The Lévy Constants Of Some Transcendental Numbers

In this paper, we mainly study the Lévy constants of transcendental numbers.Firstly, we introduce some backgrounds of the continued fractions and the current studying of the Lévy constants. Secondly, we introduce the basic knowledge of continued fractions and transcendental numbers, also give the corresponding proof. Finally, we get the main conclusion by proving some lemmas : for anyγwhich is in the interval there exist non-denumerably many, pairwise not equivalent transcendental numbersαsuch that the Lévy constants ofαare equal toγ.We apply the continuant which is an important technical tool in the process of proof. Let ,since we can prove the conclusion by two parts. On one hand, since in this case, we may discuss in the interval ( wc ,wd ). We can construct quasiperiodic continued fractionαand prove the necessary condition in whichαis an algebraic number. Then using proof by contradiction, we prove that there existγin the interval ( w_c ,w_d ) such that the Lévy constant of transcendental numbersαwhich are non-denumerably many and pairwise not equivalent are equal toγ. On the other hand, we prove that there exist non-denumerably many, pairwise not equivalent transcendental numbersαsuch thatβ(α) = log ( (1 + 5 )2).