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Advances in Difference Equations

, 2010:143521 | Cite as

Elementary Proof of Yu. V. Nesterenko Expansion of the Number Zeta(3) in Continued Fraction

  • Leonid GutnikEmail author
Open Access
Research Article
  • 961 Downloads

Abstract

Yu. V. Nesterenko has proved that Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window for Open image in new window ; Open image in new window , Open image in new window , and Open image in new window , Open image in new window for Open image in new window His proof is based on some properties of hypergeometric functions. We give here an elementary direct proof of this result.

Keywords

Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Analysis Functional Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Foreword

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, 2, 3, 4, 5, 6, 7, 8]). The interest in this area increases after Apéry's discovery of irrationality of the number Open image in new window This paper is inspired by Yu.V. Nesterenko's work [9]. My goal is to give an elementary direct proof of his expansion of the number Open image in new window in continued fraction. Let us consider a difference equation

with Open image in new window We denote by

the solutions of this equation with initial values

Then

is a sequence of convergents of the continued fraction

Accoding to the famous result of R. Apéry [10],

where Open image in new window and Open image in new window are solutions of difference equation

with initial values Open image in new window The equality (1.6) is equivalent to the equality

with

where Open image in new window Nesterenko in [9] has offered the following expansion of the number Open image in new window in continued fraction:

with

for Open image in new window

for Open image in new window

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

with

Furthermore, to avoid confusion in notations, we denote below Open image in new window for the fraction (2.1) by Open image in new window Let Open image in new window

where values Open image in new window are specified in (1.9), and Open image in new window Then

Let Open image in new window

where Open image in new window and values Open image in new window are specified in (2.2), (1.12), and (1.13). We calculate first Open image in new window and Open image in new window for Open image in new window

Since Open image in new window it follows from (2.2) that

Let Open image in new window

We want to to prove that if Open image in new window then

Note that if Open image in new window then (2.12) follows from (2.6)–(2.10). Therefore, we can consider only Open image in new window Let us consider the following difference equations:

with Open image in new window Then Open image in new window , Open image in new window with Open image in new window representing a fundamental system of solutions of (2.13), and Open image in new window , Open image in new window with Open image in new window 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

where

and Open image in new window Let

with Open image in new window be fundamental matrices of solutions of systems (2.15) and (2.16), respectively. Therefore,

for Open image in new window In view of (2.18) and (2.21), Open image in new window and therefore,

Hence

(see [11]).

Further, we have

Let Open image in new window Then, in view of (2.20),

Let Open image in new window for Open image in new window In view of (2.16) and (2.18),

where, as before, Open image in new window

In view of (2.22), (2.2), (1.12), (1.13), (2.29), and (2.28), the matrix Open image in new window is a fundamental matrix of solutions of system (2.28). The substitution Open image in new window with Open image in new window for Open image in new window transforms the system (2.28) into the system

with Open image in new window for Open image in new window We prove now that if we take Open image in new window and Open image in new window where

with Open image in new window and Open image in new window then we obtain the equality Open image in new window So, let Open image in new window Then, in view of (2.33),

In view of(1.9)

where Open image in new window Hence, in view of (2.19),

In view of (2.34)–(2.36),

In view of (2.30) and (2.33),

Since

it follows from (2.35), (2.37), and (2.38) that

for Open image in new window We prove by induction now the following equality:

for any Open image in new window In view of (2.25) and (2.32), the equality (2.41) holds for Open image in new window In view of (2.26) and (2.33), the equality (2.41) hold for Open image in new window Let Open image in new window and (2.41) holds for Open image in new window Then, in view of (2.29), (2.40), and (2.21),

So, the equality (2.41) holds for any Open image in new window In view of (2.41),

for Open image in new window Since

for Open image in new window and Open image in new window in (1.6) and Open image in new window it follows from (2.43) and (2.44), that

As it is well known, for any Open image in new window there exist Open image in new window and Open image in new window such that

We apply (2.23) now. Let Open image in new window In view of (2.2), (1.12)–(1.13), and (2.45), if Open image in new window then

In view of (2.23), (2.50), and (2.49), if Open image in new window

when Open image in new window In view of (2.45), (2.48), and (2.51), there exist Open image in new window and Open image in new window such that

where Open image in new window So, the equality (2.1) is proved. In view of (2.23),

where

Further, we have

Hence, the series (2.53) is the series of Leibnitz type. Therefore, Open image in new window decreases, when Open image in new window increases in Open image in new window and Open image in new window increases, when Open image in new window increases in Open image in new window

Notes

Acknowledgment

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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Copyright information

© Leonid Gutnik. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Moscow State Institute of Electronics and MathematicsRussia

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