ñòð. 54 |

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

ñòð. 54 |