Skip to main content
SpringerLink
Log in

Arithmetic Properties of Generalized Hypergeometric F-Series

  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

In the paper, using a generalization of the Siegel-Shidlovskii method in the theory of transcendental numbers, we prove the infinite algebraic independence of elements, generated by generalized hypergeometric series, of direct products of the fields of \(\mathbb{K}_v\)-completions of an algebraic number field\(\mathbb{K}\) of finite degree over the field of rational numbers with respect to a valuation v of the field \(\mathbb{K}\) extending the p-adic valuation of the field ℚ over all primes p except for finitely many of them.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. G. Chirskii, “Product Formula, Global Relations and Polyadic Integers,” Russ. J. Math. Phys. 26 3), 286–305 (2019).

    Article  MathSciNet  Google Scholar 

  2. V. G. Chirskii, “Arithmetic Properties of Generalized Hypergeometric F-Series,” Dokl. Akad. Nauk 483 3), 257–259 (2018) [Dokl. Math. 98 (3), 589–591 (2018)].

    MATH  Google Scholar 

  3. A. B. Shidlovskii, Transcendental Numbers (Nauka, Moscow, 1987; Walter de Gruyter & Co., Berlin, 1989).

    MATH  Google Scholar 

  4. V. Kh. Salikhov, “Algebraic Independence of Values of Hypergeometric F-Functions,” Dokl. Akad. Nauk 307 2), 284–286 (1989) [Dokl. Math. 40 (1), 71–74 (1990)].

    Google Scholar 

  5. V. Kh. Salikhov, “Irreducibility of Hypergeometric Equations, and Algebraic Independence of Values of F-Functions,” Acta Arithm. 53, 453–471 (1990) [in Russian].

    Article  MathSciNet  Google Scholar 

  6. V. Kh. Salikhov, “A Criterion for the Algebraic Independence of Values of a Class of Hypergeometric F-Functions,” Mat. Sb. 181 2), 189–211 (1990) [Math. USSR-Sb. 69 (1), 203–226 (1991)].

    Google Scholar 

  7. V. G. Sprindzhuk, “The Irrationality of the Values of Certain Transcendental Functions,” Izv. AN SSSR. Ser. Mat. 32 10), 93–107 (1968) [Math. USSR-Izv. 2 1), 89–104 (1968)].

    Google Scholar 

  8. A. I. Galochkin, “On Estimates, Unimprovable with Respect to Height, of Some Linear Forms,” Mat. Sb. 124 (7), 3 (7), 416–430 (1984) [Math. USSR-Sb. 52 (2), 407–421 (1985)].

    MathSciNet  MATH  Google Scholar 

  9. A. N. Korobov, “Estimates of Some Linear Forms,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. (6), 36–40 (1981) [in Russian].

    MathSciNet  MATH  Google Scholar 

  10. P. L. Ivankov, “On Linear Independence of the Values of Entire Hypergeometric Functions with Irrational Parameters,” Sib. Mat. Zh. 34 5), 53–62 (1993) [Siberian Math. J. 34 (5), 839–847 (1993)].

    Article  MathSciNet  Google Scholar 

  11. A. I. Galochkin, “Estimates from Below of Polynomials in the Values of Analytic Functions of a Certain Class,” Mat. Sb. 95 (137), 3 (11), 396–417 (1974) [Math. USSR-Sb. 24 (3), 385–407 (1974)].

    MathSciNet  Google Scholar 

  12. G. V. Chudnovsky, “On Applications of Diophantine Approximations,” Proc. Natl. Acad. Sci. USA 81, 7261–7265 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  13. E. Bombieri, “On G-Functions,” Rec. Progr. in Anal. Num. Theory 2, 1–68 (1981).

    MathSciNet  MATH  Google Scholar 

  14. Y. André, “G-Functions and Geometry,” Aspects of Math. 476, 95–125 (1996).

    MATH  Google Scholar 

  15. M. Hata, “Irrationality Measures of the Values of Hypergeometric Functions,” Acta Arithm. 60, 335–347 (1992).

    Article  MathSciNet  Google Scholar 

  16. Yu. V. Nesterenko, “On the Algebraic Independence of the Values of F-Functions Satisfying Nonho-mogeneous Linear Differential Equations,” Mat. Zametki 5 5), 587–599 (1969) [Math. Notes 5 (5), 352–358 (1969)].

    MathSciNet  Google Scholar 

  17. D. Bertrand, V. Chirskii, and J. Yebbou, “Effective Estimates for Global Relations on Euler-Type Series,” Ann. Fac. Sci. Toulouse 13 2), 241–260 (2004).

    Article  MathSciNet  Google Scholar 

  18. V. G. Chirskii, “On the Arithmetic Properties of Generalized Hypergeometric Series with Irrational Parameters,” Izv. RAN. Ser. Mat. 78 6), 193–210 (2014) [Izv. Math. 78 (6), 1244–1260 (2014)].

    Article  MathSciNet  Google Scholar 

  19. V. G. Chirskii, “Arithmetic Properties of Polyadic Series with Periodic Coefficients,” Dokl. Akad. Nauk 459 6), 677–679 (2014) [Dokl. Math. 90 (3), 766–768 (2014)].

    MathSciNet  MATH  Google Scholar 

  20. V. G. Chirskii, “Arithmetic Properties of Polyadic Series with Periodic Coefficients,” Izv. RAN. Ser. Mat. 81 2), 215–232 (2017) [Izv. Math. 81 (2), 444–461 (2017)].

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgment

The author is grateful to the corresponding member of the RAS Yu. V. Nesterenko for useful discussions.

Author information

Authors and Affiliations

  1. Dept. of Mech. and Math., Moscow State University, Moscow, GSP-1, 119991, Russia

    V. G. Chirskii

  2. Moscow State Pedagogical University, M. Pirogovskaya st., 1, Moscow, 119882, Russia

    V. G. Chirskii

  3. Russian Academy of National Economy and Public Administration under the President of the Russian Federation, pr. Vernadskogo, 84, Moscow, 119606, Russia

    V. G. Chirskii

Authors
  1. V. G. Chirskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. G. Chirskii.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chirskii, V.G. Arithmetic Properties of Generalized Hypergeometric F-Series. Russ. J. Math. Phys. 27, 175–184 (2020). https://doi.org/10.1134/S106192082002003X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S106192082002003X

Search

Navigation