Infinite linear and algebraic independence of values of 𝐹-series at polyadic Liouvillea points

This paper proves infinite linear and algebraic independence of the values of 𝐹-series at polyadic Liouville points using a modification of the generalised Siegel-Shidlovskii method. 𝐹-
series have form 𝑓𝑛 =Σ︀∞𝑛=0 𝑎𝑛𝑛!𝑧𝑛 whose coefficients 𝑎𝑛 satisfy some arithmetic properties.
These series converge in the field Q𝑝 of 𝑝-adic numbers and their algebraic extensions K𝑣.
Polyadic number is a series of the form Σ︀∞𝑛=0 𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z. Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (𝑝, 𝑞) with 𝑞 > 1 such that 0 <|𝑥 − 𝑝/𝑞|< 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼−𝐴|𝑝 < 𝐴−𝐷. Infinite linear (algebraic) independence means that for any nonzero linear form
(any nonzero polynomial) there are infinitely many primes 𝑝 and valuations 𝑣 extending 𝑝-adic valuation to an algebraic number field K with the following property: the result of substitution
in the considered linear form (polynomial) of the values of 𝐹 - of series instead of variables is a nonzero element of the field.
Previously, only the existence of at least one prime number 𝑝 with the properties listed above was proved.

1. Galochkin A. I. 1970, “On algebraic independence of the values of 𝐸 – functions at certain transcendental points“, Moscow University Mathematics Bulletin, iss. 1, no. 5, pp. 58-63.

2. Bombieri E., 1981, “On 𝐺 – functions“, Recent Progress in Analytic Number Theory, Academic Press (London), vol. 2, pp. 1-68.

3. Shidlovskii, A. B. 1989, “Transcendental numbers“, Studies in mathematics, Walter de Gruyter (Berlin, New York), vol. 12

4. Chirskii V. G. 2019, “Product formula, global relations and polyadic integers“, Russian Journal of Mathematical Physics, Maik Nauka/Interperiodica Publishing (Russian Federation), vol. 26, no. 2, pp. 175-184.

5. Chirskii, V. G. 2015, “Arithmetic properties of Euler series“, Moscow University Mathematics Bulletin, Allerton Press Inc.(United States), vol. 70, no. 1, pp. 41-43.

6. Chirskii, V. G. 2017, “Arithmetic properties of polyadic series with periodic coefficients“, Izvestiya Mathematics, American Mathematical Society (United States), vol. 81, no. 2, pp.

7. -461.

8. Chirskii, V. G. 2018, “Arithmetic properties of generalized hypergeometric F-series“, Doklady Mathematics, Maik Nauka/Interperiodica Publishing (Russian Federation), vol. 98, no. 3, pp. 589-591.

9. Chirskii, V. G. 2014, “Arithmetic properties of polyadic series with periodic coefficients“, Doklady Mathematics, Maik Nauka/Interperiodica Publishing (Russian Federation), vol. 90, no. 3, pp. 766-768.

10. Chirskii, V. G. 2014, “On the arithmetic properties of generalized hypergeometric series with irrational parameters“, Izvestiya Mathematics, American Mathematical Society (United States), vol. 78, no. 6, pp. 1244-1260.

11. Chirskii, V. G. 2011, “Estimates of linear forms and polynomials in polyadic numbers”, Chebyshevskii Sb., 12:4, pp.129-133.

12. Chirskii, V. G. 1990, “Global relations”, Mat. Zametki, vol. 48, no. 2, pp. 123–127

13. Chirskii, V. G. 2014, “Arithmetic properties of polyadic series with periodic coefficients“, Doklady Mathematics, Maik Nauka/Interperiodica Publishing (Russian Federation), vol. 90, no. 3, pp. 766-768.

14. Chirskii, V.,G. 2017, “Arithmetic properties of polyadic series with periodic coefficients“, Izvestiya Mathematics, American Mathematical Society (United States), vol. 81, no. 2, pp.

15. -461.

16. Chirskii, V. G. 2016, “On transformations of periodic sequences”, Chebyshevskii Sb., vol. 17, no. 3, pp. 191–196.

17. Chirskii, V. G. 2015, “Arithmetic properties of polyadic integers”, Chebyshevskii Sb., vol. 16, no. 1, pp. 254–264.

18. Andr´e Y. 2000, “S´eries Gevrey de type arithm´etique“, Annals of Mathematics, vol. 151, pp. 705-740

19. Chirskii V. G. Arithmetic properties of Generalized Hypergeometric Series // Russian Journal of Mathematical Physics, Maik Nauka/Interperiodica Publishing (Russian Federation). 2020. Vol. 27, №2, pp. 175-184.

20. Matala–Aho T. & Zudilin W. 2018, “Euler factorial series and global relations“, J. Number Theory, vol. 186, pp. 202-210.

21. Bertrand, D., Chirskii, V. G. & Yebbou, Y. 2004, “Effective estimates for global relations on Euler-type series“, Ann. Fac. Sci. Toulouse, vol. XIII, no. 2, pp. 241-260.