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e


admits another expansion
r’1 ∞
[(’B=A)(ÿ= )r’1 ]k
tk tr
f(t) = ’r ;
A + Bÿk A t= ’ ( =ÿ)k
k
k=0 k=0

where r ∈ N is chosen such that | |r |A| ¿ |ÿ|r |B|.

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

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

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

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

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

k
k i
A + Bq n+k’j
˜ (n) i(i’1)=2 i
(a) P k (t) = q (’t= ) ;
A + Bq n+2k’j
i
i=0 j=1
q

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


k
= 1:
0 q


k’1
˜ (n) (1 ’ tqj = )|6R|q|k ;
(b) | P k (t) ’ k¿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; w’1 ; w0 ∈ Z(i); for all common multiple dm of {w0 ; w1 ; : : : ; wm }

˜ (n)
(c) wn+k · · · wn+2k’1 P k ∈ Z(i)[t]; ∀n ∈ Z=n + k ’ 1¿0
M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e 237


and
˜ (n)
(d) dn+k’1 wn+k · · · wn+2k’1 Qk ∈ Z(i)[t]; ∀n ∈ Z =n + k ’ 1¿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
+ Bÿn :
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.
˜ (n) ˜ (n)
The degrees of Qk and P k are, respectively, k + n ’ 1 and k, so if we take t ∈ Q(i) with
vt ∈ Z(i), the above theorem implies that the following sequence:
˜ (n)
˜ (n)
ek; n :=f(t) — v k dn+k’1 wn+k · · · wn+2k’1 P k (t) ’ v k dn+k’1 wn+k · · · wn+2k’1 Qk (t);
where k = max{n + k ’ 1; k} is a Diophantine approximation to f(t), if
(i) ∀n ∈ Z; limk’∞ ek; n = 0,
(ii) [n + k ’ 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
˜ (0) and P k :=P (0) :
˜ ˜ ˜k
ek :=ek; 0 ; Qk :=Qk
From the asymptotics given in (3), we get
1=k 2
˜
Qk (t) 1=k 2
˜
1=k 2 k
lim sup |ek | 6 lim sup f(t) ’ lim sup v dk’1 wk · · · w2k’1 P k (t) (4)
˜
P k (t)
k k k

1=k 2
6|p|lim sup| k’1 | ; (5)
where k :=dk = k wi .
i=0
We will get limk’∞ ek = 0 if the following condition is satisÿed:
1=k 2
lim sup | k’1 | ¡ 1=|p|:
k’∞

Moreover, from the Christo el“Darboux identity between orthogonal polynomials, condition (ii) is
satisÿed since the di erence
k 2
wi’1
k
˜ k+1 (t)P k (t) ’ P k+1 (t)Qk (t) = t (’1)
˜
˜ ˜ 2k 2i’2 i i2
Q ABp ( ’ÿ) 2 2
A+B w2i’1 w2i w2i’2
i=1

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

Theorem 6. Let f be the meromorphic function deÿned by the following series:

tn
f(t) = ;
wn
n=0
238 M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e


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

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

f(t) ∈ 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 Padà approximation to the asymptotic expansion of the
e
remainder of the series

For sums of series f, Padà approximation to the function f does not always provide Diophantine
e
approximation. Although the approximation error |x ’ pn =qn | is very sharp, the value of the denom-
inator qn of the approximation may be too large such that |qn x ’ pn | does not tend to zero when n
tends to inÿnity.
Another way is the following.
Consider the series f(t) = ∞ 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 Padà 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 Padà approximation to Rn (t0 ) may not converge to Rn (t0 ),
e
(2) the denominator of the approximant computed at t0 , can converge to inÿnity 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 deÿned as

1
(s) = ; (7)
ns
n=1
M. Prà vost / Journal of Computational and Applied Mathematics 122 (2000) 231“250
e 239


where the Dirichlet series on the right-hand side of (7) is convergent for Re(s) ¿ 1 and uniformly
convergent in any ÿnite region where Re(s)¿1 + , with ¿ 0. It deÿnes an analytic function for
Re(s) ¿ 1.
Riemann™s formula

xs’1
1
(s) = d x; Re(s) ¿ 1;
(s) ex ’ 1
0

where

ys’1 e’y dy is the gamma function
(s) = (8)
0

and
e’i s
z s’1
(1 ’ s)
(s) = dz (9)
2i ez ’ 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 ∞
1 1
(s) = +
ks (n + k)s
k=1 k=1


+ 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 ∞ (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):

e’u=x
s’1
s (x) = u du; x ¿ 0;
eu ’ 1
0

whose asymptotic expansion is

Bk
(k + s ’ 1)xk+s’1 ;
s (x) =
k!
k=0

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. 54
( 83 .)



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