zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Quadratic relations for confluent hypergeometric functions. (English) Zbl 1006.33004

Let ω be a 1-form on 1 with poles x={x 1 ,,x m } of order n 1 ,,n m (n=n 1 ++n m ), and let ±ω =d±ω be the integrable connections on X= 1 x. Consider the twisted cohomology groups H 1 (Ω (x), ±ω ) for the complexes ±ω :Ω (x)Ω (x), where Ω k (x) is the vector space of rational k-forms with poles at most at x and consider u(t)=cexp( t ω) which satisfies -ω u(t)=0. An integral ϕ,γ= γ u(t)ϕ for some ϕH 1 (Ω (x), ω ) is called the hypergeometric integral, where γ is an element of twisted homology group H 1 (C ω , ω ), which is proved to be dual to H 1 (Ω (x), ω ) by the pairing ,. The authors introduce the cohomological intersection pairing for (n-2)-dimensional vector spaces H 1 (Ω (x), ω ) and H 1 (Ω (x), -ω ), and the homological intersection pairing for H 1 (C ω , ω ) and H 1 (C -ω , -ω )· For a choice of bases {ϕ μ ± } μ and {γ μ ± } μ of the groups H 1 (Ω (x), ±ω ) and H 1 (C ±ω , ±ω ), define the four matrices of size n-2:

Π + =ϕ μ + ,γ ν + μ,ν ,Π - =ϕ μ - ,γ ν - μ,ν ,I ch =ϕ μ + ,ϕ ν - μ,ν ,I h =γ μ + ,γ ν - μ,ν ·

The main result of this paper is the following.

Theorem. We have twisted period relations:

Π + t I h -1 t Π - =I ch

which give quadratic relations among confluent hypergeometric integrals.

33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1