ñòð. 53 |

(mâˆ’n+1)

Rnâˆ’1 (t):= c ; ; Rnâˆ’1 âˆˆ Pnâˆ’1 ;

xâˆ’t

where c(mâˆ’n+1) acts on the letter x, then

mâˆ’n

Ëœ (mâˆ’n+1) (t) + t mâˆ’n+1 Rnâˆ’1

Ëœ (mâˆ’n+1) (t);

ci t i P n

Nm (t) =

i=0

Ëœ (mâˆ’n+1) (t) = t nâˆ’1 Rnâˆ’1 Ëœ (mâˆ’n+1) (t) = t n Pn

(mâˆ’n+1) âˆ’1 nâˆ’m

(mâˆ’n+1) âˆ’1

(t ) and i=0 ci t i = 0; n Â¡ m.

where Rnâˆ’1 (t ); P n

(n)

The sequence of polynomials (Pk )k , of degree k, exists if and only if âˆ€n âˆˆ Z, the Hankel

determinant

cn Â· Â· Â· cn+kâˆ’1

Hk(n) := Â·Â·Â· Â·Â·Â· Â·Â·Â· = 0;

cn+kâˆ’1 Â· Â· Â· cn+2kâˆ’2

where cn = 0 if n Â¡ 0.

M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250

e 233

In that case, we shall say that the linear functional c is completely deÃ¿nite. For the noncompletely

deÃ¿nite case, the interested reader is referred to Draux [15].

For extensive applications of PadÃƒ approximants to Physics, see Bakerâ€™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

ï£« ï£¶

Ëœ (mâˆ’n+1) (x))2

t mâˆ’n+1 (P n

(mâˆ’n+1) ï£ ï£¸

h(t) âˆ’ [m=n]h (t) = c

Ëœ (mâˆ’n+1) (t))2 1 âˆ’ xt

(P n

(mâˆ’n+1)

Ëœ

t mâˆ’n+1 (x))2

mâˆ’n+1 (P n

= x d (x): (1)

Ëœ (mâˆ’n+1) (t))2 1 âˆ’ xt

(P n R

Note that if c0 = 0 then [n=n]h (t) = t[n âˆ’ 1=n]h=t (t) and if c0 = 0 and c1 = 0; then [n=n]h (t) = t 2 [n âˆ’

2=n]h=t 2 (t).

Consequence: If is a nondecreasing function on R, then

h(t) = [m=n]f (t) âˆ€t âˆˆ C âˆ’ supp( ):

1.2. Computation of PadÃƒ approximants with -algorithm

e

The values of PadÃƒ 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; : : : ;

âˆ’1 0

1

(n) (n+1)

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

k+1 kâˆ’1 (n+1) (n)

âˆ’

k k

where Sn = n ck t k .

k=0

-values are placed in a double-entry array as following:

(0)

=0

âˆ’1

(0)

= S0

0

(1) (0)

=0

âˆ’1 1

(1) (0)

= S1

0 2

(2) (1) (0)

=0

âˆ’1 1 3

..

(2) (1)

.

= S2

0 2

. ..

.

(3) (2)

.

=0 .

âˆ’1 1

. . ..

. .

(3)

.

. = S3 .

0

234 M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250

e

The connection between PadÃƒ approximant and -algorithm has been established by Shanks [26] and

e

Wynn [35]:

âˆž

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 (Markovâ€“Stieltjes function)

(see [29,6,11] for a survey).

2. Diophantine approximation of sum of series with PadÃƒ approximation

e

Sometimes, PadÃƒ 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 âˆ’ 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.

âˆš

Theorem 3. Let r =a=b; a âˆˆ Z; b âˆˆ N; b = 0; with b:e:(1âˆ’ 1 âˆ’ r)2 Â¡ 1(ln e=1) Then -algorithm

(n)

applied to the partial sums of f(r):= âˆ’ ln(1 âˆ’ r)=r = âˆž r i =(i + 1) satisÃ¿es that âˆ€n âˆˆ N; ( 2k )k

i=0

is a Diophantine approximation of f(r).

