- Research Article
- Open access
- Published:
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)
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 [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 [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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ1_HTML.gif)
with We denote by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ2_HTML.gif)
the solutions of this equation with initial values
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ3_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ4_HTML.gif)
is a sequence of convergents of the continued fraction
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ5_HTML.gif)
Accoding to the famous result of R. Apéry [10],
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ6_HTML.gif)
where and
are solutions of difference equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ7_HTML.gif)
with initial values The equality (1.6) is equivalent to the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ8_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ9_HTML.gif)
where Nesterenko in [9] has offered the following expansion of the number
in continued fraction:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ10_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ12_HTML.gif)
for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ14_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ15_HTML.gif)
Furthermore, to avoid confusion in notations, we denote below for the fraction (2.1) by
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ16_HTML.gif)
where values are specified in (1.9), and
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ17_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ23_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ24_HTML.gif)
We want to to prove that if then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ25_HTML.gif)
Note that if then (2.12) follows from (2.6)–(2.10). Therefore, we can consider only
Let us consider the following difference equations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ29_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ31_HTML.gif)
and Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ32_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ33_HTML.gif)
with be fundamental matrices of solutions of systems (2.15) and (2.16), respectively. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ34_HTML.gif)
for In view of (2.18) and (2.21),
and therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ35_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ36_HTML.gif)
(see [11]).
Further, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ38_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ39_HTML.gif)
Let Then, in view of (2.20),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ40_HTML.gif)
Let for
In view of (2.16) and (2.18),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ41_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ42_HTML.gif)
where, as before,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ44_HTML.gif)
with for
We prove now that if we take
and
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ45_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ46_HTML.gif)
with and
then we obtain the equality
So, let
Then, in view of (2.33),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ47_HTML.gif)
In view of(1.9)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ48_HTML.gif)
where Hence, in view of (2.19),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ49_HTML.gif)
In view of (2.34)–(2.36),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ50_HTML.gif)
In view of (2.30) and (2.33),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ51_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ52_HTML.gif)
it follows from (2.35), (2.37), and (2.38) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ53_HTML.gif)
for We prove by induction now the following equality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ54_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ55_HTML.gif)
So, the equality (2.41) holds for any In view of (2.41),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ56_HTML.gif)
for Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ57_HTML.gif)
for and
in (1.6) and
it follows from (2.43) and (2.44), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ58_HTML.gif)
As it is well known, for any there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ59_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ60_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ61_HTML.gif)
We apply (2.23) now. Let In view of (2.2), (1.12)–(1.13), and (2.45), if
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ62_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ63_HTML.gif)
In view of (2.23), (2.50), and (2.49), if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ64_HTML.gif)
when In view of (2.45), (2.48), and (2.51), there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ65_HTML.gif)
where So, the equality (2.1) is proved. In view of (2.23),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ66_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ67_HTML.gif)
Further, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F143521/MediaObjects/13662_2009_Article_1247_Equ68_HTML.gif)
Hence, the series (2.53) is the series of Leibnitz type. Therefore, decreases, when
increases in
and
increases, when
increases in
References
Kim T, Hwang K-W, Kim Y-H: Symmetry properties of higher-order Bernoulli polynomials. Advances in Difference Equations 2009, 2009:6.
Kim T, Hwang K-W, Lee B: A note on the
-Euler measures. Advances in Difference Equations 2009, 2009:8.
Park K-H, Kim Y-H: On some arithmetical properties of the Genocchi numbers and polynomials. Advances in Difference Equations 2008, 2008:14.
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.
Jang LC: Multiple twisted
-Euler numbers and polynomials associated with
-adic
-integrals. Advances in Difference Equations 2008, 2008:11.
Kim T:
-Bernoulli numbers associated with
-Stirling numbers. Advances in Difference Equations 2008, 2008:10.
Rachidi M, Saeki O: Extending generalized Fibonacci sequences and their Binet-type formula. Advances in Difference Equations 2006, 2006:11.
Jaroma JH: On the appearance of primes in linear recursive sequences. Advances in Difference Equations 2005,2005(2):145-151. 10.1155/ADE.2005.145
Nesterenko YuV: Some remarks on
(3). RossiÄskaya Akademiya Nauk. Matematicheskie Zametki 1996,59(6):865-880.
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.
Perron O: Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche. B. G. Teubner Verlagsgesellschaft, Stuttgart, Germany; 1954:vi+194.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/143521