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e

rвҲ’1 вҲһ
[(вҲ’B=A)(Гҝ= )rвҲ’1 ]k
tk tr
f(t) = вҲ’r ;
A + BГҝk A t= вҲ’ ( =Гҝ)k
k
k=0 k=0

where r вҲҲ N is chosen such that | |r |A| Вҝ |Гҝ|r |B|.

With the notations of Section 1.1, the PadГғ approximant [n + k вҲ’ 1=k]f is
e

Лң (n)
Qk (t)
[n + k вҲ’ 1=k]f (t) = (n) ;
Лң
P k (t)

Лң (n) (n)
where P k (t) = t k Pk (t вҲ’1 ).
In a previous papers by the author [24,25], it has been proved that for all n вҲҲ Z, the sequence of
PadГғ approximants ([n + k вҲ’ 1=k])k to f converges on any compact set included in the domain of
e
meromorphy of the function f; with the following error term:
Гҝ
2
вҲҖt вҲҲ C \ { ( =Гҝ)j ; j вҲҲ N}; вҲҖn вҲҲ N; lim sup |f(t) вҲ’ [n + k вҲ’ 1=k]f (t)|1=k 6 ; (3)
k

and Гҝ are the two solutions of z 2 вҲ’ sz + p = 0; | | Вҝ |Гҝ|.
where

Лң (n) Лң (n)
Theorem 5. If Qk (t)= P k (t) denotes the PadГғ approximant [n + k вҲ’ 1=k]f ; then
e

k
k i
A + Bq n+kвҲ’j
Лң (n) i(iвҲ’1)=2 i
(a) P k (t) = q (вҲ’t= ) ;
A + Bq n+2kвҲ’j
i
i=0 j=1
q

where
k (1 вҲ’ qk ) : : : (1 вҲ’ qkвҲ’i+1 )
q := Гҝ= ; := ; 16i6k (Gaussian binomial coe cient);
(1 вҲ’ q)(1 вҲ’ q2 ) : : : (1 вҲ’ qi )
i q

k
= 1:
0 q

kвҲ’1
Лң (n) (1 вҲ’ tqj = )|6R|q|k ;
(b) | P k (t) вҲ’ kВҝK0
j=0

for some constant R independent of k and K0 is an integer depending on A; B; q; n.
Moreover; if s; p; wвҲ’1 ; w0 вҲҲ Z(i); for all common multiple dm of {w0 ; w1 ; : : : ; wm }

Лң (n)
(c) wn+k В· В· В· wn+2kвҲ’1 P k вҲҲ Z(i)[t]; вҲҖn вҲҲ Z=n + k вҲ’ 1Вҝ0
M. PrГғ vost / Journal of Computational and Applied Mathematics 122 (2000) 231вҖ“250
e 237

and
Лң (n)
(d) dn+kвҲ’1 wn+k В· В· В· wn+2kвҲ’1 Qk вҲҲ Z(i)[t]; вҲҖn вҲҲ Z =n + k вҲ’ 1Вҝ0:

Proof. (a) is proved in  and (b) is proved in . (c) and (d) comes from expression (a).

The expression of wn is
n
+ BГҝn :
wn = A
If A or B is equal to 0 then f(t) is a rational function, so without loss of generality, we can assume
that AB = 0.
Лң (n) Лң (n)
The degrees of Qk and P k are, respectively, k + n вҲ’ 1 and k, so if we take t вҲҲ Q(i) with
vt вҲҲ Z(i), the above theorem implies that the following sequence:
Лң (n)
Лң (n)
ek; n :=f(t) Г— v k dn+kвҲ’1 wn+k В· В· В· wn+2kвҲ’1 P k (t) вҲ’ v k dn+kвҲ’1 wn+k В· В· В· wn+2kвҲ’1 Qk (t);
where k = max{n + k вҲ’ 1; k} is a Diophantine approximation to f(t), if
(i) вҲҖn вҲҲ Z; limkвҶ’вҲһ ek; n = 0,
(ii) [n + k вҲ’ 1=k]f (t) = [n + k=k + 1]f (t).
For sake of simplicity, we only display the proof for the particular case n = 0.
We set
Лң (0) and P k :=P (0) :
Лң Лң Лңk
ek :=ek; 0 ; Qk :=Qk
From the asymptotics given in (3), we get
1=k 2
Лң
Qk (t) 1=k 2
Лң
1=k 2 k
lim sup |ek | 6 lim sup f(t) вҲ’ lim sup v dkвҲ’1 wk В· В· В· w2kвҲ’1 P k (t) (4)
Лң
P k (t)
k k k

1=k 2
6|p|lim sup| kвҲ’1 | ; (5)
where k :=dk = k wi .
i=0
We will get limkвҶ’вҲһ ek = 0 if the following condition is satisГҝed:
1=k 2
lim sup | kвҲ’1 | ВЎ 1=|p|:
kвҶ’вҲһ

Moreover, from the Christo elвҖ“Darboux identity between orthogonal polynomials, condition (ii) is
satisГҝed since the di erence
k 2
wiвҲ’1
k
Лң k+1 (t)P k (t) вҲ’ P k+1 (t)Qk (t) = t (вҲ’1)
Лң
Лң Лң 2k 2iвҲ’2 i i2
Q ABp ( вҲ’Гҝ) 2 2
A+B w2iвҲ’1 w2i w2iвҲ’2
i=1

is di erent from 0.
The following theorem is now proved.

