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where Bk are the Bernoulli numbers.
Outline of the method: In (10), we replace the unknown value s (1=n) by some Padà -approximant
e
to s (x), at the point x = 1=n. We get the following approximation:
n
1 1
(s) ≈ + [p=q] s (x = 1=n): (11)
ks (s)
k=1

We only consider the particular case p = q.
240 M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e


Case (2): If s = 2 then (10) becomes
n
1
(2) = + 2 (1=n);
k2
k=1
and its approximation (11):
n
1
(2) ≈ + [p=p] 2 (x = 1=n); (12)
k2
k=1
where

Bk xk+1 = B0 x + B1 x2 + B2 x3 + · · ·
2 (x) = (asymptotic expansion): (13)
k=0
The asymptotic expansion (13) is Borel-summable and its sum is

e’u=x
2 (x) = uu du:
e ’1
0
Computation of [p=p] 2 (x)=x : We apply Section 1.1, where function f(x) = 2 (x)=x. The Padà ap- e
proximants [p=p]f are linked with the orthogonal polynomial with respect to the sequence B0 ; B1 ; B2 : : :.
As in Section 1, we deÿne the linear functional B acting on the space of polynomials by
B : P’R

xi ’ B; xi = Bi ; i = 0; 1; 2; : : : :
The orthogonal polynomials satisfy
p

B; xi p (x) = 0; i = 0; 1; : : : ; p ’ 1: (14)
These polynomials have been studied by Touchard ([31,9,28,29]) and generalized by Carlitz ([12,13]).
The following expressions
2
2x + p ’ 2r x
p (x) =
p ’ 2r r
2r6p


p p+k x+k p p+k x
p p
p k
= (’1) (’1) = (15)
k k k k k k
k=0 k=0

hold (see [34,12]).
Note that the p ™s are orthogonal polynomials and thus satisfy a three-term recurrence relation.
The associated polynomials p of degree p ’ 1 are deÿned as
p (x) ’ p (t)
p (t) = B; ;
x’t
where B acts on x.
From expression (15) for p , we get the following formula for p :
x t

p p+k
p
k k
p (t) = B; :
x’t
k k
k=0
M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e 241


The recurrence relation between the Bernoulli numbers Bi implies that
x (’1)k
B; = :
k +1
k
x t
Using the expression of the polynomial (( k ) ’ ( k ))=(x ’ t) on the Newton basis on 0; 1; : : : ; k ’ 1;
x t x

t k
k k i’1
= ;
x’t t
k i=1
i
i
we can write a compact formula for p:

p p+k t
p k
(’1)i’1
p (t) = ∈ Pp’1 :
t
k k k i=1
k=1
i2
i
Approximation (12) for (2) becomes
˜ p (t)
n n
1 1 p (n)
(2) ≈ +t = + :
˜ p (t)
k2 k2 p (n)
k=1 k=1
t=1=n

n
Using partial decomposition of 1= with respect to the variable n, it is easy to prove that
i

dn
∈ N; ∀i ∈ {1; 2; : : : ; n} (16)
n
i
i
with dn :=LCM(1; 2; : : : ; n).
A consequence of the above result is
d2 p (n) ∈ N; ∀p ∈ N
n

and
d2 (2) ’ d2 (Sn
p (n) p (n) + p (n)) (17)
n n

is a Diophantine approximation of (2), for all values of integer p, where Sn denotes the partial
sums Sn = n 1=k 2 . It remains to estimate the error for the Padà approximation:
e
k=1

2 (t) ’ [p=p] 2 (t) = 2 (t) ’ [p ’ 1=p] (t):
2 =t


Touchard found the integral representation for the linear functional B:
+i∞
dx
k
xk
B; x :=Bk = ’i ; ’1 ¡ ¡ 0:
sin2 ( x)
2 ’i∞
242 M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e


Thus, formula (1) becomes
2
p (x)
t 2p +i∞
dx
’1
t 2 (t) ’ [p ’ 1=p] 2 =t (t) = ’i ;
2 ˜ 2 (t) 1 ’ xt sin2 ( x)
’i∞
p

and we obtain the error for the Padà approximant to
e 2:
2
p (x)
+i∞
t dx
2 (t) ’ [p=p] 2 (t) = ’i
1 ’ xt sin2 ( x)
2 p (t ’1 )
2
’i∞

and the error for formula (17):
2
p (x)
+i∞
1 dx
d2 d2 (Sn ’d2 i
p (n) (2) ’ p (n) + p (n)) = : (18)
n n n
1 ’ x=n sin2 ( x)
2n p (n) ’i∞

If p = n, we get Apà ry™s numbers [4]:
e
2
n n+k
n
bn = n (n) =
k k
k=0

and
2
n n+k
n n k
(’1)i’1
1
an = Sn n (n) + n (n) = bn + :
k2 n
k k i=1
k=1 k=1
i2
i
The error in formula (18) becomes
+i∞ 2
1 n (x)
dx
d2 bn d2 an ’d2 i
(2) ’ = (19)
n n n
1 ’ x=n sin2 x
2n bn ’i∞

In order to prove the irrationality of (2), we have to show that the right-hand side of (19) tends
to 0 when n tends to inÿnity, and is di erent from 0, for each integer n.
We have
1
2
n (’ 2
+ iu) du
’1=2+i∞ 2 +∞
n (x) dx 1 2
6 6 B; n (x)
1 ’ x=n sin2 x 1 + 1=2n cosh2 u 1 + 1=2n
’1=2’i∞ ’∞


since cosh2 u is positive for u ∈ R and n (’ 1 + iu) real positive for u real ( n has all its roots on
2
2
the line ’ 2 + iR; because n (’ 2 + iu) is orthogonal with respect to the positive weight 1=cosh2 u
1 1
2
on R). The quantity B; n (x) can be computed from the three term recurrence relation between
the n s [31]:
(’1)n
2
B; n (x) = :
2n + 1
The Diophantine approximation (19) satisÿes
1
|d2 bn (2) ’ d2 an |6d2 — :
n n n
(2n + 1)2 bn
M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e 243

In [14], it is proved that bn ∼ A ((1 + 5)=2)5n n’1 when n ’ ∞, for some constant A . From a
result concerning dn = LCM(1; 2; : : : ; n): (dn = e(n(1+o(1)) ), we get
lim |d2 bn (2) ’ d2 an | = 0; (20)
n n
n’∞

where d2 bn and d2 an are integers.
n n
Relation (20) proves that (2) is not rational.

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