Rev.Acad.Canar.Cienc., IV (núms. 1 y 2) , 111 - 124 (1992)
PADE-TYPE APPROXIMANTS FOR A FORMAL LAURENT SERIES
M. Camacho and P. González-Vera
Departamento de Análisis Matemático
Universidad de La Laguna
ABSTRACT
In this paper, we obtain the Laurent Padé Approximats (LPA) to a given
formal Laurent series, as a Laurent Padé-Type Aproximant (LPTA) of higher
order, in a similar way as the process carried out by Brezinski ((!])., for the
classic case. For this purpose a new concept of Padé-Type Approximant to a
Laurent series is introduced.
Keywords: Padé-Type Approximants, Formal Laurent series, Laurent polynomials.
AMS Classification: 40G; 41
l. PRELIMINARY CONSIDERATIONS.
Let {ck}: 00 be a bi-infinite sequence of complex numbers, and L the set of
all the formal series G(z) of the form,
(l.l)
We shall denote by TI n the linear space of the polynomials of degree n at
most, and by TI = U II n the space of all polynomials. For every pair of integers
p,q with p ~ q, we shall denote by A the linear space of the Laurent
p,q
polynomials {L-polynomials) or functions of the form
L(z) =fa./
J=p
andA=UA the linear space of all L-polynomials
p,q
(obviously JI =A ).
n O,n
For each p,q e Z with p~q. the projection
ílp,q G(z) = f e/
operator is defined by
J::p
(l.2)
For a formal Laurent series, if the nonzero coefficients extend only to
one side, which can be +oo or -oo, it is necessary to have a notation to
indicate which part of the formal Laurent series is zero. So, the notations O+
and O need to be defined as follows
"' G(z) = o.czm) .. n...,,m-IG(z) = o* G(z) = L e/
J~m
G(z) = O_(zm) "ílm+t,.,GC,zl =O" G(z) -~e/
being G(z) e L.
As a natural extension of the classical concept of Padé Approximant to a
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formal power series, in [S] the so-called Laurent Padé Approximants (LPA) to a
formal Laurent series have been defined. On the other hand Bultheel, in [21
makes an algorithmic and algebraic study for such those approximations.
lndeed, splitting G(z) in two, that is G(z) = dº1Czl + G101Czl, belng .,
G {0) (z) = e /2 + e z l k
o k
(l.3a)
~31
"'(O) l -k G (z) = e /2 + e z o -k
(l.3b)
k=l
we have f!rst the following
Definition l ([2], p.26)
Let m and n be nonnegatlve integers and the ratlonal functlons R(z)
p m(z)
--:
p m(z)
R(z) = -.--, where P m (z), Q (z),P m (z) , Q0 (z) e /\, such that
Qn(z)
l) Q (z) e /\ and Q (z) e /\
n O,n n -n.o
2) G101Cz) - R(z) = O ,<zm•n•I¡
3) ¿co>(z) - Rczl = O (z -<m+n+l))
4) R(z) + R(z) is equivalent with the quotient of L-polynomials P(z)/Q(z),
with P(z) e A and Q(z) e /\ . We shall then call the pair (R(z),R(z)) a
-m,m -n,n
Laurent Padé Approxtmant (LPA) of type (m,n).
These approximants are obviously an extension of the classic Padé
Approximants because the Laurent expansion of R(z)+R(z) matches the
coefficients of zk in L for 1k1 !S m+n.
On the other hand, as is well known, Padé-Type Approximants (PTA) are an
special interesting case in the study of the rational approximations, because
of the free election of the denominator mean, that a lesser number of
coefficients in the initial series is required to be known. Such approximants
used to provide, in sorne cases, better estimations than Padé Approximants (see
[l] for more details).
Thus, we can also consider rational approximants to F(z) with a given
denominator in the sense of the following
Definition 2
Let m and n be nonnegative integers and Qn (z) e '\,n ,Qn (z) e A _n.o two
• P (z)
given L-poiynomials. The pair (R(z),R(z)) where R(z)~
Qn (z)
R(z)=
p m(z) .
-.-- ,1s a
Qn (z)
Laurent Padé-Type Approximant (LPTA) of type (m,n) for F(z) if the foiiowing
conditions are satisfied
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1) P (z), P (z) e A
2) ;coi(z) ".'. R(z) = O (zm+l)
3) c'º'(z) - R(z) = o (z =1m•ui
4) R(z) + R(z) is equivalent with P(z)/Q(z), where P(z)eA and Q(z)eA .
