ALGEBRAIC INDEPENDENCE OF CERTAIN ALMOST POLYADIC SERIES

The paper describes the arithmetic nature of the values at integer points of series from the so-called class of \(F\)--series which constitute a solution of a system of linear differential equations with coefficients --- rational functions in z.

We consider a subclass of the series consisting of the series of the form

\sum_{n=0}^\infty a_n\cdot n!\; z^n

where \(a_n\in\mathbb Q\), \(|a_n|\leq e^{c_1 n}\), \(n=0,1,\ldots\) with some constant \(c_1\). Besides there exists a sequence of positive integers \(d_n\) such that \(d_n\; a_k\in\mathbb Z\), \(k=0,\ldots,n\) and \(d_n=d_{0,n} d_n\), \(d_{0,n}\in\mathbb N\), \mbox{\(n=0,1,\ldots,d\in\mathbb N\)} and for any \(n\) the number \(d_{0,n}\) is divisible only by primes \(p\) such that \(p\leqslant c_2 n\). Moreover

$$ord_p n \leq c_3\left(\log_p n+\frac{n}{p^2}\right).$$

We say then that the considered series belongs to the class \(F(\mathbb{Q},c_1,c_2,c_3,d)\).
Such series converge at a point \(z\in\mathbb Z\), \(z\ne 0\) in the field \(\mathbb Q_p\) for almost all primes \(p\).

The direct product of the rings \(\mathbb Z_p\) of \(p\)--adic integers over all primes \(p\) is called the ring of polyadic integers. It's elements have the form
$$\mathfrak{a} = \sum_{n=0}^\infty a_n\cdot n!,\quad a_n\in\mathbb Z$$

and they can be considered as vectors with coordinates \(\mathfrak{a}^{(p)}\) which are equal to the sum of the series \(\mathfrak{a}\) in the field \(\mathbb Q_p\) (This direct product is infinite).

For any polynomial \(P(x)\) with integer coefficients we define \(P(\mathfrak{a})\) as the vector with coordinates \(P(\mathfrak{a}^{(p)})\) in \(\mathbb Q_p\). According to the classification, described in V. G. Chirskii's works we call polyadic numbers \(\mathfrak{a}_1,\ldots,\mathfrak{a}_m\) infinitely algebraically independent, if for any nonzero polynomial \(P(x_1,\ldots,x_m)\) with integer coefficients there exist infinitely many primes \(p\) such that
$$P\left(\mathfrak{a}_1^{(p)},\ldots,\mathfrak{a}_m^{(p)}\right)\ne 0 $$
in \(\mathbb Q_p\).

The present paper states that if the considered \(F\)--series \(f_1,\ldots,f_m\) satisfy a system of differential equations of the form
$$P_{1,i}y_i^\prime + P_{0,i}y_i = Q_i, i=1,\ldots,m$$
where the coefficients \(P_{0,i}, P_{1,i}, Q_i\) are rational functions in \(z\) and if \(\xi\in\mathbb Z\), \(\xi\ne 0\), \(\xi\) is not a pole of any of these functions and if
$$\exp\left(\int\left(\frac{P_{0,i}(z)}{P_{1,i}(z)}-\frac{P_{0,j}(z)}{P_{1,j}(z)}\right)dz\right)\not\in\mathbb C(z)$$
then
\(f_1(\xi),\ldots,f_m(\xi)\) are infinitely algebraically independent almost polyadic numbers.

For the proof we use a modification of the Siegel-Shidlovsky's method and V. G. Chirskii's. Salikhov's approach to prove the algebraic independence of functions, constituting a solution of the above system of differential equations.

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