# Pfpcoh (cohomology/homology groups for p F q) Manual

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# 1 Cohomology group associated to pFq

This section describes functions to evaluate invariants associated to (co)homology groups of the hypergeometric functions pFq

In order to use the functions in this section in OpenXM/Risa/Asir, executing the commands

```load("pfpcoh.rr")\$ load("pfphom.rr")\$
```

is necessary at first.

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### 1.0.1 `pfp_omega`

pfp_omega(P)

: It returns the Gauss-Manin connection Omega for the generalized hypergeometric function P F P-1 (aa1,aa2, ...; cc1, cc2, ...;x) .

Description:

Define a vector valued function Y of which elements are generalized hypergeometric function f_1=F and f_2=xdf_1/dx, f3=xd f_2/dx, ... It satisfies dY/dx= Omega Y. Generalized hypergeometric function is defined by the series p F p-1(aa1,aa2, ...; cc1, cc2, ...;x) = sum(k=0,infty; (aa1)_k (aa2)_k .../( (1)_k (cc1)_k (cc2)_k ... ) x^k)

Example:

``` pfp_omega(3);
```

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### 1.0.2 `pfpcoh_intersection`

pfpcoh_intersection(P)

: pfpcoh_intersection(P) returns an intersection matrix for cocycles associated to the generalized hypergeometric function p F_(p-1).

Description:

This program pfpcoh.rr computes an intersection matrix S of cocycles of p F p-1 and compares it with the matrix obtained by solving a differential equation for intersection matrix.

Algorithm:

Ohara, Sugiki, Takayama, Quadratic Relations for Generalized Hypergeometric Functions p F p-1

Example:

```load("pfpcoh.rr")\$
S=pfpcoh_intersection(3);

```

Author : K.Ohara

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### 1.0.3 `pfphom_intersection`

pfphom_intersection(P)

: intersection matrix of homology cycles.

Description:

Computing intersection matrix of cycles associated to p F_(p-1). As to the meaning of parameters c1, c2, c3, ..., see the paper Ohara, Kyushu J. Math. Vol. 51 PP.123.

Algorithm:

Ohara, Sugiki, Takayama, Quadratic Relations for Generalized Hypergeometric Functions p F p-1

Example:

```         SS = pfphom_intersection(3)\$
```

You get the intersection matrix of homologies for 3 F 2.

Author : K.Ohara

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### 1.0.4 `pfphom_monodromy_pair_kyushu`

pfphom_monodromy_pair_kyushu(P)

:

Description:

It returns the pair of monodromy matrices.

Algorithm:

Ohara, Kyushu J. Math. Vol.51 PP.123 (1997)

Example:

```	     MP = pfphom_monodromy_pair_kyushu(3)\$
```

You get a pair of monodromy matricies for 3F2 standing for two paths encircling 0 and 1.

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# Index

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