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Transformations of some Gauss hypergeometric functions. (English) Zbl 1076.33002
Hypergeometric identities derived from transformations are a long standing and highly valuable issue in the study of hypergeometric functions. The classical transformations have been first stated by Gauss, Euler, Kummer, Pfaff, just to mention some of them. Thus, the question of deriving all transformations of a certain type is an interesting and valuable problem. An algebraic transformation of the Gauss hypergeometric function is an identity of the form $$ F(\alpha,\eta;\gamma;x) \, = \, \theta(x) F(a,b;c;\varphi(x)), $$ where $\varphi$ is a rational function of $x$, and $\theta$ is a product of powers of rational functions. The author studies transformations of what he calls {\it hyperbolic hypergeometric functions}. These are Gauss hypergeometric functions which are solutions of the hypergeometric equation with the local exponent differences $1/k$, $1/l$, $1/m$, where $k,l,m$ are positive integers and $1/k+1/l+1/m<1$. The author uses the correspondence between algebraic transformations and pull-back transformations of the hypergeometric differential equation with respect to the finite covering $\varphi:{\Bbb P}^1\to{\Bbb P}^1$ determined by the function $\varphi$. He then explicitly calculates all algebraic transformations of hyperbolic hypergeometric functions by computing suitable coverings $\varphi:{\Bbb P}^1\to{\Bbb P}^1$ with certain branching patterns. Since rational functions on Riemann surfaces are Belyi functions, he can give an algorithm to compute certain Belyi functions explicitly. The results include well-known hypergeometric transformations of degrees 2,3,4, and 6, as well as new transformations of higher degrees. It is an interesting point that some of the derived transformations correspond to Coxeter decompositions of Schwarz triangles.

MSC:
33C05Classical hypergeometric functions, ${}_2F_1$
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References:
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