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By Kentaro Yano

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2 n=1 n k=1 1 1 2 + − a+k−1 b+k−1 k . Notice that this solution can be obtained by taking the difference of solutions z 1−c F (a − c + 1, b − c + 1, 2 − c|z) − F (a, b, c|z), divide it by c − 1 and take the limit as c → 1. Later it will turn out that Riemann’s approach to hypergeometric functions gives a remarkably transparent insight into these formulas as well as the quadratic transformations of Kummer and Goursat. Examples of such transformations are F (a, b, a + b + 1/2|4z − 4z 2 ) = F (2a, 2b, a + b + 1/2|z) 26 Frits Beukers and F (a, b, a + b + 1/2|z 2/(4z − 4)) = (1 − z)a F (2a, a + b, 2a + 2b|z).

4) The condition of regular singularity sees to it that the functions qi (z) are holomorphic near z = 0. The indicial equation of (3) at z = 0 is defined as X n + q1 (0)X n−1 + · · · + qn−1 (0)X + qn (0) = 0. 28 Frits Beukers Suppose we introduce a local parameter t at 0 given by z = c1 t + c2 t2 + c3 t3 + · · · with c1 = 0. The differential equation can be rewritten in the new variable t. We d obtain, writing Dt = t dt , Dtn y + q˜1 (t)Dtn−1 y + · · · + q˜n−1 (t)Dt y + q˜n (t)y = 0, with new functions q˜i (t) holomorphic at t = 0.

25. Let ρ, σ be two geodesics intersecting in a point P with an angle πλ. Let r, s be the reflections in ρ, σ respectively. Then the group D generated by r, s is a dihedral group consisting of rotations (rs)n around P with angles 2nπλ, n ∈ Z and reflections in the lines (rs)n (ρ), (rs)n (σ). In particular D is finite of order 2m if and only if λ = q/m for some q ∈ Z with q = 0 and gcd(m, q) = 1. Furthermore, D is discrete if and only if λ is either zero or a rational number. 26. For any geodesic triangle Δ we have S = denotes the closure of Δ in S.

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