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Electronic Transactions on Numerical Analysis.
Volume 25, pp. 302-308, 2006.
Copyright 2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
MORE EXAMPLES ON GENERAL ORDER MULTIVARIATE PADÉ
APPROXIMANTS FOR PSEUDO-MULTIVARIATE FUNCTIONS
PING ZHOU
Dedicated to Ed Saff on the occasion of his 60th birthday
Abstract. Although general order multivariate Padé approximants have been introduced some decades ago, very
few explicit formulas have been given so far. We show in this paper that, for any given pseudo-multivariate function,
we can compute its general order multivariate Padé approximant for some given index sets with the
usage of Maple or other software. Examples include a multivariate form of the sine function
'(
*)
$&%!
!#"
)
+,-*/.*%!0
a multivariate form of the logarithm function
1
#
'(324
-56.
a multivariate form of the inverse tangent function
7
8#98
'(
:;"
<$&%!
*)
)
+=-*6.>(%
and many others.
Key words. multivariate Padé approximant; pseudo-multivariate function
AMS subject classification. 41A21
1. Introduction. Multivariate Padé approximants have been extensively investigated in
the past few decades. The existence, uniqueness and non-uniqueness for homogeneous and
general order multivariate Padé approximants and some convergence theorems have been established [3], [4], [5]. Despite all these activities, there are very few explicit constructions of
multivariate Padé approximants. By using the residue theorem and the functional equation
method, several researchers have successfully constructed multivariate Padé approximants to
some functions which satisfy functional equations [2], [11], [12], [13]. Unfortunately, not
many functions satisfy those functional equations. Besides, because the index sets for the
numerator and denominator polynomials can not be chosen freely, most numerators of the
approximants look complicated. In Cuyt-Tan-Zhou [7], we explicitly construct multivariate
Padé approximants to so-called pseudo-multivariate functions, by using the Padé approximants of particular univariate functions which, in most of the cases, are the univariate projections of the pseudo-multivariate functions obtained by letting all but one variable be zero.
Examples given in [7] are the general order multivariate Padé approximants for the multivariate form of the exponential function
?A@BDCFE;GH
J I
KL M!N=O
W
B
K E M
@PRQ9STGVU
C
Received February 25, 2005. Accepted for publication November 16, 2005. Recommended by D. Lubinsky.
Research
supported by NSERC of Canada.
Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, NS,
Canada, B2G 2W5 (pzhou@stfx.ca).
302
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303
GENERAL ORDER MULTIVARIATE PADE APPROXIMANTS
the multivariate form of the XY exponential function (see also [1], [12])
B K E M
J I
?[Z&@<B\C!E]G^H
K<L M!N=O
_ PRQ9S*` Z U
C
a
abdc*C
X
the Appell function (see also [6], [9])
@i]G
J I
egfh@<iCjc(Cjc(k!l5k!B\CFE;G&H
KmM
@ljG
K<L M!N=O
B K E M
Cnl
boipbrq;C
KmM
and the multivariate form of the partial theta function (see also [12], [8], [1])
s,Zg@B\C!E]GhH
J I
K8mM!u
K<L M!N=O
f
K8mM!m
XTt
uv:w B
t
K E M C
a
a]xyc(z
X
Our aim in this paper is to show that, by using Theorem 2.1 in [7] and Maple, we can compute
@{dC:|}G
the
general order multivariate Padé approximant to any given pseudo-multivariate
{dC:|~C!?
function for
defined in Theorem 2.1 in [7] (and stated as Theorem 1.3 in this paper).
D EFINITION 1.1. Let
J
e@BDCFE;GH
l
B K E M Cl
KM
KM
K<L M!u5
]
t
{dC!|~C:?
w
H
z
{dC!|
be a formal power series, and let
be index sets in d
The (
) general
e@BDCFE;G
?
order multivariate Padé approximant to
on the set is a rational function
@<B\CFE;G
_ {V|`@BDCFE;GH
@<B\CFE;G
where the polynomials
J
@<B\CFE;GH
i
B K E M Cyi
KM
C
KM~
KL M!u(
t
J
@BDCFE;GH
KM
B K E M C
K<L M!u*~
KM
C
t
are such that
(1.1)
@e
Y
J
Gj@B\C!E;GhH
K<L M!u5
!
/
KM
B K E M C
KM/
t
and
?
(1.2)
satisfying the inclusion property
@PCSG
?ChqroP:C;qr ¡nS¢H=£¤@RC! G
D EFINITION 1.2. A multivariate function
the coefficients of its formal power series
er@BDCFE;GH
e@<B\CFE;G
J I
K<L M!N=O
l
KM
?z
is said to be pseudo-multivariate if
B K E M
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304
P. ZHOU
satisfy
l
Ho¥¦@P=Q}SGRC§PCS¨HAq#Cjc(Cj©j©>©VC
KM
¥
@#G
Rz
where
is a certain function of
{dC!|
Please see [7] foremore
details on definitions
of the (
) general order multivariate
@<B\C!E]G
?C
Padé approximant to
on the set
which is slightly different from the one given in
earlier papers, and the pseudo-multivariate functions.
