By Holzapfel R.-P.

This quantity contains lecture notes, survey and study articles originating from the CIMPA summer season university mathematics and Geometry round Hypergeometric services held at Galatasaray collage, Istanbul in the course of June 13-25, 2005. a variety of issues concerning hypergeometric features is roofed, hence giving a vast viewpoint of the cutting-edge within the box.

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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.