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Elementary Proof of Yu. V. Nesterenko Expansion of the Number Zeta(3) in Continued Fraction
Advances in Difference Equations volume 2010, Article number: 143521 (2010)
Yu. V. Nesterenko has proved that , , , , , , and for ; , , and , for His proof is based on some properties of hypergeometric functions. We give here an elementary direct proof of this result.
Applications of difference equations to the Number Theory have a long history. For example, one can find in this journal several articles connected with the mentioned applications (see [1–8]). The interest in this area increases after Apéry's discovery of irrationality of the number This paper is inspired by Yu.V. Nesterenko's work . My goal is to give an elementary direct proof of his expansion of the number in continued fraction. Let us consider a difference equation
with We denote by
the solutions of this equation with initial values
is a sequence of convergents of the continued fraction
Accoding to the famous result of R. Apéry ,
where and are solutions of difference equation
with initial values The equality (1.6) is equivalent to the equality
where Nesterenko in  has offered the following expansion of the number in continued fraction:
The halved convergents of continued fraction (1.10) compose a sequence containing convergents of continued fraction (1.8). I give an elementary proof of Yu.V. Nesterenko expansion in Section 2.
2. Elementary Proof of Yu. V. Nesterenko Expansion
Instead of expansion (1.10) with (1.11), it is more convenient for us to prove the equivalent expansion
Furthermore, to avoid confusion in notations, we denote below for the fraction (2.1) by Let
where values are specified in (1.9), and Then
where and values are specified in (2.2), (1.12), and (1.13). We calculate first and for
Since it follows from (2.2) that
We want to to prove that if then
Note that if then (2.12) follows from (2.6)–(2.10). Therefore, we can consider only Let us consider the following difference equations:
with Then , with representing a fundamental system of solutions of (2.13), and , with representing a fundamental system of solutions of (2.14). Making use of standard interpretation of a difference equation as a difference system, we rewrite the equalities (2.13) and (2.14), respectively in the form
with be fundamental matrices of solutions of systems (2.15) and (2.16), respectively. Therefore,
for In view of (2.18) and (2.21), and therefore,
Further, we have
Let Then, in view of (2.20),
Let for In view of (2.16) and (2.18),
where, as before,
In view of (2.22), (2.2), (1.12), (1.13), (2.29), and (2.28), the matrix is a fundamental matrix of solutions of system (2.28). The substitution with for transforms the system (2.28) into the system
with for We prove now that if we take and where
with and then we obtain the equality So, let Then, in view of (2.33),
In view of(1.9)
where Hence, in view of (2.19),
In view of (2.34)–(2.36),
In view of (2.30) and (2.33),
it follows from (2.35), (2.37), and (2.38) that
for We prove by induction now the following equality:
for any In view of (2.25) and (2.32), the equality (2.41) holds for In view of (2.26) and (2.33), the equality (2.41) hold for Let and (2.41) holds for Then, in view of (2.29), (2.40), and (2.21),
So, the equality (2.41) holds for any In view of (2.41),
for and in (1.6) and it follows from (2.43) and (2.44), that
As it is well known, for any there exist and such that
We apply (2.23) now. Let In view of (2.2), (1.12)–(1.13), and (2.45), if then
In view of (2.23), (2.50), and (2.49), if
when In view of (2.45), (2.48), and (2.51), there exist and such that
where So, the equality (2.1) is proved. In view of (2.23),
Further, we have
Hence, the series (2.53) is the series of Leibnitz type. Therefore, decreases, when increases in and increases, when increases in
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The author would like to express his thanks to the reviewer of this article for his efforts, his criticism, his advices, and indications of misprints. Ravi P. Agarwal had expressed a useful suggestion, which the author realized in foreword and references. He is grateful to him in this connection.
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Gutnik, L. Elementary Proof of Yu. V. Nesterenko Expansion of the Number Zeta(3) in Continued Fraction. Adv Differ Equ 2010, 143521 (2010). https://doi.org/10.1155/2010/143521
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
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- Functional Equation