Polyadic Liouville numbers

V. G. Chirskii

Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 245-255

Résumé

We study here polyadic Liouville numbers, which are involved in a series of recent papers. The author considered the series $$ f_{0}(\lambda)=\sum_{n=0}^\infty (\lambda)_{n}\lambda^{n}, f_{1}(\lambda)=\sum_{n=0}^\infty (\lambda +1)_{n}\lambda^{n},$$ where $ \lambda $ is a certain polyadic Liouville number. The series considered converge in any field $ \mathbb{\mathrm{Q}}_p $. Here $(\gamma)_{n}$ denotes Pochhammer symbol, i.e. $(\gamma)_{0}=1$, and for $n\geq 1$ we have$ (\gamma)_{n}=\gamma(\gamma+1)\dots(\gamma+n-1)$. The values of these series were also calculated at polyadic Liouville number. The canonic expansion of a polyadic number $\lambda$ is of the form $$ \lambda= \sum_{n=0}^\infty a_{n} n!, a_{n}\in\mathbb{\mathrm{Z}}, 0\leq a_{n}\leq n.$$ This series converges in any field of $p$-adic numbers $ \mathbb{\mathrm{Q}}_p $. We call a polyadic number $\lambda$ a polyadic Liouville number, if for any $n$ and $P$ there exists a positive integer $A$ such that for all primes $p$, satisfying $p\leq P$ the inequality $$\left|\lambda -A \right|_{p}^{-n}$$ holds. The paper gives a simple proof that the Liouville polyadic number is transcendental in any field $\mathbb{\mathrm{Q}}_p.$ In other words,the Liouville polyadic number is globally transcendental. We prove here a theorem on approximations of a set of $p$-adic numbers and it's corollary — a sufficient condition of the algebraic independence of a set of $p$-adic numbers. We also present a theorem on global algebraic independence of polyadic numbers.

Détail

Keywords: polyadic number, polyadic Liouville number.

Bibliographie

[1] Andrei B. Shidlovskii, Transcendental Numbers, W.de Gruyter, Berlin-New York, 1989, 467 pp. | MR | MR

[2] Adams W., “On the algebraic independence of certain Liouville numbers”, J. Pure and Appl. Algebra, 13 (1978), 41–47 | DOI | MR | Zbl

[3] Waldschmidt M., “Independance algebrique de nombres de Liouville”, Lect. Notes Math., 1415, 1990, 225–235 | DOI | MR | Zbl

[4] Chiskii V. G., “Arithmetic Properties of Euler-Type Series with a Liouvillean Polyadic Parameter”, Dokl. Math., 102 (2020), 412–413 | DOI | MR

[5] Chirskii V. G., “Arifmeticheskie svoistva znachenii v poliadicheskoi liuvillevoi tochke ryadov eilerova tipa s poliadicheskim liuvillevym parametrom”, Chebyshevskii sbornik, 22:2 (2021), 304–312 | MR | Zbl

[6] Chirskii V. G., “Obobschenie ponyatiya globalnogo sootnosheniya”, Trudy po teorii chisel, Zap. nauchn. sem. POMI, 322, POMI, Spb., 2005, 220–232

[7] Chirskii V. G., “O ryadakh, algebraicheski nezavisimykh vo vsekh lokalnykh polyakh”, Vestn. Mosk. un-ta. Ser. Matem., mekh., 1994, no. 3, 93–95 | Zbl

[8] Chirskii V. G., “Product Formula, Global Relations and Polyadic Integers”, Russ. J. Math. Phys., 26:3 (2019), 286–305 | DOI | MR | Zbl

[9] Chirskii V. G., “Arithmetic properties of generalized hypergeometric $F$– series”, Russ. J. Math. Phys., 27:2 (2020), 175–184 | DOI | MR | Zbl

[10] Yudenkova E. Yu., “Arifmeticheskie svoistva ryadov nekotorykh klassov v poliadicheskoi liuvillevoi tochke”, Chebyshevskii sbornik, 22:2 (2021), 536–542 | MR | Zbl

[11] Yudenkova E. Yu., “Beskonechnaya lineinaya i algebraicheskaya nezavisimost znachenii F-ryadov v poliadicheskikh liuvillevykh tochkakh”, Chebyshevskii sbornik, 22:2 (2021), 334–346 | MR | Zbl

[12] Matveev V. Yu., “Algebraicheskaya nezavisimost nekotorykh pochti poliadicheskikh ryadov”, Chebyshevskii sbornik, 17:3 (2018), 156–167

[13] Matveev V. Yu., “Svoistva elementov pryamykh proizvedenii polei”, Chebyshevskii sbornik, 20:2 (2019), 383–390 | MR | Zbl

[14] Krupitsyn E. S., “Arifmeticheskie svoistva ryadov nekotorykh klassov”, Chebyshevskii sbornik, 20:2 (2019), 374–382 | MR

[15] Samsonov A. S., “Arifmeticheskie svoistva elementov pryamykh proizvedenii p-adicheskikh polei, II”, Chebyshevskii sbornik, 22:2 (2021), 334–346 | MR

[16] Munos Vaskes A. Kh., “Arifmeticheskie svoistva nekotorykh gipergeometricheskikh F-ryadov”, Chebyshevskii sbornik, 22:2 (2021), 519–527 | MR | Zbl