Infinite algebraic independence of some almost polyadic numbers
https://doi.org/10.22405/2226-8383-2024-25-3-365-372
Abstract
The paper considers 𝐹-series 𝑓𝑖,𝑗(𝑧) = Σ︀∞
𝑛=0 (𝛼𝑖)𝑛(𝛽𝑗𝑧)𝑛, where 𝛼𝑖, 𝛽𝑗 are some rational
numbers. These series satisfy a system of first-order linear differential equations with coefficients from C(𝑧). Using previous results obtained using the approach proposed in one of the works of V.Kh. Salikhov, the algebraic independence of these series over C(𝑧) is established. Application of the general theorem on the arithmetic properties of 𝐹-series from the works of V.G. Chirsky, allows us to assert the infinite algebraic independence of the values of these series. This means that for any polynomial 𝑃 (𝑥1,1, . . . , 𝑥𝑚,𝑛) with integer coefficients other than the identical zero and any integer 𝜉 ̸= 0, there is an infinite set of prime numbers 𝑝 such that in the field Q𝑝 the inequality
Here the symbols 𝑓(𝑝) 𝑖𝑗 (𝜉) denote the sums of the
series Σ︀∞ 𝑛=0 (𝛼𝑖)𝑛 (𝛽𝑗𝜉)𝑛 in the field Q𝑝.
About the Author
Vladimir Yur’evich Matveev
Russian Presidential Academy of National Economy and Public Administration
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
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Review
For citations:
Matveev V.Yu. Infinite algebraic independence of some almost polyadic numbers. Chebyshevskii Sbornik. 2024;25(3):365-372. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-3-365-372
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