Linear independence of values of hypergeometric functions and arithmetic Gevrey series

Summary · qwen2.5:32b

The study establishes new linear independence results for values of generalized hypergeometric functions ${}_pF_q$ at multiple algebraic points over various number fields, using a uniform construction of Padé approximants and a novel non-vanishing argument for Hermite-type Wronskians, extending previous single-point findings to multi-points in both complex and $p$-adic settings.

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Discovering new linear independence results for hypergeometric functions and arithmetic Gevrey series

Excerpt

arXiv:2511.06534v2 Announce Type: replace Abstract: We prove new linear independence results for the values of generalized hypergeometric functions ${}_pF_q$ at several distinct algebraic points, over suitable algebraic number fields. Our approach provides a uniform construction of Pad\'{e} approximants of type II, together with a novel non-vanishing argument for generalized Wronskians of Hermite type. This method applies uniformly across all parameter regimes. Even in the case $p = q+1$, we extend known results from single-point to multi-points settings over general number fields, in both complex and $p$-adic settings. When $p < q+1$, we establish linear independence results over arbitrary number fields; and for $p > q+1$, we confirm that the values do not satisfy global linear relations in the $p$-adic setting in a framework of arithmetic Gevrey series. The results generalize and strengthen earlier works, demonstrating the flexibility of our Pad\'{e} construction for families of contiguous hypergeometric functions, through a new non-vanishing proof for the determinant, that is crucial for the universality.