-m,m -n,n
Q(z) will be called the generating L-polynomial of the approximant. Such
approximants have been studied by us from an algebraic point of view (see [3],
[4]).
In this paper, we shall obtain the LPA as a LPT A of higher arder, making
use of an analogous process followed in [l) far the classic case. Far this
purpose it is necessary to define the concept of Padé-Type Approximant to a
Laurent series in a similar way as that used by Bultheel in (21.
2. PADE-TYPE APPROXIMANTS TO A FORMAL LAURENT SERIES.
In ((2]. p.25) the concept of Padé Approximant far a Laurent series was
defined as follows
Definition 3
Let m and n be integers and n~O,. then the pair (Am(z)/Bn(z)) with Bn(z) e íln
and A m (z) = o_ (zm) is a Padé Aμproximant of type (m,n) far the formal Laurent
series (1.1) and we shall denote by PAL, if
(2.1)
Observe that A m (z)/B n {z) should be understood as a notation fer a pair of
elements rather as a quotient.
In the particular case in which ck = O for k=-1,-2, ... (that is, G(z) is
a formal power series) and mi!::O then Am(z) e Tlm and the pair (Am(z)/Bn(z))
coincides with the classic [m/n] PA for G(z).
It can be easily seen that a necessary and sufflcient condition fer the
existence of Am(z) and Bn(z) (except a multiplicatlve factor) is IT~mll ~ O,
meZ; nelN, T~m) being the Toeplitz matrices represented by
Tn{ml= ( l n ºm•l-J •
1,J::O
e e
m m-1
cm+l cm e m-n+l
e e e
m+n m+n-1. m
In such case it will be said that G(z) is a normal series.
Since we have only taken into account in the correspondence conditions
the iñcreasing character (i.e. toward +co) we might wonder if an alternative
definition with decreasing correspondence conditions (toward -o::i) could be
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given. In this way, the rational approximants defined in (2.1) wlll be called
Padé Approximants in an lncreasing sense. Thus, to define' the PAL in
decreasing sense the so-called dual series F(z) to G(z) will be required. Thus
we have
and if we denoted by s~m)
following
Proposition
w!th d =e , keZ (F(z)=G(z-1))
k -k
= [d ln == [e ln we
m+l-J -m-l+J
1,J•O 1,J"'O
G(z) is normal iff F(z) is normal too.
Proof.
One has
[S(ml)T- [d Jn -
n - m+J-1 l,J=O-have
first the
m=0,±1,±2, ... ; n=0,1,2, ...
and the proof follows .•
Let (Am(z)/Bn(z)) be the PAL of type (m,n) in !ncreasing sense for F(z),
with
Am(z) = O (zm)
then, by (2.1) we have
Thus,
~ a z) and B (z) = {\ b ZJ; B (0) =b ,oQ _fu J n J~O J n o
F(z-1l'8n(z-1) - Am(z-1) = O_(z-(m+n+l))
and consequently if we write
it follows that
G(zJ8n(z) - Am(z) = O_(z-(m+n•U)
The pair (;.\m(z)/í3n(z)) it will be called the PAL in decreasing sense of
type (m,n) for G(z) the following immediately holds
Proposition 2
The pair cAmcz-1)/8ncz-1)) coincides with the PAL in increasing sense for the
dual series F(z).
Proof.
It is enough to take into account the unicity property of the PAL for
G(z) .•
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In [2], connectlon between the LPA and PAL.. (see [2), Theorems 4.2 and
4.3) is exhibited, On the other hand, lf (R(z),R(z)) represents the LPA of
type (m,n) for G(z) and (S(z),S(z)) is the LPA of type (m,n) for the dual
series F(z), then we have
Proposition 3
(R(z-1),R(z-1)) coincides with (S(z),S(z))
Proof.
Since (R(z),R(z)) is the LPA for G(z), and from condltlons 2 and 3 in the
Definition 1 we have,
GCOl(z-1) _ R(z-1) = O+((z-1)m+n+1) = O_(z-Cm+n+ll)
¿coJ(z -1) _ R(z-1) = 0 ((z-lf(m+n+l)) = 0 + (zm+n+l)
Now, taking F(z) = F'°'(z) + r'º'Czl, with
"'(Ol r -k
F (z) * c0/2 + k~I ckz , we have
m
F{OJ(z) = c0/2 + l c_kzk and
G10'Cz-1)=F'º'Czl and ~(Ói(z-1 )=F'º'Czl and replacing in (2.2) one has
the pair (R(z - 1).R(z - 1)) constitutes the LPA far the dual series F(z).•
(2.2a)
(2.2b)
that
From the two last propositions we can conclude that it does not matter
which definition of PAL (increasing or decreasing) can be taken. We shall
consider throughout the paper the PAL in the increasing sense.
