Recurrent sequences and the measure of irrationality of values of elliptic integrals. (English. Russian original) Zbl 0917.11031

Zudilin, V. V.

Math. Notes 61, No. 5, 657-661 (1997); translation from Mat. Zametki 61, No. 5, 785-789 (1997).

The following main result is proved: Let \(\wp(z)\) be the Weierstrass function with invariants \(g_2=4b\), \(g_3=-1\), where \(b\in\mathbb Z\), \(b \geq 8\), and let \(y_2,y_3\) with \(y_2<y_3\) be positive roots of the polynomial \(4y^3-g_2y-g_3\). Then \(\alpha= \int_0^{y_2} d\xi/ \sqrt {4 \xi^3- g_2\xi-g_3}\) (the distance from the center of the basic parallelogram of periods to one of the nearest zeros of \(\wp(z))\) has the measure of irrationality \(\mu=1 -(1-\log y_2)/ (1-\log y_3)\), that is, for any \(\varepsilon >0\), there exists \(q_0=q_0 (\varepsilon)\) such that \(|\alpha-p/q |\geq q^{-(\mu+ \varepsilon)}\) for all integers \(q\geq q_0\) and for all \(p\in\mathbb Z\). In order to prove this result, the author generally considers the third-order recurrent sequence \[ (n+1)t_{n+1}- b_1(n+1/2) t_n+b_2 nt_{n-1}-b_3(n-1/2) t_{n-2}=0, \quad n=2,3 \dots, \] with arbitrary initial data \(t_0,t_1, t_2\), where \(b_1,b_2,b_3 \in\mathbb Q\), and a result by G. V. Chudnovsky [Lect. Notes Math. 925, 299–322 (1982; Zbl 0518.41014)] is used.

Reviewer: Zhu Yaochen (Beijing)

MSC:

11J89	Transcendence theory of elliptic and abelian functions
11J82	Measures of irrationality and of transcendence

Keywords:

values of elliptic integrals; measure of irrationality; third-order recurrent sequence

Citations:

Zbl 0518.41014

References:

[1]	A. O. Gel’fond,Calculus of Finite Differences [in Russian], 3 d ed., Nauka, Moscow (1967).
[2]	G. V. Chudnovsky,Lecture Notes in Math.,925, 299–322 (1982).
[3]	V. V. Zudilin, ”On the measure of irrationality of values of elliptic integrals,” in:Materials of International Scientific Lectures on Analytic Number Theory and Applications [in Russian], Moscow State University, Moscow (1997), pp. 20–21. · Zbl 0917.11031
[4]	V. V. Zudilin, ”Difference equations and the measure of irrationality of numbers,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],318 (to appear). · Zbl 0910.11032