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(m’n+1) (m’n+1)
(m’n+1)
Rn’1 (t):= c ; ; Rn’1 ∈ Pn’1 ;
x’t
where c(m’n+1) acts on the letter x, then
m’n
˜ (m’n+1) (t) + t m’n+1 Rn’1
˜ (m’n+1) (t);
ci t i P n
Nm (t) =
i=0

˜ (m’n+1) (t) = t n’1 Rn’1 ˜ (m’n+1) (t) = t n Pn
(m’n+1) ’1 n’m
(m’n+1) ’1
(t ) and i=0 ci t i = 0; n ¡ m.
where Rn’1 (t ); P n
(n)
The sequence of polynomials (Pk )k , of degree k, exists if and only if ∀n ∈ Z, the Hankel
determinant
cn · · · cn+k’1
Hk(n) := ··· ··· ··· = 0;
cn+k’1 · · · cn+2k’2
where cn = 0 if n ¡ 0.
M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e 233


In that case, we shall say that the linear functional c is completely deÿnite. For the noncompletely
deÿnite case, the interested reader is referred to Draux [15].
For extensive applications of Padà approximants to Physics, see Baker™s monograph [5].
e
If c admits an integral representation by a nondecreasing function , with bounded variation
xi d (x);
ci =
R
then the theory of Gaussian quadrature shows that the polynomials Pn orthogonal with respect to c,
have all their roots in the support of the function and
« 
˜ (m’n+1) (x))2
t m’n+1 (P n
(m’n+1)  
h(t) ’ [m=n]h (t) = c
˜ (m’n+1) (t))2 1 ’ xt
(P n
(m’n+1)
˜
t m’n+1 (x))2
m’n+1 (P n
= x d (x): (1)
˜ (m’n+1) (t))2 1 ’ xt
(P n R

Note that if c0 = 0 then [n=n]h (t) = t[n ’ 1=n]h=t (t) and if c0 = 0 and c1 = 0; then [n=n]h (t) = t 2 [n ’
2=n]h=t 2 (t).
Consequence: If is a nondecreasing function on R, then
h(t) = [m=n]f (t) ∀t ∈ C ’ supp( ):

1.2. Computation of Padà approximants with -algorithm
e

The values of Padà approximants at some point of parameter t, can be recursively computed with
e
the -algorithm of Wynn. The rules are the following:
(n) (n)
= 0; = Sn ; n = 0; 1; : : : ;
’1 0

1
(n) (n+1)
= + ; k; n = 0; 1; : : : (rhombus rule);
k+1 k’1 (n+1) (n)

k k

where Sn = n ck t k .
k=0
-values are placed in a double-entry array as following:
(0)
=0
’1
(0)
= S0
0
(1) (0)
=0
’1 1
(1) (0)
= S1
0 2
(2) (1) (0)
=0
’1 1 3

..
(2) (1)
.
= S2
0 2

. ..
.
(3) (2)
.
=0 .
’1 1

. . ..
. .
(3)
.
. = S3 .
0
234 M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e


The connection between Padà approximant and -algorithm has been established by Shanks [26] and
e
Wynn [35]:


ci t i ; then
Theorem 2. If we apply -algorithm to the partial sums of the series h(t) = i=0
(n)
= [n + k=k]h (t):
2k


Many convergence results for -algorithm has been proved for series which are meromorphic func-
tions in some complex domain, or which have an integral representation (Markov“Stieltjes function)
(see [29,6,11] for a survey).


2. Diophantine approximation of sum of series with Padà approximation
e

Sometimes, Padà approximation is su cient to prove irrationality of values of a series, as it can
e
be seen in the following two results.

2.1. Irrationality of ln(1 ’ r)

We explain in the following theorem, how the old proof of irrationality of some logarithm number
can be re-written in terms of -algorithm.

Theorem 3. Let r =a=b; a ∈ Z; b ∈ N; b = 0; with b:e:(1’ 1 ’ r)2 ¡ 1(ln e=1) Then -algorithm
(n)
applied to the partial sums of f(r):= ’ ln(1 ’ r)=r = ∞ r i =(i + 1) satisÿes that ∀n ∈ N; ( 2k )k
i=0
is a Diophantine approximation of f(r).

