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

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

(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