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On certain arithmetical power series. (Sur certaines séries entières arithmétiques.) (French) Zbl 0512.10004

Groupe Étude Anal. Ultramétrique, 9e Année, 1981/82, No. 1, Exp. No. 16, 2 p. (1983).
The author introduces the idea of global contraction \(\mathbb Z[[x_1,\ldots,x_r]]\rightarrow \mathbb Z[[t]]\) which consists of replacing \(x_1\cdots x_n)^n\) by \(t^n\) and replacing all other monomials by \(0\). An important feature is that property \((\Gamma)\) saying that \(f(x)/f(x^p)\) is a polynomial mod \(p\) is conserved under global contraction. The author then observes that the generating function of the numbers \(1,3,19,\ldots\) and \(1,5,73,\ldots\) occurring in his irrationality proof of \(\zeta(2)\) and \(\zeta(3)\) can be obtained by contraction of certain power series in 3 and 4 variables. This procedure is quite reminiscent of taking periods of certain forms on algebraic curves and surfaces. [See the reviewer, Approximations diophantiennes et nombres transcendants, Colloq. Luminy/Fr. 1982, Prog. Math. 31, 47–66 (1983; Zbl 0518.10040), and together with C. Peters, “A family of \(K3\) surfaces and irrationality of \(\zeta(3)\)”, J. Reine Angew. Math. 351, 42–54 (1984; Zbl 0541.14007)].

MSC:

11B83 Special sequences and polynomials
11B37 Recurrences
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