T HEOREM 1.3. (Theorem 2.1 in [7]) Let
J I
er@<B\CFE;GªH«
J
¥­@;G
¬ N=O
CF¯
be a pseudo-multivariate function. For ®
B K E M
¬
K8mM!N
, let
±
(1.3)
°±
L²
±
L²
X
be the
@
®
CF¯\G
³ M
M!N=OD¶ M
²
³ M
M!N,O\· M
@³TGªHµ´
´
C
H¸c
· O
Padé approximant of the function
¹
J I
@<³GH
¥
@;G;³ ¬ C
¬ N=O
let º
HA»¼¾½R¿
®
CF¯gÀ(C
and let
|ÁTH¸¿T@<P:CSG(qÂPCS¯gÀTC
(1.4)
(1.5)
{ÃTH¸¿T@<P:CSG(qÂPCS
À^Ä¿T@PCSTGª(qÅPRQ9S
º
?ÇTH¸¿T@<P:CSG(qÂP=Q}Sp
(1.6)
w
be index sets in . Then the (
?
on the index set is
{dC!|
®
QƯgÀ(C
QƯ&C&PCSÈrq]À
®
) general order multivariate Padé approximant to
@BDCFE;G
_ {3|`@BDCFE;GHµ
e@B\C!E;G
C
@BDCFE;G
where
(1.7)
@BDCFE;GH
±
X
L²
@<BG
X
±
@<E;GRC
L²
and
@BDCFE;G
B °±
H
@BG
ÉÊ\Ë&Ì ÍÎ <
K L ²(Ï
J
±
J
H
L²
¶
K8N=O
M!N
Ë&Ì ÍÎ KL ²(Ï!m
L²
@<E]G
B
B K E M
E
Y
Y
E
X
QÆB K<Ñ
±
f
L²
@BRG °±
E M!m
f
L²
@E;G
Q©j©j©QÒB M E KÓ
M!N=O
J
Y
±
K · MÐ
²
(1.8)
X
f
¶
K · MÐ
B K8m
f
E MVÑ
f
QÆB K8m,w E MVÑRw
QA©>©j©QÆB MVÑ
f
E K8m
f
ÓVÕÔ
z
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305
GENERAL ORDER MULTIVARIATE PADE APPROXIMANTS
The proof of this theorem in [7] uses the fact that if
B ¹
er@BDCFE;GhH
and if
@<BG
B
E ¹
Y
BnA
ÖH
ERC
@<E]G
C
E
Y
BHAEC
er@BDCFBRGH
B
@B
¹
@BG:GhH
E
@<E ¹
@E]G!GHAer@<EC!E;GRz
@<BDCFE;G#
R EMARK 1.4. It is easy to see that
is irreducible.
@B\C!E]G
is irreducible, as °
±
L²
@<³G#
X
±
L²
@³TG
2. More Explicit Constructions. The importance of finding more explicit formulas of
multivariate Padé approximants to some given functions is obvious as very few@{d
of C!them
are
|}G
known so far. For any given pseudo-multivariate function, we can compute its
gen?
CF¯&C
{dC!|C!?
eral order multivariate Padé approximant on the set for given ®
where
, and
C!¯
are defined in Theorem 1.3. For some pseudo-multivariate functions whose one variable
®
projection functions have general explicit formulas of Padé approximants, like the ones given
in [7], we can use Theorem 1.3 to write their general order multivariate Padé approximants.
For those pseudo-multivariate functions whose one variable projection functions don’t have
general explicit formulas of Padé approximants, we can use Maple or other software to com@{dC!|9G
?pz
pute the
general order multivariate Padé approximant on the set
In what @follows
{dC!|}G
we present an example of a short procedure in Maple, called mpa, to compute the
general order multivariate Padé approximant for two variable pseudo-multivariate functions.
C!B\C
C!¯
®
In the procedure mpa( ×
), × is the one variable projection function of the pseudoe
B
¯
multivariate function , is the variable of × , ® and are non-negative integers. It computes
@{dC!|}G
e@<B\C!E]G
?
the
general order multivariate Padé approximant to
on the set , where
{dC!|
?
, and are defined in Theorem 1.3.