Now, as a new extension of the classical concept of Padé-Type Approximant
we shall define this one for a Laurent series, as follows
Def inition 4
Let m and n be integers with n?:::O, and the polynomial Bn(z) e rrn with Bn(Q):;!:Q,
The pair (Am(z)/Bn(z)), being Am(z) = O_(zm), is said to be a Padé-Type
Approximant of type (m,n) fer the formal Laurent series (1.1) with generator
polynomial 8 n (z) and we shall represent it by PTAL, if
G(z)Bn(z) - Am(z) = o.Czm+t) ; (2.3)
It is readily seen that the coefficients of A m (z) can be obtained from
the coefficients of the series and from the generator polynomial. lndeed, if
we write A m (z)
are given by
-~ a zJ and 8 n (z) = f b zJ by (2.3), then the coefficients
.., j J=O J
n
ªi .t cJ-kbk, j= .. .,m-2,m-l,m (2.4)
When m~O and G(z) is a formal power series (ck =O if k::s-1), then by (2.4),
one has, ªtº for j::s-1, that is, Am(z) e Ilm and the pair (Am(z)/Bn{z)) takes
sense as a quotient, and represents the (m/n) PTA for the formal power series
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G(z) with denominator 8 n (z).
On the seque! we shall show how an LPTA can be deduced from two PTAL. For
that, we shall use a more general decomposition than (1.3) for G(z). So if
meZ, then we write
with
Theorem l.
.,
o<m>(z) =l/2cmzm + [ ckzk and
k::m+l
m-1
¿cm>(z) =V2c zm + \ e zk
m -~ k
(2.Sa)
(2.Sb)
a) Let (P!m>(z)/Q!m>(z)) be a PTAL of type (m+n,n) for the normal series
(1.1) being Q!m>(z) E Jin its generator polynomial such that Q!ml(O) = 1, and
define A!m)(z) by
zmA!m)(z) = P~ml(z) - G(m)(z)Q!ml(z}
A!m1(z)
Then A!m1(z) is a polynomial of degree n and R~m 1 (z) = --o!
m>(z)
to the formal series z-mG(ml(z).
(2.6a)
is a (n/n) PTA
b) If (p<m>(z)/Q 1ml(z)) is a PTA of type (m-1,m) far G(z) with generator
n n ... ~-nA.(m){z)
polynomial Qn(ml(z), then R<m>(z) = --. "-- represents a (n/n) PTA to the
n 2 -nQ~m)(z)
formal power series z-m¿<m>(z). Now A~m1 (z) is given by
(2.6b)
Proof.
a) If we denote by E!ml the residual of the PTAL (P!ml(z)/Q!ml(z)), then
E~ml(z) = G(z) Q~m)(z) _ p~ml(z) = º• (zm+n+l)
hence, by (2.5),
and consequently,
Glml(z) Q(ml(z) - Elml(z) = p<m>(z) - ¿<m>(z) Q(ml(z) (2.?a)
n n n n
Now, taking into account that P~ml(z) = o.Czm•n•l). G(ml(z) o.<zm),
the left-hand side of (2. 7a) is O• (zm), meanwhile the right-hand side is
O (zm•n). thus (2. 7a) represents a polynomial of degree m+n, denoted by
zmA~ml(z), with A~ml(z) E TTn, hence
(2.?b)
and
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G(ml(z) Q!m>(z) - zmA~m>(z) = E!m>(z) = O+(zm+n+l). (2.8)
After dividing by zmQ:m1Cz) we get
A(ml(z)
z-mc<m>(z) - -"-- = o.(zn+l) (2.9)
QCml(z)
A!m>(;)
which preves that is an (n/n) PTA to the series z-mG'm>(z)
Q!ml(z)
b) Let now (PCml(z)/QCml(z)) be a PTAL far G(z) of type (m-1 .• nl, · and
denote by E:m1(z) th: residu:l, then we have
(2.!0al
Hence,
being the
therefore
left-hand side O+ (zm) and the right-hand side o_ (zm+n), and
zmA!ml{z) = G(m)(Z) Q~m)(z) - P!m1(z)
with A.~ml(z) E íln. Consequently
G(m)(z) o:ml(z) - zmA:m}(z) = P!ml(z) = O_{zm-1)
and
and the proof follows .•
Fer the sake ef simplicity, if we set m=O then one has
Corollary 1
Under the same cenditiens as in Theerem 1, the pair
[A~º'Czl • z-n~~º'Czll
Q:01(z) z-nQ!º1(z)
represents a LPTA ef type (n,n) for the series G(z).
