Abstract
We present solutions for general theorems regarding algebraic independence of solutions of hypergeometric equation ensembles and the values of these solutions at algebraic points. The conditions of the theorems are necessary and sufficient. Furthermore, errors in theorems from F. Beukers and others are corrected.
Highlights
Let Z+ = N ∪ {0}, Z− = Z \ N, let A be the set of all algebraic numbers, M (q, K )be the set of all matrices of size q × q with elements from a ring K, GL(q, K ) be the set of j all invertible matrices in M (q, K ), C[z±1 ] be the ring C[z, z−1 ], δi be the Kronecker delta, Fhv1, . . . , vn i be the smallest differential field containing the field F and the functions v1, . . . , vn
Necessary and sufficient conditions for the irreducibility of the equations L(~ν; ~λ; zq−l ) y = 0 were obtained by V.Kh
In the theorems of this article, we find necessary and sufficient conditions for the algebraic independence of solutions of “almost all” generalized hypergeometric equations, except those whose sets of parameters are, in a sense, a set of measure zero
Summary
The function l φq (z) satisfies the (generalized) hypergeometric differential equation:. Putting in the main statement of the article [9] α1 = i/2, α2 = 2i, b1 = i, 4λ ∈ Q \ Z, we obtain a contradiction with identity (7) This error in article [9] ( repeated in [5,12]), comes from the fact that when proving its basic statement, the possibility of cogredience and contragredience of equations for hypergeometric functions with equal values of q, but different l is not considered. In Example 1, we have that if 2λ ∈ C \ Z, α ∈ C, and Φ1 , Φ2 are fundamental matrices of Equations (4) and (5), in which ν → 2λ + 1, μ → λ + 1/2, z → 2iz, corresponding to sets of functions, respectively,. ~λ1 − λ1,j ∼ ±2(~λ2 − λ2,1 ), 1 6 j 6 2, 2ν1,1 − λ1,1 − λ1,2 ∈ Z
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