By Rolf-Peter Holzapfel, Muhammed Uludag, M. Yoshida

ISBN-10: 376438283X

ISBN-13: 9783764382834

ISBN-10: 3764382848

ISBN-13: 9783764382841

This quantity includes lecture notes, survey and learn articles originating from the CIMPA summer time institution mathematics and Geometry round Hypergeometric features held at Galatasaray collage, Istanbul, June 13-25, 2005. It covers quite a lot of subject matters regarding hypergeometric capabilities, hence giving a large standpoint of the state-of-the-art within the box.

**Read Online or Download Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005 PDF**

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**Additional resources for Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005**

**Sample text**

Let ai = lim (z − P )i pi (z) z→P for i = 1, . . , n. The indicial equation at P is given by X(X − 1) · · · (X − n + 1) + a1 X(X − 1) · · · (X − n + 2) + · · · + an−1 X + an = 0. When ∞ is regular or a regular singularity, let ai = limz→∞ z i pi (z) for i = 1, . . , n. The indicial equation at ∞ is given by X(X + 1) · · · (X + n − 1) − a1 X(X + 1) · · · (X + n − 2) + · · · +(−1)n−1 an−1 X + (−1)n an = 0. Proof. Exercise. 2 (Cauchy). Suppose P ∈ C is a regular point of (5). Then there exist n C-linear independent Taylor series solutions f1 , .

When c ≥ a + b we get λ = 1 − c, μ = c − a − b, ν = b − a and the inequalities hold. Moreover, λ + μ + ν = 1 − 2a < 1. When c ≤ a + b we get λ = 1 − c, μ = a + b − c, ν = b − a and the inequalities hold. Moreover, λ + μ + ν = 1 + 2b − 2c < 1. Case iii) 0 ≤ c < a ≤ b < 1. We take a = a, b = b, c = c + 1. Then, λ = c, μ = c + 1 − a − b, ν = b − a and the inequalities are readily veriﬁed. Moreover, λ + μ + ν = 1 + 2c − 2a < 1. As to uniqueness we note that an integral shift in the a, b, c such that the corresponding values of λ, μ, ν stay below 1 necessarily gives the substitutions of the form λ → 1 − λ, μ → 1 − μ, ν → ν and similar ones where two of the parameters are replaced by 1 minus their value.

Then there exists a holomorphic power series g(z) with non-zero constant term such that z ρ g(z) is a solution of (3). Let ρ1 , . . , ρn be the set of local exponents ordered in such a way that exponents which diﬀer by an integer occur in decreasing order. Then there exists a nilpotent n × n matrix N , and functions g1 , . . , gn , analytic near z = 0 with gi (0) = 0, such that (z ρ1 g1 , . . , z ρn gn )z N is a basis of solutions of (3). Moreover, Nij = 0 implies i = j and ρi − ρj ∈ Z≥0 . 7.

### Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005 by Rolf-Peter Holzapfel, Muhammed Uludag, M. Yoshida

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