Proef.
Use Theorem 1 and Definition 2 .•
(2.IOb)
(2.11)
(2.12)
As a result, we have seen how from two PTAL of types (-1,n) and (n,n) we
can ebtain an LPTA of type (n,n) fer the same formal Laurent series G(z).
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3. HIGHER ORDER APPROXIMANTS
Since to obtain the LPTA for a formal Laurent series G(z) we need two
arbitrary polynomials Q~o>(z) and Q~01(z), we might think how to choose these
ones, so that the arder of correspondence were increased as much as possible.
Thus, the so-called higher arder approximant arise (see (1)). On the sequel,
when m=O, upperindex (O) will be deleted.
then,
Let us suppose that the pair (Pn(z)/Qn(z)) is a PTAL of type (n,n), where
P (z) = r a ZJ Q (z) = r b ZJ , b ~Q
n _fu j • !\. J~O J O
By (2.4), one has
j= .. ,n-2,n-l,n
Moreover, because of {Pn(z)/Qn(z)) is a PTAL of type (-1,nl, with
-r ~ZJ = 0 (z-1), Q {z) = r bzJ, b :;tQ _fu J - n J~O J O
re b. L J-k k
k=O
j=-1,-2, ..
Now, if En(z) is the residual of the PTAL (Pn(z)/Qn(z)), one has
~
En(z) = G(z) Qn(z) - Pn(z) = O (z"•1¡ = le zJ
• J=n+IJ
and the following Theorem holds.
Theorem 2
With the above notations if the coefficients bJ and bJ of the denominator
polynomials satisfy the conditions
ln - - e b = a = O i=l,2, .. ,p (lspsn) (3.la)
J=O -1-J J -1 f e b = e = O i=n+l,n+2,.. ,n+q (3. lb)
J=O 1-J J 1
then, for the pair [-- , ----- one has,
An(z) z-"An(z)l
Q (z) z-"Q (z)
n n •
A (z) z-"A (z)
dº)(z) - -"- = o.(zn•q•l) and G(O'(z} - --.-"-=o (z-n-p-l) (3.2)
Qn(z) z-"Qn(z) -
Proof.
Let us first suppose that (3.la) holds, then Pn(z)=O (z-p-l) and by
(2. lll one has
and hencef orth
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.. z-nA. (z)
G(O)(z) - --.-"-=O (z-n-p- I)
z-nQn(z) -
"(OJ In other words, we see that the arder of correspondence far the PTAL to
G (z) has been increased up to n+p+l.
From (3.lb), we have En(z) = o.(zn+q+I} so, by using (2.8), it results
dºl(z) Qn(z) - An(z) = En(z) = o.(zn+q+I}
and consequently
In such that case, it will be said that
represents an LPTA of higher arder.
Obviously, the highest arder of correspondence will be reached
n
when p=q=n, and the coefficients of the polynomial Qn (z) ¿bzJ, will be
J=O J
determined from the linear system of equations
f e b = O i=n+l,n+2, .. ,2n
J=O 1-J J
(3.3)
Since G(z) is a normal series, then the polynomial Qn (z) satisfying the
relations (3.3) is uniquely given by
n
z
e e e
n<l n 1
e e e
Qn(z) = K
n•2 n•I 2
(K*O)
e e
Zn Zn-1
In short, Qn (z) coincides apart a multiplicative factor with the
denominator polynomial of the [n/nJ PA to the formal power series G101(z) (see
[!]).
