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5=32 2=13 53=344
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(we have only displayed the odd columns), 1+1=23 +1=2âˆ— 8 =351=292=a2 =b2 : -algorithm is a par-
ticular extrapolation algorithm as PadÃƒ approximation is particular case of PadÃƒ -type approximation.
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Generalization has been achieved by Brezinski and Havie, the so-called E-algorithm. Diophantine
approximation using E-algorithm and PadÃƒ -type approximation are under consideration.
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3.1.2. Irrationality of ln(1 + )
In this part, we use the same method as in the preceding section:
n âˆž
k
(âˆ’1)k+n+1
k+1 k+n
We set ln(1 + ) = (âˆ’1) + : (25)
k k +n
k=1 k=1
âˆž
eâˆ’(k+n)v dv, we get an integral representation for the remainder
From the formula 1=(k + n) = 0
term in (25):
âˆž k+n âˆž
eâˆ’nv
k+n+1 n n+1
(âˆ’1) = (âˆ’1) dv:
k +n ev +
0
k=1
M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250
e 247

If we expand the function
âˆž
vk
1+
= Rk (âˆ’ ) ;
ev + k!
k=0

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

Let us set
âˆž
Rk (âˆ’ )xk :
1 (x) =
k=0

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

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

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

The orthogonal polynomial Pn satisÃ¿es [12]
+iâˆž âˆ’z
+2i
2
(âˆ’ )n+1 ;
Pn (z) dz =
sin z i+
âˆ’iâˆž
248 M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250
e

and since Re( âˆ’z sin z) Â¿ 0 for z âˆˆ âˆ’ 1 + iR, we obtain a majoration of the error for the PadÃƒ
e
2
approximation to 1 :
n
x Â¿ 0; 1 (x) âˆ’ [n âˆ’ 1=n] 1 (x) 6
|1 + x=2|
and if x = 1=n, we get
| |n
1
âˆ’ [n âˆ’ 1=n] 1 (1=n) 6 :
1
n 1 + 1=2n
Let us replace in (25) the remainder term by its PadÃƒ approximant:
e
n k
(âˆ’1)n n+1
k+1
ln(1 + ) â‰ˆ (âˆ’1) + [n âˆ’ 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) âˆ’ dn q Tn (n) 6 ; (28)
q (n + 2)Pn (n)
where Tn (n) = Pn (n) n (âˆ’1)k+1 pk =kqk + (âˆ’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 PadÃƒ 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
âˆž
(âˆ’1)kâˆ’1
(2) = 2 :
k2
k=1

Let us write it as a sum
n âˆž
(âˆ’1)kâˆ’1 (âˆ’1)k+n+1
(2) = 2 +2 :
k2 (k + n)2
k=1 k=1
âˆž
Rk (âˆ’1)(k + 1)xk . So
Let be deÃ¿ned by 2 (x) =
2 k=0
n
(âˆ’1)kâˆ’1 (âˆ’1)n
(2) = 2 + 2 (1=n):
k2 n2
k=1

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

integral representation for the sequence (kRkâˆ’1 (âˆ’1))k :
+iâˆž
(1 + ) cos z
zk 2
kRkâˆ’1 (âˆ’1) = âˆ’ d z; kÂ¿1:
2i sin z
âˆ’iâˆž

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 âˆ’ r), for q an integer greater than 2, and
r a non zero rational (di erent from q n ; for any nÂ¿1); by using similar method. It is as follows:
Set
âˆž âˆž
xn
x
Lq (x):= = ; |q| Â¿ 1:
q n âˆ’ x n=1 q n âˆ’ 1
n=1

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

pn =qn deÃ¿ned by pn =qn := N r=(q n âˆ’ 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 [17â€“19].
Di erent authors used PadÃƒ or PadÃƒ Hermite approximants to get Diophantine approximation, see
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for example [8,20â€“23,27].

References

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