A determinantal approach to irrationality. (English) Zbl 1367.11062

Zudilin, Wadim

Constr. Approx. 45, No. 2, 301-310 (2017).

If we want to prove the irrationality of the real number \(\zeta\) then we usually consider sequences \(\{ p_n\}_{n=1}^\infty\) and \(\{ q_n\}_{n=1}^\infty\) of integers such that \(\lim_{n\to\infty}(q_n\zeta - p_n)=0\) where \(\zeta -\frac{ p_n}{q_n}\not= 0\) for all positive integers \(n\). The author of this interesting paper deals with the modification of this method and presents a new proof of the irrationality of the number \(\pi\) and other numbers. The method makes use of special integrals as well as properties of Hankel and Vandermonde determinants.

Reviewer: Jaroslav Hančl (Ostrava)

MSC:

11J72	Irrationality; linear independence over a field
30C10	Polynomials and rational functions of one complex variable

Keywords:

irrationality; number \(\pi\); Vandermonde and Hankel determinants

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