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e

admits another expansion
râˆ’1 âˆž
[(âˆ’B=A)(Ã¿= )râˆ’1 ]k
tk tr
f(t) = âˆ’r ;
A + BÃ¿k A t= âˆ’ ( =Ã¿)k
k
k=0 k=0

where r âˆˆ N is chosen such that | |r |A| Â¿ |Ã¿|r |B|.

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

Ëœ (n)
Qk (t)
[n + k âˆ’ 1=k]f (t) = (n) ;
Ëœ
P k (t)

Ëœ (n) (n)
where P k (t) = t k Pk (t âˆ’1 ).
In a previous papers by the author [24,25], it has been proved that for all n âˆˆ Z, the sequence of
PadÃƒ approximants ([n + k âˆ’ 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:
Ã¿
2
âˆ€t âˆˆ C \ { ( =Ã¿)j ; j âˆˆ N}; âˆ€n âˆˆ N; lim sup |f(t) âˆ’ [n + k âˆ’ 1=k]f (t)|1=k 6 ; (3)
k

and Ã¿ are the two solutions of z 2 âˆ’ sz + p = 0; | | Â¿ |Ã¿|.
where

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

k
k i
A + Bq n+kâˆ’j
Ëœ (n) i(iâˆ’1)=2 i
(a) P k (t) = q (âˆ’t= ) ;
A + Bq n+2kâˆ’j
i
i=0 j=1
q

where
k (1 âˆ’ qk ) : : : (1 âˆ’ qkâˆ’i+1 )
q := Ã¿= ; := ; 16i6k (Gaussian binomial coe cient);
(1 âˆ’ q)(1 âˆ’ q2 ) : : : (1 âˆ’ qi )
i q

k
= 1:
0 q

kâˆ’1
Ëœ (n) (1 âˆ’ tqj = )|6R|q|k ;
(b) | P k (t) âˆ’ kÂ¿K0
j=0

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

Ëœ (n)
(c) wn+k Â· Â· Â· wn+2kâˆ’1 P k âˆˆ Z(i)[t]; âˆ€n âˆˆ Z=n + k âˆ’ 1Â¿0
M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250
e 237

and
Ëœ (n)
(d) dn+kâˆ’1 wn+k Â· Â· Â· wn+2kâˆ’1 Qk âˆˆ Z(i)[t]; âˆ€n âˆˆ Z =n + k âˆ’ 1Â¿0:

Proof. (a) is proved in [16] and (b) is proved in [25]. (c) and (d) comes from expression (a).

The expression of wn is
n
+ BÃ¿n :
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.
Ëœ (n) Ëœ (n)
The degrees of Qk and P k are, respectively, k + n âˆ’ 1 and k, so if we take t âˆˆ Q(i) with
vt âˆˆ Z(i), the above theorem implies that the following sequence:
Ëœ (n)
Ëœ (n)
ek; n :=f(t) Ã— v k dn+kâˆ’1 wn+k Â· Â· Â· wn+2kâˆ’1 P k (t) âˆ’ v k dn+kâˆ’1 wn+k Â· Â· Â· wn+2kâˆ’1 Qk (t);
where k = max{n + k âˆ’ 1; k} is a Diophantine approximation to f(t), if
(i) âˆ€n âˆˆ Z; limkâ†’âˆž ek; n = 0,
(ii) [n + k âˆ’ 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
Ëœ (0) and P k :=P (0) :
Ëœ Ëœ Ëœk
ek :=ek; 0 ; Qk :=Qk
From the asymptotics given in (3), we get
1=k 2
Ëœ
Qk (t) 1=k 2
Ëœ
1=k 2 k
lim sup |ek | 6 lim sup f(t) âˆ’ lim sup v dkâˆ’1 wk Â· Â· Â· w2kâˆ’1 P k (t) (4)
Ëœ
P k (t)
k k k

1=k 2
6|p|lim sup| kâˆ’1 | ; (5)
where k :=dk = k wi .
i=0
We will get limkâ†’âˆž ek = 0 if the following condition is satisÃ¿ed:
1=k 2
lim sup | kâˆ’1 | Â¡ 1=|p|:
kâ†’âˆž

Moreover, from the Christo elâ€“Darboux identity between orthogonal polynomials, condition (ii) is
satisÃ¿ed since the di erence
k 2
wiâˆ’1
k
Ëœ k+1 (t)P k (t) âˆ’ P k+1 (t)Qk (t) = t (âˆ’1)
Ëœ
Ëœ Ëœ 2k 2iâˆ’2 i i2
Q ABp ( âˆ’Ã¿) 2 2
A+B w2iâˆ’1 w2i w2iâˆ’2
i=1

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

Theorem 6. Let f be the meromorphic function deÃ¿ned by the following series:
âˆž
tn
f(t) = ;
wn
n=0
238 M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250
e

where (wn )n is a sequence of Z(i) satisfying a three-term recurrence relation
wn+1 = s wn âˆ’ p wnâˆ’1 ; s; p âˆˆ Z(i)
with the initial conditions: wâˆ’1 ; w0 âˆˆ Z(i). If for each integer m; there exists a common multiple
dm for the numbers {w0 ; w1 ; : : : ; wm } such that m deÃ¿ned by
dm
m := m
i=0 wi
satisÃ¿es the condition
2
1=m
lim sup | m| Â¡ 1=|p|; (6)
m

then for t âˆˆ Q(i); t = ( =Ã¿) j ; j = 0; 1; 2; : : : we have

f(t) âˆˆ Q(i):
See [25] 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 [2].)

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

For sums of series f, PadÃƒ approximation to the function f does not always provide Diophantine
e
approximation. Although the approximation error |x âˆ’ pn =qn | is very sharp, the value of the denom-
inator qn of the approximation may be too large such that |qn x âˆ’ pn | does not tend to zero when n
tends to inÃ¿nity.
Another way is the following.
Consider the series f(t) = âˆž 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 PadÃƒ 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 PadÃƒ approximation to Rn (t0 ) may not converge to Rn (t0 ),
e
(2) the denominator of the approximant computed at t0 , can converge to inÃ¿nity 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 deÃ¿ned as
âˆž
1
(s) = ; (7)
ns
n=1
M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250
e 239

where the Dirichlet series on the right-hand side of (7) is convergent for Re(s) Â¿ 1 and uniformly
convergent in any Ã¿nite region where Re(s)Â¿1 + , with Â¿ 0. It deÃ¿nes an analytic function for
Re(s) Â¿ 1.
Riemannâ€™s formula
âˆž
xsâˆ’1
1
(s) = d x; Re(s) Â¿ 1;
(s) ex âˆ’ 1
0

where
âˆž
ysâˆ’1 eâˆ’y dy is the gamma function
(s) = (8)
0

and
eâˆ’i s
z sâˆ’1
(1 âˆ’ s)
(s) = dz (9)
2i ez âˆ’ 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 âˆž
1 1
(s) = +
ks (n + k)s
k=1 k=1

âˆž
+ 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 âˆž (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):
âˆž
eâˆ’u=x
sâˆ’1
s (x) = u du; x Â¿ 0;
eu âˆ’ 1
0

whose asymptotic expansion is
âˆž
Bk
(k + s âˆ’ 1)xk+sâˆ’1 ;
s (x) =
k!
k=0

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