Proof. From the connection between Padà approximation, orthogonal polynomials and -algorithm,
e
the following expression holds:
˜ (n+1)
n
Rk’1 (r)
ri Nn+k (r)
(n)
+ r n+1 (n+1)
= = (n+1) ;
2k
i+1 ˜ ˜
P k (r) P k (r)
i=0

where
k k +n+1
k
˜ (n+1) (n+1)
k
Pk (t ’1 ) (1 ’ t)i
P k (t) =t =
k ’i i
i=0

˜ (n+1)
is the reversed shifted Jacobi polynomial on [0,1], with parameters = 0; ÿ = n + 1; and Rk’1 (t) =
P (n+1) (x)’P (n+1) (t)
(n+1) (n+1)
t k’1 Rk’1 (t ’1 ) with Rk’1 (t) = c(n+1) ; k ( c(n+1) ; xi :=1=(n + i + 2)) (c acts on the
k
x’t
variable x ).
˜ (n+1) ˜ (n+1)
Since P k (t) has only integer coe cients, bk P k (a=b) ∈ Z:
˜ (n+1)
(n+1)
The expression of Rk’1 (t) shows that dn+k+1 bk Rk’1 (a=b) ∈ Z, where dn+k+1 :=LCM(1; 2; : : : ; n +
k + 1) (LCM means lowest common multiple).
(n)
We prove now that the sequence ( 2k )k is a Diophantine approximation of ln(1 ’ a=b):
M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e 235

˜ (n+1) (n)
The proof needs asymptotics for dn+k+1 , for P k (a=b) and for ( 2k ’ f(r)) when k tends to
inÿnity. dn = en(1+o(1)) follows from analytic number theory [1].
˜ (n+1)
limk (P k (x))1=k =x(y + y2 ’ 1); x ¿ 1, y =2=x ’1, comes from asymptotic properties of Jacobi
(n)
polynomials (see [30]), and limk’+∞ ( 2k ’ f(r))1=k = (2=r ’ 1 ’ (2=r ’ 1)2 ’ 1)2 (error of PadÃ
e
approximants to Markov“Stieltjes function).
So
1=k
(n+1)
˜
lim sup dn+k+1 bk P k (a=b)f(r) ’ dn+k+1 bk Nn+k (a=b)
k’+∞

1=k 1=k
˜ (n+1) (n)
1=k k
6 lim sup(dn+k+1 ) lim sup b P k (a=b) lim sup + 1=rln(1 ’ r)
2k
k’+∞ k’+∞
k


(2=r ’ 1)2 ’ 1)2
6e:b:r:(2=r ’ 1 + (2=r ’ 1)2 ’ 1)(2=r ’ 1 ’

1 ’ r)2 ¡ 1
=e:b:(2=r ’ 1 ’ (2=r ’ 1)2 ’ 1) = e:b:(1 ’
by hypothesis, which proves that
˜ (n+1)
lim (dn+k+1 bk P k (a=b)f(r) ’ dn+k+1 bk Nn+k (a=b)) = 0:
∀n ∈ N;
k’+∞

Moreover,
(n+1)
r 2k+n+1 1
(Pk (x))2
(n)
(1 ’ x)n+1 d x = 0:
+ 1=r ln(1 ’ r) = ’
2k (n+1)
1 ’ xr
˜
(P k (r))2 0

(n)
1 ’ a=b)2 ¡ 1.
So the sequence ( 2k )k is a Diophantine approximation of ln(1 ’ a=b); if b:e:(1 ’


t n =wn
2.2. Irrationality of
∞n
The same method as previously seen provides Diophantine approximations of f(t):= n=0 t =wn
when the sequence (wk )k satisÿes a second-order recurrence relation
wn+1 = swn ’ pwn’1 ; n ∈ N; (2)
where w0 and w’1 are given in C and s and p are some complex numbers.
We suppose that wn = 0; ∀n ∈ N and that the two roots of the characteristic equation z 2 ’ sz +
p = 0, and ÿ satisfy | | ¿ |ÿ|.
So wn admits an expression in term of geometric sequences: wn = A n + Bÿn ; n ∈ N.
The roots of the characteristic equation are assumed to be of distinct modulus (| | ¿ |ÿ|), so there
exists an integer r such that | =ÿ|r ¿ |B=A|.

Lemma 4 (see [25]). If ; ÿ; A; B are some complex numbers; and | | ¿ |ÿ|; then the function

tk
f(t):=
A + Bÿk
k
k=0
236 M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250

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