Arithmetic properties of the values some hypergeometric 𝐹-series
https://doi.org/10.22405/2226-8383-2021-22-2-519-527
Abstract
Generalized hypergeometric series are of the form
$$𝑓(𝑧) =∞Σ︁𝑛=0((𝑎1)𝑛 . . . (𝑎𝑙)𝑛)/((𝑏1)𝑛 . . . (𝑏𝑚)𝑛)𝑧𝑛$$
If 𝑙 < 𝑚 and if the parameters are rational, they are closely related to Siegel’s 𝐸-functions. If 𝑙 = 𝑚 and if the parameters are rational, they are 𝐺-functions. For 𝑙 > 𝑚 and if the parameters
are rational, they are 𝐹-series.
The arithmetic properties values of generalized hypergeometric series is an actual problem with a long history. We shall only mention Siegel C. L., Shidlovskii A. B., Salikhov V. Kh.,
Beukers F., Brownawell W. D., Heckman G., Galochkin A. I., Oleinikov V. A., Ivankov P. L., Gorelov V. A., Chirskii V. G., Zudilin W., Matala–Aho T. etc. We consider the so–called 𝐹-series. Chirskii V.G. proved the infinitу algebraic independence
of the corresponding values.
Here we obtain lower estimates of polynomials and linear forms in the values of these series and their derivatives in a concrete 𝑝-adic field.
About the Author
Vaskes Ankhel Khorkheevich Munos
Moscow State Pedagogical University
Russian Federation
Russian Federation
graduate student
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Review
For citations:
Munos V.A. Arithmetic properties of the values some hypergeometric 𝐹-series. Chebyshevskii Sbornik. 2021;22(2):519-527. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-2-519-527
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