)**2): # which is, in fact, “360*((Ci(x) - ln(x) + 19/18 - gamma)/x**2 - sin(x)/x**3 + (cos(x) + 5/3)/x**4 - 8/3*sin(x)/x**5)”
expand(%);
=

simplify(%);

```

Is it possible to obtain a non-hypergeometric form of it? E.g.,

Here are more examples that `simplify/hypergeom` (as well as expand) is unable to deal with:

1. hypergeom([1, 1, 3/2], [5/2, 5/2, 3, 3], -(x/2)**2):
2. hypergeom([1, 1], [5/2, 3, 3], -(x/2)**2):
3. hypergeom([1, 1], [3/2, 2, 2], -(x/2)**2):
4. hypergeom([3/2], [5/2, 5/2], -(x/2)**2):
5. … (to be continued)

I have read something like How to simplify this hypergeometric function? - MaplePrimes, yet those tricks do not appear to work here. Is there any workaround?

Edit. There seems to be a bug in `simplify/hypergeom`:

```'hypergeom([-1/3, 1/3, -1/2 - m, -m], [1/2, 1/3 - m, 2/3 - m], 1)':
simplify(`%`, hypergeom) assuming m::nonnegint;
=
-1
--
2

simplify(eval(`%%`, m = 0), 'constant');
=
1

```

The expected result should be a piecewise function.

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