Proof. From the connection between PadÃƒ approximation, orthogonal polynomials and -algorithm,

e

the following expression holds:

Ëœ (n+1)

n

Rkâˆ’1 (r)

ri Nn+k (r)

(n)

+ r n+1 (n+1)

= = (n+1) ;

2k

i+1 Ëœ Ëœ

P k (r) P k (r)

i=0

where

k k +n+1

k

Ëœ (n+1) (n+1)

k

Pk (t âˆ’1 ) (1 âˆ’ t)i

P k (t) =t =

k âˆ’i i

i=0

Ëœ (n+1)

is the reversed shifted Jacobi polynomial on [0,1], with parameters = 0; Ã¿ = n + 1; and Rkâˆ’1 (t) =

P (n+1) (x)âˆ’P (n+1) (t)

(n+1) (n+1)

t kâˆ’1 Rkâˆ’1 (t âˆ’1 ) with Rkâˆ’1 (t) = c(n+1) ; k ( c(n+1) ; xi :=1=(n + i + 2)) (c acts on the

k

xâˆ’t

variable x ).

Ëœ (n+1) Ëœ (n+1)

Since P k (t) has only integer coe cients, bk P k (a=b) âˆˆ Z:

Ëœ (n+1)

(n+1)

The expression of Rkâˆ’1 (t) shows that dn+k+1 bk Rkâˆ’1 (a=b) âˆˆ 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 âˆ’ a=b):

M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250

e 235

Ëœ (n+1) (n)

The proof needs asymptotics for dn+k+1 , for P k (a=b) and for ( 2k âˆ’ f(r)) when k tends to

inÃ¿nity. dn = en(1+o(1)) follows from analytic number theory [1].

Ëœ (n+1)

limk (P k (x))1=k =x(y + y2 âˆ’ 1); x Â¿ 1, y =2=x âˆ’1, comes from asymptotic properties of Jacobi

(n)

polynomials (see [30]), and limkâ†’+âˆž ( 2k âˆ’ f(r))1=k = (2=r âˆ’ 1 âˆ’ (2=r âˆ’ 1)2 âˆ’ 1)2 (error of PadÃƒ

e

approximants to Markovâ€“Stieltjes function).

So

1=k

(n+1)

Ëœ

lim sup dn+k+1 bk P k (a=b)f(r) âˆ’ dn+k+1 bk Nn+k (a=b)

kâ†’+âˆž

1=k 1=k

Ëœ (n+1) (n)

1=k k

6 lim sup(dn+k+1 ) lim sup b P k (a=b) lim sup + 1=rln(1 âˆ’ r)

2k

kâ†’+âˆž kâ†’+âˆž

k

(2=r âˆ’ 1)2 âˆ’ 1)2

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

âˆš

1 âˆ’ r)2 Â¡ 1

=e:b:(2=r âˆ’ 1 âˆ’ (2=r âˆ’ 1)2 âˆ’ 1) = e:b:(1 âˆ’

by hypothesis, which proves that

Ëœ (n+1)

lim (dn+k+1 bk P k (a=b)f(r) âˆ’ dn+k+1 bk Nn+k (a=b)) = 0:

âˆ€n âˆˆ N;

kâ†’+âˆž

Moreover,

(n+1)

r 2k+n+1 1

(Pk (x))2

(n)

(1 âˆ’ x)n+1 d x = 0:

+ 1=r ln(1 âˆ’ r) = âˆ’

2k (n+1)

1 âˆ’ xr

Ëœ

(P k (r))2 0

(n)

1 âˆ’ a=b)2 Â¡ 1.

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

t n =wn

2.2. Irrationality of

âˆžn

The same method as previously seen provides Diophantine approximations of f(t):= n=0 t =wn

when the sequence (wk )k satisÃ¿es a second-order recurrence relation

wn+1 = swn âˆ’ pwnâˆ’1 ; n âˆˆ N; (2)

where w0 and wâˆ’1 are given in C and s and p are some complex numbers.

We suppose that wn = 0; âˆ€n âˆˆ N and that the two roots of the characteristic equation z 2 âˆ’ sz +

p = 0, and Ã¿ satisfy | | Â¿ |Ã¿|.

So wn admits an expression in term of geometric sequences: wn = A n + BÃ¿n ; n âˆˆ N.

The roots of the characteristic equation are assumed to be of distinct modulus (| | Â¿ |Ã¿|), so there

exists an integer r such that | =Ã¿|r Â¿ |B=A|.

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

âˆž

tk

f(t):=

A + BÃ¿k

k

k=0

236 M. PrÃƒ vost / Journal of Computational and Applied Mathematics 122 (2000) 231â€“250

ñòð. 53 |