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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.
1.0.1 pfp_omega | ||
1.0.2 pfpcoh_intersection | ||
1.0.3 pfphom_intersection | ||
1.0.4 pfphom_monodromy_pair_kyushu |
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pfp_omega
: 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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pfpcoh_intersection
: 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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pfphom_intersection
: 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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pfphom_monodromy_pair_kyushu
:
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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