# zbMATH — the first resource for mathematics

Monodromy for the hypergeometric function $$_ nF_{n-1}$$. (English) Zbl 0663.30044
Consider the $$n$$-dimensional complex vector space of analytic solutions of the differential equation $$D(\alpha,\beta)f(z)=0$$, where $D(\alpha,\beta):=(\theta +\beta_ 1-1)\dots(\theta +\beta_ n-1)-z(\theta +\alpha_ 1)\dots(\theta +\alpha_ n),\quad \theta:=z(d/dz),$ and $$\alpha_ 1,\dots,\alpha_ n,\beta_ 1,\dots,\beta_ n\in\mathbb C$$. This is the hypergeometric equation with regular singularities at $$0, 1, \infty$$.
More specifically, let $$V(\alpha,\beta)$$ be the space of solutions holomorphic around $$z=1/2$$; this space has a basis of solutions expressed as generalized hypergeometric series $${}_ nF_{n-1}$$ provided the numbers $$\beta_ 1,\dots,\beta_ n$$ are distinct mod $$\mathbb Z$$. The fundamental group G of the punctured complex plane $$\mathbb C\setminus \{0,1\}$$ (base point at $$z=1/2)$$ acts on $$V(\alpha,\beta)$$, inducing a homomorphism of $$G$$ into $$\text{GL}(n,\mathbb C)$$ whose image is called the monodromy group. This paper gives a complete analysis of which subgroups of $$\text{GL}(n,\mathbb C)$$ are thus obtained, as a function of the parameters $$\alpha_ 1,\dots,\beta_ 1,\dots$$. The group $$G$$ is generated by $$g_ 0$$, $$g_ 1$$ (positively oriented simple closed paths around $$0, 1$$, respectively). Let $$g_{\infty}=(g_ 1b_ 0)^{- 1}$$. The monodromy groups are called “hypergeometric groups” and are subgroups $$H$$ of $$\text{GL}(n,\mathbb C)$$ generated by $$h_ 0$$, $$h_ 1$$, $$h_{\infty}$$ satisfying: $$h_{\infty}h_ 1h_ 0=I$$, the eigenvalues of $$h_{\infty}$$, $$h_ 0$$ are $$e^{2\pi i\alpha_ j}$$, $$e^{-2\pi i\beta_ j}$$ $$(1\leq j\leq n)$$, respectively, and $$h_ 1$$ is a pseudo-reflection (rank of $$h_ 1-I$$ is 1). Under the assumption $$\alpha_ j- \beta_ k\not\in\mathbb Z$$ (all $$j,k$$), $$H$$ is an irreducible subgroup of $$\text{GL}(n,\mathbb C)$$.
If all the $$\alpha_ j$$, $$\beta_ k$$ are real, then there is an Hermitian form invariant under $$H$$ (an explicit formula for the signature is derived). The situation of $$H$$ being finite is described in detail. There are tables of the parameter values corresponding to the classification of finite unitary reflection groups by G. C. Shephard and J. A. Todd [Can. J. Math. 6, 274–304 (1954; Zbl 0055.14305)].

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 20H15 Other geometric groups, including crystallographic groups 33C05 Classical hypergeometric functions, $${}_2F_1$$
Full Text:
##### References:
 [1] [Ba] Bailey, W.N.: On certain relations between hypergeometric series of higher order. J. London Math. Soc.8, 100-107 (1933) · Zbl 0006.31305 · doi:10.1112/jlms/s1-8.2.100 [2] [Bo] Borel, A.: Linear algebraic groups. New York: Benjamin 1969 · Zbl 0186.33201 [3] [Bou] Bourbaki, N.: Groupes et Alg?bres de Lie, Chap. 4, 5, 6. Paris: Hermann 1981 [4] [Co] Cohen, A.M.: Finite complex reflection groups. Ann. Sci. ?c. Norm. Super., IU. Ser.9, 379-436 (1976); erratum in11, 613 (1978) · Zbl 0359.20029 [5] [E] Erd?lyi, A.: Higher transcendental functions, Vol I. Bateman Manuscript Project, New York: McGraw-Hill 1953 [6] [Ho] Honda, T.: Algebraic differential equations. INDAM Symposia Math.XXIV, 169-204 (1981) [7] [Hu] Humphreys, J.E.: Introduction to Lie algebras and representation theory, Berlin-Heidelberg-New York: Springer 1972 · Zbl 0254.17004 [8] [I] Ince, E.L.: Ordinary differential equations. Dover publ. 1956 [9] [Kat] Katz, N.M.: Algebraic solutions of differential equations. Invent. Math.18, 1-118 (1972) · Zbl 0278.14004 · doi:10.1007/BF01389714 [10] [Kap] Kaplansky, I.: An introduction to differential algebra. Paris: Hermann 1957 · Zbl 0083.03301 [11] [Kl] Klein, F.: Vorlesungen ?ber die hypergeometrische Funktion. Berlin-Heidelberg-New York: Springer 1933 · Zbl 0007.12202 [12] [La] Landau, E.: Eine Anwendung des Eisensteinschen Satz auf die Theorie der Gausschen Differentialgleichung. J. Reine Angew. Math. 127 92-102 (1904); (reprinted in Collected Works, Vol. II, pp. 98-108, Thales Verlag, Esssen 1987 · JFM 35.0463.01 · doi:10.1515/crll.1904.127.92 [13] [Le] Levelt, A.H.M.: Hypergeometric functions. Thesis, University of Amsterdam 1961 [14] [Mi] Mitchell, H.H.: Determination of all primitive collineation groups in more than four variables which contain homologies. Am. J. Math.36, 1-21 (1914) · JFM 45.0253.01 · doi:10.2307/2370513 [15] [Mo] Mostow, G.D.: Braids, hypergeometric functions and lattices. Bull. Am. Math. Soc.16, 225-246 (1987) · Zbl 0639.22005 · doi:10.1090/S0273-0979-1987-15510-8 [16] [Pl] Plemelj, J.: Problems in the sense of Riemann and Klein. Interscience Publ. 1964 · Zbl 0124.28203 [17] [Po] Pochhammer, L.: Zur Theorie der allgemeineren hypergeometrische Reihe. J. Reine Angew. Math. 102, 76-159 (1988) [18] [R] Riemann, B.: Gesammelte mathematische Werke, Teubner, Leipzig 1892 [19] [Sc] Schwarz, H.A.: ?ber diejenigen F?lle in welchen einer algebraische Funktion ihres vierten Elementes darstellt. Crelle J75, 292-335 (1873) · doi:10.1515/crll.1873.75.292 [20] [ST] Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math.6, 274-304 (1954) · Zbl 0055.14305 · doi:10.4153/CJM-1954-028-3 [21] [T] Thomae, J.: ?ber die h?heren hypergeometrischen Reihen. Math. Ann.2, 427-444 (1870) · JFM 02.0122.01 · doi:10.1007/BF01448236
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.