Geodesic
Arithmetic properties of Euler-type series with a Liouvillian polyadic parameter
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 494 (2020), pp. 68-70
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper states that, for any nonzero linear form $h_0f_0(1)+h_1f_1(1)$ with integer coefficients $h_0,h_1$, there exist infinitely many $p$-adic fields where this form does not vanish. Here, $f_0(1)=\sum\limits_{n=0}^\infty (\lambda)_n$, $f_1(1)=\sum\limits_{n=0}^\infty(\lambda+1)_n$, $\lambda$ where $\lambda$ is a Liouvillian polyadic number and $(\lambda)_n$ stands for the Pochhammer symbol. This result shows the possibility of studying the arithmetic properties of values of hypergeometric series with transcendental parameters.
Keywords: infinite linear independence, polyadic numbers
Mots-clés : Hermite–Padé approximations. 
@article{DANMA_2020_494_a15,
     author = {V. G. Chirskii},
     title = {Arithmetic properties of {Euler-type} series with a {Liouvillian} polyadic parameter},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {68--70},
     year = {2020},
     volume = {494},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_494_a15/}
}
TY  - JOUR
AU  - V. G. Chirskii
TI  - Arithmetic properties of Euler-type series with a Liouvillian polyadic parameter
JO  - Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
PY  - 2020
SP  - 68
EP  - 70
VL  - 494
UR  - http://geodesic.mathdoc.fr/item/DANMA_2020_494_a15/
LA  - ru
ID  - DANMA_2020_494_a15
ER  - 
%0 Journal Article
%A V. G. Chirskii
%T Arithmetic properties of Euler-type series with a Liouvillian polyadic parameter
%J Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
%D 2020
%P 68-70
%V 494
%U http://geodesic.mathdoc.fr/item/DANMA_2020_494_a15/
%G ru
%F DANMA_2020_494_a15
V. G. Chirskii. Arithmetic properties of Euler-type series with a Liouvillian polyadic parameter. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 494 (2020), pp. 68-70. http://geodesic.mathdoc.fr/item/DANMA_2020_494_a15/

[1] Postnikov A.G., Vvedenie v analiticheskuyu teoriyu chisel, Nauka, M., 1971, 416 pp. | MR

[2] Chirskii V.G., “Product formula, global relations and polyadic integers”, Russian Journal of Mathematical Physics, 26:3 (2019), 286–305 | DOI | MR | Zbl

[3] Bertrand D., Chirskii V., Yebbou J., “Effective Estimates for Global Relations on Euler-Type Series”, Ann. Fac. Sci. Toulouse, 13:2 (2004), 241–260 | DOI | MR | Zbl

[4] Matala-aho T., Zudilin W., “Euler factorial series and global relations”, J. Number Theory, 186 (2018), 202–210 | DOI | MR | Zbl

[5] Shidlovskii A.B., Transtsendentnye chisla, Nauka, M., 1987, 448 pp. | MR

[6] Salikhov V.Kh., “Kriterii algebraicheskoi nezavisimosti znachenii odnogo klassa gipergeometricheskikh E-funktsii”, Mat. sbornik, 181:2 (1990), 189–211

[7] Bombieri E., “On G-functions”, Recent progress in Analytic Number Theory, v. 2, Academic Press, L., 1981, 1–68 | MR

[8] Yves Andre, G-functions and Geometry, Aspects of Math., 13, Vieweg, 1989 | MR | Zbl

[9] Flicker Yu., “On p -adic G-functions”, J. London Math. Soc., 15:3 (1977), 395–402 | DOI | MR | Zbl

[10] Chirskii V.G., “Arithmetic properties of Generalized Hypergeometric Series”, Russian J. Mathematical Physics, 27:2 (2020), 175–184 | DOI | MR | Zbl

[11] Chudnovsky G.V., “On applications of Diophantine approximations”, Proc. Natl. Acad. Sci. USA, 81 (1985), 7261–7265 | DOI | MR

[12] Ivankov P.L., “O lineinoi nezavisimosti znachenii nekotorykh gipergeometricheskikh funktsii s irratsionalnymi parametrami”, Sib. mat. zhurn., 34:1 (1993), 53–62 | Zbl

[13] Chirskii V.G., “Ob arifmeticheskikh svoistvakh obobschennykh gipergeometricheskikh ryadov s irratsionalnymi parametrami”, Izvestiya RAN. Ser. matem., 78:6 (2014), 193–210 | DOI | MR | Zbl

[14] Nesterenko Yu.V., “Priblizheniya Ermita-Pade obobschennykh gipergeometricheskikh funktsii”, Mat. sb., 185:3 (1994), 39–72 | Zbl