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

##### MSC:
 11J89 Transcendence theory of elliptic and abelian functions 11J82 Measures of irrationality and of transcendence
Zbl 0518.41014
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##### 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
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