zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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 $ \omega$ be a $ 1 $-form on $ \Bbb{P}^1$ with poles $x=\{ x_1, \dots, x_m \}$ of order $ n_1, \dots, n_m$ $(n = n_1 + \cdots + n_m)$, and let $ \nabla_{\pm\omega} = d \pm \omega \wedge $ be the integrable connections on $ X=\Bbb{P}^1 \setminus x$. Consider the twisted cohomology groups $ H^1(\Omega^\bullet(x),\nabla_{\pm\omega}) $ for the complexes $\nabla_{\pm\omega}:\Omega^\bullet(x) \to \Omega^\bullet(x) $, where $ \Omega^k(x) $ is the vector space of rational $ k$-forms with poles at most at $x$ and consider $ u(t) = c \exp (\int^t \omega)$ which satisfies $\nabla_{ - \omega}u(t)=0 $. An integral $ \langle \varphi, \gamma \rangle= \int_\gamma u(t)\varphi $ for some $ \varphi \in H^1(\Omega^\bullet(x),\nabla_\omega) $ is called the hypergeometric integral, where $ \gamma $ is an element of twisted homology group $ H_1(C_\bullet^{\omega}, \partial_\omega)$, which is proved to be dual to $H^1(\Omega^\bullet(x),\nabla_\omega) $ by the pairing $ \langle\ , \ \rangle $. The authors introduce the cohomological intersection pairing for $ (n-2)$-dimensional vector spaces $H^1(\Omega^\bullet(x),\nabla_{\omega}) $ and $ H^1(\Omega^\bullet(x),\nabla_{-\omega}) $, and the homological intersection pairing for $ H_1(C_\bullet^{\omega}, \partial_{\omega}) $ and $ H_1(C_\bullet^{-\omega}, \partial_{-\omega}) .$ For a choice of bases $\{ \varphi^\pm_\mu\}_\mu$ and $\{ \gamma^\pm_\mu\}_\mu$ of the groups $H^1(\Omega^\bullet(x),\nabla_{\pm\omega}) $ and $ H_1(C_\bullet^{\pm\omega},\partial_{\pm \omega}) $, define the four matrices of size $ n-2:$ $$ \Pi^+ = \langle \varphi^+_\mu, \gamma^+_\nu \rangle_{\mu, \nu}, \quad \Pi^- = \langle \varphi^-_\mu, \gamma^-_\nu \rangle_{\mu, \nu}, \quad I_{\text{ch}}= \langle \varphi^+_\mu, \varphi^-_\nu \rangle_{\mu, \nu}, \quad I_{\text{h}}= \langle \gamma^+_\mu, \gamma^-_\nu \rangle_{\mu, \nu}. $$ The main result of this paper is the following. Theorem. We have twisted period relations: $$ \Pi^+ {}^t I_{\text{h}}^{-1} {}^t\Pi^- = I_{\text{ch}} $$ which give quadratic relations among confluent hypergeometric integrals.
MSC:
33C15Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
WorldCat.org
Full Text: DOI
References:
[1] K. CHO AND K. MATSUMOTO, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Math.J. 139 (1995), 67-86. · Zbl 0856.32015
[2] P. DELIGNE, Equations differentielles a points singuliers reguliers, Lecture Notes in Math. 163, Springer Verlag, Berlin-New York, 1970. · Zbl 0244.14004 · doi:10.1007/BFb0061194
[3] G. DERHAM, Differentiable manifolds, Hermann, Paris, 1960
[4] O. FORSTER, Lectures on Riemann surfaces, Grad. Texts in Math. 81, Springer-Verlag, New York-Berlin, 1981. · Zbl 0475.30002
[5] P. GRIFFITHS AND J. HARRIS, Principles of Algebraic Geometry, Wiley-Intersci. Publ., New York, 1978 · Zbl 0408.14001
[6] Y. HARAOKA, Quadratic relations for confluent hypergeometric functions on Z2n+i, Funkcial. Ekvac. 4 (1999), 435-490. · Zbl 1142.33303
[7] N. KACHI, K. MATSUMOTO AND M. MIHARA, The perfectness of the intersection pairings for twiste cohomology and homology groups with respect to rational 1-forms, Kyushu J. Math. 53 (1999), 163-188. · Zbl 0933.14009 · doi:10.2206/kyushujm.53.163
[8] M. KASHIWARA, H. KAWAI AND T. KIMURA, Foundations of algebraic analysis, Princeton Univ. Press, Princeton, 1986. · Zbl 0605.35001
[9] M. KASHIWARA AND P. SCHAPIRA, Sheaves on Manifolds, Grundlehren Math.Wiss. 292, Springer-Verlag, New York, 1990. · Zbl 0709.18001
[10] H. KIMURA, On rational de Rham cohomology associated with the generalized confluent hypergeometri functions I, P case, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 145-155. · Zbl 0883.33009 · doi:10.1017/S0308210500023544
[11] M. KITA AND M. YOSHIDA, Intersection theory for twisted cycles I, II, Math. Nachr. 166 (1994), 287-304, 168(1994), 171-190. · Zbl 0847.32043 · doi:10.1002/mana.19941660122
[12] H. MAJIMA, V-Poincare’s Lemma and V-de Rham Cohomology for an Integrable Connectionwith Irregula Singular Points, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 4, 150-153. · Zbl 0533.32005 · doi:10.3792/pjaa.59.150
[13] H. MAJIMA, Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes i Math. 1075, Springer-Verlag, Berlin-New York, 1984. · Zbl 0546.58003 · doi:10.1007/BFb0071550
[14] B. MALGRANGE, Remarques sur les equations differentielles a points singuliers irreguliers, Equation differentielles et systemes de Pfaff dans le champ complexe, 77-86, Lecture notes in Math. 712, Springer-Verlag, Berlin, 1979. · Zbl 0423.32014
[15] B. MALGRANGE, Equations differentielles a coefficients polynomiaux, Progress Math.96, Birkhauser, Basel, Boston, 1991. · Zbl 0764.32001
[16] K. MATSUMOTO, Intersection numbers for logarithmic forms, Osaka J. Math. 35 (1998), 873-893 · Zbl 0937.32013
[17] K. MATSUMOTO, Intersectionnumbers for 1-forms associated with confluent hypergeometric functions, Funk cial. Ekvac. 41 (1998), 291-308. · Zbl 1140.33303
[18] K. MATSUMOTO AND N. TAKAYAMA, Braid group and a confluent hypergeometric function, J. Math. Sci Univ. Tokyo 2 (1995), 589-610. · Zbl 0844.33010
[19] J. P. RAMIS, Theoremes d’indices Gevrey pour les equations differentielles ordinaires, Memoirs of the Amer ican Mathematica Society, 296, 1984.
[20] C. SABBAH, On thecomparison theorem for elementary irregular V-modules, Nagoya Math. J. 141(1996), 107-124. · Zbl 0858.32013
[21] T. SASAKI AND M. YOSHIDA, Tensor products of linear differential equations II, Funkcial. Ekvac. 33(1990), 527-549. · Zbl 0738.34010
[22] L. SCHWARTZ, Theorie desdistributions, Herman, Paris, 1966 Zentralblatt MATH: · Zbl 0962.46025 · http://www.zentralblatt-math.org/zmath/en/search/?q=an:0399.46028&format=complete