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