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Transcendence of certain infinite sums involving rational functions. (English) Zbl 1234.11094

Author’s summary: “The main result of this paper is the method which shows how to find by a simple technique the sum of the infinite series of the form \[ \sum_{n=0}^\infty \frac{\alpha^n P(n)}{Q(n)} \] where \(P(n)\) is a polynomial with algebraic coefficients, \(Q(n)\) is a reduced polynomial with integer coefficients and \(\alpha\in\overline{\mathbb Q}^*\). The connection between algebraic and transcendental infinite series and linear forms of logarithms via Baker’s theorem is included.”
Here the author uses only integration and some simple techniques to obtain some of the results of S. D. Adhikari, N. Saradha, T. N. Shorey and R. Tijdeman et al. [Indag. Math., New Ser. 12, 1–14 (2001; Zbl 0991.11043)].
Theorem. Let \((a,w)\in\mathbb N^2\), \(a\geq 2\), \(\{b_v\}_{v=1}^w\) be an increasing sequence of distinct positive integers and let \(b_v\leq a\) for all \(v=1,\dots,w\). Suppose that \(P(X)\in\overline{\mathbb Q}[X]\) be a polynomial and \(\alpha\in\overline{\mathbb Q}^*\). If
(1) \(|\alpha|>1\), then the sum of the series
\[ S_{P(X)}:=\sum_{n=0}^\infty \frac{P(n)}{\alpha^n \prod_{v=1}^w (an+b_v)} \] is a transcendental or a computable number;
(2) \(|\alpha|=1\), \(\deg P(n)< w+1\), then the sum of the series (1) is a transcendental number or zero.
The author also observes an error in Theorem 1.5 of the paper of Adhikari et al. cited above, but as one of the authors of that paper (Saradha) pointed out in his MR review of the present paper “the error is only in the statement of Theorem 1.5 and not in the proof”. In his review Saradha also states a corrected version of that theorem.

MSC:

11J81 Transcendence (general theory)

Citations:

Zbl 0991.11043
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References:

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