One application on hypergeometic series and values of 𝑔-adic functions algebraic independence investigation methods
https://doi.org/10.22405/2226-8383-2021-22-2-528-535
Abstract
The article takes a look at transcendence and algebraic independence problems, introduces statements and proofs of theorems for some kinds of elements from direct product of 𝑝-adic fields and polynomial estimation theorem. Let Q𝑝 be the 𝑝-adic completion of Q, Ω𝑝 be the completion of the algebraic closure of Q𝑝, 𝑔 = 𝑝1𝑝2 . . . 𝑝𝑛 be a composition of separate prime numbers, Q𝑔 be the 𝑔-adic completion of Q, in other words Q𝑝1 ⊕. . .⊕Q𝑝𝑛. The ring Ω𝑔 ∼=Ω𝑝1⊕...⊕Ω𝑝𝑛, a
subring Q𝑔, transcendence and algebraic independence over Q𝑔 are under consideration. Also, hypergeometric series
$$𝑓(𝑧) =∞Σ︁𝑗=0((𝛾1)𝑗 . . . (𝛾𝑟)𝑗)/((𝛽1)𝑗 . . . (𝛽𝑠)𝑗)(𝑧𝑡)^𝑡𝑗 $$,
and their formal derivatives are under consideration. Sufficient conditions are obtained under which the values of the series 𝑓(𝛼) and formal derivatives satisfy global relation of algebraic
independence, if 𝛼 =∞Σ︀𝑗=0 𝑎_𝑗𝑔^(𝑟_𝑗), where 𝑎𝑗 ∈ Z𝑔, and non-negative rationals 𝑟𝑗 increase strictly unbounded.
About the Author
Aleksei Sergeevich Samsonov
Moscow State Pedagogical University
Russian Federation
Russian Federation
graduate student
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Review
For citations:
Samsonov A.S. One application on hypergeometic series and values of 𝑔-adic functions algebraic independence investigation methods. Chebyshevskii Sbornik. 2021;22(2):528-535. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-2-528-535
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