Theorem 6. Let f be the meromorphic function deГҝned by the following series:
вҲһ
tn
f(t) = ;
wn
n=0
238 M. PrГғ vost / Journal of Computational and Applied Mathematics 122 (2000) 231вҖ“250
e

where (wn )n is a sequence of Z(i) satisfying a three-term recurrence relation
wn+1 = s wn вҲ’ p wnвҲ’1 ; s; p вҲҲ Z(i)
with the initial conditions: wвҲ’1 ; w0 вҲҲ Z(i). If for each integer m; there exists a common multiple
dm for the numbers {w0 ; w1 ; : : : ; wm } such that m deГҝned by
dm
m := m
i=0 wi
satisГҝes the condition
2
1=m
lim sup | m| ВЎ 1=|p|; (6)
m

then for t вҲҲ Q(i); t = ( =Гҝ) j ; j = 0; 1; 2; : : : we have

f(t) вҲҲ Q(i):
See  for application to Fibonacci and Lucas series. (If Fn and Ln are, respectively, Fibonacci
and Lucas sequences, then f(t) = t n =Fn and g(t) = t n =Ln are not rational for all t rational, not
a pole of the functions f or g, which is a generalization of .)

3. Diophantine approximation with PadГғ approximation to the asymptotic expansion of the
e
remainder of the series

For sums of series f, PadГғ approximation to the function f does not always provide Diophantine
e
approximation. Although the approximation error |x вҲ’ pn =qn | is very sharp, the value of the denom-
inator qn of the approximation may be too large such that |qn x вҲ’ pn | does not tend to zero when n
tends to inГҝnity.
Another way is the following.
Consider the series f(t) = вҲһ ci t i = n ci t i + Rn (t): If, for some complex number t0 , we know
i=0 i=0
the asymptotic expansion of Rn (t0 ) on the set {1=ni ; i = 1; 2; : : :}; then it is possible to construct an
approximation of f(t0 ), by adding to the partial sums Sn (t0 ):= n ci t0 ; some PadГғ approximation
i
e
i=0
to the remainder Rn (t0 ) for the variable n.
But it is not sure that we will get a Diophantine approximation for two reasons.
(1) the PadГғ approximation to Rn (t0 ) may not converge to Rn (t0 ),
e
(2) the denominator of the approximant computed at t0 , can converge to inГҝnity more rapidly
than the approximation error does converge to zero.
So, this method works only for few cases.

1=(q n + r)
3.1. Irrationality of (2); (3) , ln(1 + ) and n

3.1.1. Zeta function
The Zeta function of Riemann is deГҝned as
вҲһ
1
(s) = ; (7)
ns
n=1
M. PrГғ vost / Journal of Computational and Applied Mathematics 122 (2000) 231вҖ“250
e 239

where the Dirichlet series on the right-hand side of (7) is convergent for Re(s) Вҝ 1 and uniformly
convergent in any Гҝnite region where Re(s)Вҝ1 + , with Вҝ 0. It deГҝnes an analytic function for
Re(s) Вҝ 1.
RiemannвҖ™s formula
вҲһ
xsвҲ’1
1
(s) = d x; Re(s) Вҝ 1;
(s) ex вҲ’ 1
0

where
вҲһ
ysвҲ’1 eвҲ’y dy is the gamma function
(s) = (8)
0

and
eвҲ’i s
z sвҲ’1
(1 вҲ’ s)
(s) = dz (9)
2i ez вҲ’ 1
C

where C is some path in C, provides the analytic continuation of (s) over the whole s-plane.
If we write formula (7) as
n вҲһ
1 1
(s) = +
ks (n + k)s
k=1 k=1

вҲһ
+ kx))s then
and set s (x):= (s) k=1 (x=(1

n
1 1
(s) = + s (1=n): (10)
ks (s)
k=1

The function вҲһ (x=(1 + kx))s ) is known as the generalized zeta-function (s; 1 + 1=x) [32, Chapter
k=1
XIII] and so we get another expression of s (x):
вҲһ
eвҲ’u=x
sвҲ’1
s (x) = u du; x Вҝ 0;
eu вҲ’ 1
0

whose asymptotic expansion is
вҲһ
Bk
(k + s вҲ’ 1)xk+sвҲ’1 ;
s (x) =
k!
k=0

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