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 
 1=4 
1=6 2=13
 
 
 1=8 
3=20 2=13 2=13
 
 
 (0) 
 5=32 5=32 21=136 37=240 45=292 = 8 
 
 
 5=32 
5=32 2=13 53=344
 
 
 59=384 
59=384 37=240
 
 
 59=384 
59=384
° »
79=512
(0)
(we have only displayed the odd columns), 1+1=23 +1=2— 8 =351=292=a2 =b2 : -algorithm is a par-
ticular extrapolation algorithm as Padà approximation is particular case of Padà -type approximation.
e e
Generalization has been achieved by Brezinski and Havie, the so-called E-algorithm. Diophantine
approximation using E-algorithm and Padà -type approximation are under consideration.
e

3.1.2. Irrationality of ln(1 + )
In this part, we use the same method as in the preceding section:
n ∞
k
(’1)k+n+1
k+1 k+n
We set ln(1 + ) = (’1) + : (25)
k k +n
k=1 k=1

e’(k+n)v dv, we get an integral representation for the remainder
From the formula 1=(k + n) = 0
term in (25):
∞ k+n ∞
e’nv
k+n+1 n n+1
(’1) = (’1) dv:
k +n ev +
0
k=1
M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e 247


If we expand the function

vk
1+
= Rk (’ ) ;
ev + k!
k=0

where the Rk (’ ) s are the Eulerian numbers [12], we get the following asymptotic expansion:
∞ ∞
k+n
(’1)n n+1
k+n+1
Rk (’ )xk
(’1) = :
k +n n(1 + )
k=1 k=0 x=1=n

Let us set

Rk (’ )xk :
1 (x) =
k=0

Carlitz has studied the orthogonal polynomials with respect to R0 (’ ), R1 (’ ); : : : .
If we deÿne the linear functional R by
R; xk :=Rk (’ );
then the orthogonal polynomials Pn with respect to R; i.e.,
R; xk Pn (x) = 0; k = 0; 1; : : : ; n ’ 1;
satisfy Pn (x) = n (1 + )k k n x
[12].
k=0 k
The associated polynomials are
n x t
n
’k
k k
Qn (t) = (1 + ) R; : (26)
x’t
k
k=0
x
Carlitz proved that R; ( k ) = (’ ’ 1)’k and thus, using (26),
n t
n k i’1
1 ’1
k
Qn (t) = (1 + ) :
t
i +1
k k i
i=1
k=0

If we set = p=q, p and q ∈ Z and t = n, then
q n dn Qn (n) ∈ Z:
An integral representation for Rk (’ ) is given by Carlitz:
+i∞ ’z
1+ k
Rk (’ ) = ’ z d z; ’1 ¡ ¡ 0; (27)
2i sin z
’i∞

and thus
+i∞ ’z
1+ 1
1 (x) = ’ d z:
2i 1 ’ xz sin z
’i∞

The orthogonal polynomial Pn satisÿes [12]
+i∞ ’z
+2i
2
(’ )n+1 ;
Pn (z) dz =
sin z i+
’i∞
248 M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e


and since Re( ’z sin z) ¿ 0 for z ∈ ’ 1 + iR, we obtain a majoration of the error for the PadÃ
e
2
approximation to 1 :
n
x ¿ 0; 1 (x) ’ [n ’ 1=n] 1 (x) 6
|1 + x=2|
and if x = 1=n, we get
| |n
1
’ [n ’ 1=n] 1 (1=n) 6 :
1
n 1 + 1=2n
Let us replace in (25) the remainder term by its Padà approximant:
e
n k
(’1)n n+1
k+1
ln(1 + ) ≈ (’1) + [n ’ 1=n] 1 (1=n);
k (1 + )n
k=1

we obtain a Diophantine approximation for ln(1 + p=q):
2n
dn q2n
p 2n 2n
ln 1 + dn q Pn (n) ’ dn q Tn (n) 6 ; (28)
q (n + 2)Pn (n)
where Tn (n) = Pn (n) n (’1)k+1 pk =kqk + (’1)n+1 Qn (n)q n .
k=1
From the expression of Pn (x) we can conclude that
2
n
n
2
(1 + )k n
Pn (n) = = Legendre n; +1 ;
k
k=0

where Legendre (n; x) is the nth Legendre polynomial and thus
Tn (n)
= [n=n]ln(1+x) (x = 1):
Pn (n)
So, the classical proof for irrationality of ln(1 + p=q) based on Padà approximants to the function
e
ln(1 + x) is recovered by formula (28).
Proof of irrationality of (2) with alternated series: Another expression for (2) is

(’1)k’1
(2) = 2 :
k2
k=1

Let us write it as a sum
n ∞
(’1)k’1 (’1)k+n+1
(2) = 2 +2 :
k2 (k + n)2
k=1 k=1

Rk (’1)(k + 1)xk . So
Let be deÿned by 2 (x) =
2 k=0
n
(’1)k’1 (’1)n
(2) = 2 + 2 (1=n):
k2 n2
k=1

With the same method, we can prove that the Padà approximant [2n=2n] 2 (x) computed at
e
x = 1=n leads to Apà ry™s numbers an and bn and so proves the irrationality of (2) with the
e
M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e 249


integral representation for the sequence (kRk’1 (’1))k :
+i∞
(1 + ) cos z
zk 2
kRk’1 (’1) = ’ d z; k¿1:
2i sin z
’i∞

obtained with an integration by parts applied to (27).

1=(q n + r)
3.1.3. Irrationality of
In [7], Borwein proves the irrationality of L(r) = 1=(q n ’ r), for q an integer greater than 2, and
r a non zero rational (di erent from q n ; for any n¿1); by using similar method. It is as follows:
Set
∞ ∞
xn
x
Lq (x):= = ; |q| ¿ 1:
q n ’ x n=1 q n ’ 1
n=1

Fix N a positive integer and write Lq (r) = N r=(q n ’ r) + Lq (r=qN ).
n=1
Then, it remains to replace Lq (r=qN ) by its Padà approximant [N=N ]Lq (r=qN ).
e
The convergence of [N=N ]Lq to Lq is a consequence of the following formula:
2
∀t ∈ C \ {qj ; j ∈ N}; lim sup |Lq (t) ’ [N=N ]Lq (t)|1=3N 61=q:
∀n ∈ N;
N

pn =qn deÿned by pn =qn := N r=(q n ’ r) + [N=N ]Lq (r=qN ) leads to Diophantine approximation of
n=1
Lq (r) and so proves the irrationality of Lq (r).
For further results concerning the function Lq , see [17“19].
Di erent authors used Padà or Padà Hermite approximants to get Diophantine approximation, see
e e
for example [8,20“23,27].

References

[1] K. Alladi, M.L. Robinson, Legendre polynomials and irrationality, J. Reine Angew. Math. 318 (1980) 137“155.
[2] R. Andrà -Jeannin, Irrationalità de la somme des inverses de certaines suites rà currentes, C.R. Acad. Sci. Paris Sà r.
e e e e
I 308 (1989) 539 “541.
[3] G.E. Andrews, R. Askey, Classical orthogonal polynomials, in: C. Brezinski, A. Draux, A.P. Magnus, P. Maroni, A.
Ronveaux (Eds.) Polynˆ mes Orthogonaux et applications, Lecture notes in Mathematics, Vol. 1171, Springer, New
o
York, 1985, pp. 36 “ 62.
[4] R. Apà ry, Irrationalità de (2) et (3), J. Arith. Luminy, Astà risque 61 (1979) 11“13.
e e e
[5] G.A. Baker Jr., Essentials of Padà approximants, Academic Press, New York, 1975.

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