On the other hand, if Qn (z) satisfies the conditions (3.la) with p=n,
then the coefficients bJ are now obtained by solving the linear system
n • l c _1_JbJ = O i=l,2, .. ,n
j=O
(3.4)
Taking into account that f b z-J
J=O n-J
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hence
F (z)
1
-1
_ z An(z)
z-1F1(z) = O+(zn)
Qn (z)
and z-1A {z) e TI , that is,
n n-1
z- 1 A (z)
G<01(z) - --"- = O (z0 )
Qn (z} •
(3.5)
Let (P n (z)/Qn (z)) be now the PTAL to F(z) of type (-1,n), being Qn (z)
the generator polynomial, then by (b) in Theorem 1, there exists A n (z) e TT n
such that
F (z)
1
z-0 A
0
(z)
z-ºQn (z)
= O (z-n-1)
Moreover, the highest degree coefficient is zero, because of
d0=0, that is, A0(z) e TI0 _1, and then ..
z - Cn-1> An(z)
z F1Czl - ---.~-z-
1:Q" (z)
and due to the fact that zF (z)
1
z-cn-I 1 An(z)
{;'º'¡zi -
z-"Q 0 (z)
G'°>(z), one has
from (3.5) and (3.6) the proof follows .•
REMARK 1:
(3.6)
[
z - 1 A (z) z-!n-ll A (z)]
According to Definition 2, the pair --"- , -n ... " is not
Qn(z) z Qn(z)
estrictly an LPTA of type (n-1,n), because when adding both components, the
resulting rational function has as numerator and denominator L-polynomials
both in the linear space A_010• However, taking into account that Q0 (0)Qn(Q):;tQ
then by ([4],p.11), we can obtain two constants ªn+ and ªn such that the
pair
[a~z"+ z- 1An(z) a~z-0+ z-<n-OA.0 (z)]
Qn (z) ' z-"Qn(z)
is in effect an LPTA of type (n- 1,n).
in the next two Theorems, we shall see how obtain LPTA of arbitrary type
(m,n) with m and n nonnegative integers. For this purpose, we first define the
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is the denominator of the second component of the LPTA of type (n,n), and
because of Qn(z) Jt bn_/ (b1 solutions of (3.4)), represents the
denominator of the PA fer the series o:>
¿o>(z-1) =co/2 + l c_kzk,
k=l
then Qn (z-1) = z-nQn (z) will be the denominator of the classic PA to G(O)(z).
In other words, we have proved the following
Theorem 3
satisfied by the coefficients
pair so that conditions (3.3) and (3.4)
of the polynomials Qn (z) = f b ZJ
J=O j
are
and
Q (z)= f b z1 respectively then, such pair represents the LPA of type (n,n) to
n j:Q J
the Laurent series G(z).
Nextly, we shall concentrHe on how obtaining the LPTA of arder (n-1,n)
"' for G(z). Fer this purpose we define F(z) = _fu dkzk, with d o= O·• d 1= d -1 =e o/ 2·•
d1=cl-l if i=2,3,.. and d1=c1• 1 if i=-2,-3, ... Then one has
Proposition 4
Let ~ , --,-"-
A (z) z-"A (z)l
be a LPTA of type (n,n) for F(z), with n>ol then, the
Qn (z) z-"Qn (z)
[
z- 1 A (z) z-(n-ll A (z)l
pair --"- , -n ,.. n represents the LPTA of
Q (z) z Q (z)
n n ,
with the sarne generator polynomial , being Qn (O) Qn (Q):;i!;Q.
Proof.
Set F (z)
l
F (z)
- l
so that F(z) = F (z) + F (z).
l l
ca z
= O + 2 z + c1z + ..
e
= O + ¡. z-1+ c1z-2+ ...
type (n-1,n) for G(z)
Let (Pn(z)/Qn(z)) be the PTAL for F(z) with Qn (0)=1. If we denote by
En (z) the residual, that is,
E (z) = f(z) Q (z) - P (z) = O (zn•l¡
n n n +
using (a) in Theorem 1, we can obtain An(z) = F1(z)Qn(z) - En(z), An(z) being
a polynomial of degree n. Furthermore, since d0 =O, one has A n (O)=O and
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.,
formal Laurent series F(z) = -~ dkzk, with d1=0; -(k-l):Si::!ik-1
d±k =c0/2 and d1=c1•k if i=-(k+l),-(k+2), ...
ct1=c1_k if i=k+l,k+2, ...
clearly, taking
then F(z) = Fk(z) + F,(z) (lsksn), and
z-'F,(z) = G'º'<zl and z'Fk(z) = G'º'<zl (3.7)
one has
Theorem
If the pair [
ACkl(z) z-"A.'"<zll
-"-- , _:... represents the LPTA for F(z) of type (n,n),
Q0 (z) z Q0 (z)
with denominator polynomials Qn (z) and Qn {z), then the pair
[
z-k<"'(z)·z-Cn::~ A~"(z)l
º·(z) z o.(z)
(3.8)
represents an LPTA of type (n-k,n) for G(z).
