Open Access

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

Advances in Difference Equations20102010:143521

DOI: 10.1155/2010/143521

Received: 12 August 2009

Accepted: 10 January 2010

Published: 2 March 2010

Abstract

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.

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 [18]). 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 [9]. My goal is to give an elementary direct proof of his expansion of the number in continued fraction. Let us consider a difference equation

(1.1)

with We denote by

(1.2)

the solutions of this equation with initial values

(1.3)

Then

(1.4)

is a sequence of convergents of the continued fraction

(1.5)

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

(1.6)

where and are solutions of difference equation

(1.7)

with initial values The equality (1.6) is equivalent to the equality

(1.8)

with

(1.9)

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

(1.10)

with

(1.11)
(1.12)

for

(1.13)

for

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

(2.1)

with

(2.2)

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

(2.3)

where values are specified in (1.9), and Then

(2.4)

Let

(2.5)

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

(2.6)
(2.7)
(2.8)
(2.9)
(2.10)

Let

(2.11)

We want to to prove that if then

(2.12)

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

(2.13)
(2.14)

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

(2.15)
(2.16)

where

(2.17)
(2.18)

and Let

(2.19)
(2.20)

with be fundamental matrices of solutions of systems (2.15) and (2.16), respectively. Therefore,

(2.21)

for In view of (2.18) and (2.21), and therefore,

(2.22)

Hence

(2.23)

(see [11]).

Further, we have

(2.24)
(2.25)
(2.26)

Let Then, in view of (2.20),

(2.27)

Let for In view of (2.16) and (2.18),

(2.28)
(2.29)

where, as before,

(2.30)

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

(2.31)

with for We prove now that if we take and where

(2.32)
(2.33)

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

(2.34)

In view of(1.9)

(2.35)

where Hence, in view of (2.19),

(2.36)

In view of (2.34)–(2.36),

(2.37)

In view of (2.30) and (2.33),

(2.38)

Since

(2.39)

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

(2.40)

for We prove by induction now the following equality:

(2.41)

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),

(2.42)

So, the equality (2.41) holds for any In view of (2.41),

(2.43)

for Since

(2.44)

for and in (1.6) and it follows from (2.43) and (2.44), that

(2.45)

As it is well known, for any there exist and such that

(2.46)
(2.47)
(2.48)

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

(2.49)
(2.50)

In view of (2.23), (2.50), and (2.49), if

(2.51)

when In view of (2.45), (2.48), and (2.51), there exist and such that

(2.52)

where So, the equality (2.1) is proved. In view of (2.23),

(2.53)

where

(2.54)

Further, we have

(2.55)

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

Declarations

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.

Authors’ Affiliations

(1)
Moscow State Institute of Electronics and Mathematics

References

  1. Kim T, Hwang K-W, Kim Y-H: Symmetry properties of higher-order Bernoulli polynomials. Advances in Difference Equations 2009, 2009:6.Google Scholar
  2. Kim T, Hwang K-W, Lee B: A note on the -Euler measures. Advances in Difference Equations 2009, 2009:8.Google Scholar
  3. Park K-H, Kim Y-H: On some arithmetical properties of the Genocchi numbers and polynomials. Advances in Difference Equations 2008, 2008:14.Google Scholar
  4. Simsek Y, Cangul IN, Kurt V, Curt V, Kim D: -Genocchi numbers and polynomials associated with -Genocchi-type -functions. Advances in Difference Equations 2008, 2008:12.Google Scholar
  5. Jang LC: Multiple twisted -Euler numbers and polynomials associated with -adic -integrals. Advances in Difference Equations 2008, 2008:11.Google Scholar
  6. Kim T: -Bernoulli numbers associated with -Stirling numbers. Advances in Difference Equations 2008, 2008:10.Google Scholar
  7. Rachidi M, Saeki O: Extending generalized Fibonacci sequences and their Binet-type formula. Advances in Difference Equations 2006, 2006:11.Google Scholar
  8. Jaroma JH: On the appearance of primes in linear recursive sequences. Advances in Difference Equations 2005,2005(2):145-151. 10.1155/ADE.2005.145MathSciNetView ArticleMATHGoogle Scholar
  9. Nesterenko YuV: Some remarks on (3). Rossiĭskaya Akademiya Nauk. Matematicheskie Zametki 1996,59(6):865-880.MathSciNetView ArticleGoogle Scholar
  10. Apéry R: Interpolation des fractions continues et irrationalité de certaines constantes. Bulletin de la Section des Sciences du C.T.H 1981, 3: 37-53.Google Scholar
  11. Perron O: Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche. B. G. Teubner Verlagsgesellschaft, Stuttgart, Germany; 1954:vi+194.Google Scholar

Copyright

© 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.