Proof.
Taking into account the correspondence conditions far the LPTA
[
ACkl (z) z-"A."'(z)l
-"--, _ ~,. and by (3. 7), it is easy to see that
Q0 (z) z Q0 (z)
z-kA!kl(z)
G(O)(z) - n (3.9) º• (z)
z-n•k A.<kl(z)
G'º'(z) - = O (z-Cn-kl-1) (3.10)
z-"Qn(z)
Now, from (2.7b) one has
<k'(z) = F,(z) Q0 (z) - E0 (z), being E0 (z) = 0,(z".1 )
and since the k-1 terms far F (z) are
k
(k) n J
zeros, A (z) = [ a z ,
n J :::k j
that is,
z-kA~k1(z) is a polynomial of degree :S n-k.
Moreover, from (2.lOb), the polynomial A~kl(z) = fk(z) Qn(z) - ¡;n(z)
will be of degree !!ó n-k, and by this, z-ln-k)A~k)(z)
the proof follows .•
122
A and
-(n-k),O
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In order to determine an LPTA of type (n+k,n), with k i!'; 1, we must
introduce the formal Laurent series H(z) = Hk (z) + Hk (z) being
H (z) = e + e z+ ...
H\z) = ck +e k+t z-1+ ...
k -k -k-1
(3.lla)
(3.llb)
Let [--- , n... be an LPTA of type (n,n) for H(z) obtained
<kl(z) z-n;.(kl(z)l
Qn(z) z-nQn{z)
from the PTAL of types (n ...n. ) and {-1,n) for Hk(z) and Hk{z) and with
denominator polynomials Qn (z), Qn (z) respectively Let
A (z) k-1, A(kl(z)
~ = l CZJ + Zk -"-- (3.12)
Qn(z) J=O J Qn(z)
be a rational function, being A n+k (z) e Il n+k and consider
z-(n+k) A (z) k- 1, z-(n+k) A(k)(z)
-__ _;nc_•cc.k_ \ -J n
= L e z +
Z-nQn(z) J::O -j Z-nQn(z)
k ... k-1, -j ... {k)
and An+k(z) = z Qn(z) l c_1z + An (z)
J::O
With these notations we have the next
E Jl
n•k
(3.13)
Theorem 5.A~
k 1(z)
Let
z-n;.(k)(z) l
, -~" be the LPTA of type (n,n) for H(z) with generator
z Qn(z)
polynomials Qn (z) and Qn (z), then the pair
[
An•k(z) z-<n•kl An•k(z)l
Qn(z)' z-"Qn(z)
represents a LPTA for G(z) of type (n+k,n) (with the same generator
polynomials), with An+k(z) and An+k(z) given by (3.12) and (3.13)
respectively.
Proof.
By using the definition of LPTA, one has
zk A (kl(z)
zkH (z) - --"-- = O (zn+k+t)
k Q (z) •
n •
z-(n+kl A~kl(z)
z-kH_k (z) = O (z-n-k-t)
z-nQn(z)
Now, because of
(0) k-1, J k
G (z) = L e z + z H (z)
J=O J k
k-1, ...
and G<ol = [ e z-J + z-kH (z)
J=O -J k
123
(3.14a)
(3.14b)
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from (3.12), (3.13) and (3.14), it yieids
A (z)
G<o>(z) - ~ = O (zMk•l) and
·coi z-Cn+k l A (z)
G - --~-•-•_•_ = O (z -n-k-1)
o.<zJ • z-"Q0 (zl
and the Theorem follows .•
REFERENCES
[l) C. Brezinski, Padé-Type approx!mants and general orthogonaL polynomtals,
(Birkhaüser, Base!, 1980).
[21 A. Bultheel, Laurent series and their Padé Approx!mattons (OT, Vol. 27
BirkhaUser, Basel 1987)
(3] M. Camacho, P. González-Vera, "Aproximaciones racionales a series
doblemente infinitas", XIII Jornadas HCspanoLusas de Matemáticas (Universidad
de Valladolid, 1988) to appear.
( 4 J M. Camacho, Sobre funciones racionales asociadas a sucesiones doblemente
infinitas, (Doctoral Thesis; Unive:sidad de La Laguna, 1991)
(5) W .B. Gragg, G.D. Johnson, "The Laurent Padé Table". Informatf.on Processing
74, (North Holland, Amsterdam, 1974) pp. 78-89.
124
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