b
b
b
b
b
b
b
b
b
b
b
with(numapprox):
mpa:=proc(f,x,m,n)
local g,px,py,qx,qy,PP,QQ,PQ;
g:=pade(f,x,[m,n]);
px:=numer(g); qx:=denom(g);
py:=subs(x=y,px); qy:=subs(x=y,qx);
PP:=simplify((x*px*qy-y*qx*py)/(x-y));
QQ:=simplify(qx*qy);
PQ:=simplify(PP/QQ);
return (PQ);
end proc:
E XAMPLE 2.1. The function
Ø
J I
@B\C!E]GhHÙ@BQÒE;GÙ
@
K<L M!N=O
Y
B
cG KmM
w!K E wM
@ÚÛ@P=Q}SG\QcGVU
is a pseudo-multivariate function with
¹
J I
@<³GgHÃ
²¾N=O
@
Y
cG ²
³ w:²¾m
f
@Ú¾¯QÜcGU
HÝ!Þß³àz
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306
P. ZHOU
Running mpa
@<ÝFÞ8ß\@BRG3C!B\C!á#C:Ú(G
cjâTq¾BãQÂä5BàãVE w
in Maple gives
QcjâTq¾B w EQoäB w E]ã
cÚ¾q*B¨QAc>â(q¾BàE w
Y
á[@<Ú¾q[QÆB w =
G @Ú*qQÆE w G
Y
@{dC:|}G
which is?the
C
the set
with
cÚ*q*q*E
QAc>â(q¾E]ã
Y
general order multivariate Padé approximant to the function
Ø
C
@<B\C!E]G
on
|§H¸¿T@PCSTGª(qÅPCSprÚ]À(C
{åH¸¿T@PCSTGª(qÅPCSpoá#C¡P=Q}Spræ]À(C
?çH¸¿T@PCSTGª(qÅP,Q}SpoæÀTz
E XAMPLE 2.2. A multivariate form of the logarithm series is
B
J
è[@<B\C!E]GhH
z
PRQ9S
f
KmRMVé
K E M
It is a pseudo-multivariate function with
¹
²¾N,O
Running mpa(
ê8ß­@!c
³GC!³àC!á#Cjc
Y
Y
Y
@{dC:|}G
HAêß/@!c
¯pQAc
³Gz
Y
) gives
Ú5â(B w Qcjâ(B w /
E QÆá*B w E w QÅë*ì*B
Y
ì @<á*B
â G=@<á*E
âTG
Y
Y
ë*ì¾BàE­QÜcjâ(B#E w QÆá*EãjB
â*E]ã
Ú¾â*E w QÆë(ì¾E
Y
Y
ì[@á¾B
âTGR@á¾E
âG
Y
Y
â(BàãQÅá¾BàãjE
Y
which is the
?C
the set
with
f
³ ²¾m
J I
@<³G&H
C
general order multivariate Padé approximant to the function
|§H¸¿T@PCSTGª(qÅPCSpyc*À(C
{åH¸¿T@PCSTGª(qÅPCSpoá#C¡P=Q}SpÂâ#À(C
?çH¸¿T@PCSTGª(qÅP,Q}SpÅâ;ÀTz
E XAMPLE 2.3. The function
ío@BDCFE;GhH
J I
@
K<L M!N=O
cÛ©jᦩ*©>©j©¾©*@ÚÛ@P=Q}SG
Y
cG KmRM
@PRQ9SG3U
Y
cG
B K E M C[a Bga3Ca E,ax
is a pseudo-multivariate function with
¹
@
J I
@³TG&H
Y
cG ²
cÛ©jᦩ歩¾©>©j©¾©T@Ú¾¯
¯&U
²¾N=O
Hdî
c^QÅÚ*³;C[a ³aTx
Ú
c
z
Y
cG
³ ²
Ú
c
è@<B\C!E]G
on
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GENERAL ORDER MULTIVARIATE PADE APPROXIMANTS
@
Similarly, mpa ºXï
@<B\C!E]Gà
proximant
@ðc^QÂÚ¾³GC!³àC!á;CÚ*G
@BDCFE;G
to
@{dC:|}G
gives the
general order multivariate Padé ap?
a BgaVCa E,axyc*Ú]C
on the index set for
where
í¢@<B\C!E]G
QÆñ*B ã E/QÅá¾B ã E w
@B\CFE;GhHâ*B ã
QÆá(ì¾B w
Q
â(ñ*B6QAcÚ¾q¾BàE­Qoä5ì¾BàE w
Qoä5ì*B w E¦QÅá(æ5B w E w
QÆâ(ñ¾E¦QÆá(ì¾E w
QÆñ*B#E ã
QÅá¾B w E ã
QÒâ(E ã
Qc>ì#C
and
@B\C!E]G^H
Ð
â
QÆñ*B¢QÆá*B w Ó
â
QÆñ*E¦QÅá¾E w Ó
Ð
C
{dC!|~C:?
with the same
as given in Example 2.1.
E XAMPLE 2.4. The function
J I
sn@B\C!E;GhHç@<B¢QÒE;G
@
Y
K<L M!N=O
cG KmRM
B w:K E wðM
Ú@PRQ9SG\Qc
is a pseudo-multivariate function with
¹
J I
@<³GhH
@
²*N=O
@¼*ó:ôVò:¼*ß,@BGVCFBDC
Y
cG ²
f
³ w!²*m
Ñ
Hrò¼¾ß
Ú¾¯Qc
CF¯\G
f
B\z
@{dC!|}G
®
Then we can use mpa
to compute the
general order multivariate
s6@<B\C!E]G
?
CF¯&z
Padé approximant to
on the index set for any given positive integers ®
E XAMPLE 2.5. The function
c۩ᦩ¾©j©>©¾©(@ÚÛ@P=Q}SG
J I
õÇ@B\C!E;GhHç@<B¢QÒE;Gö
K<L M!N=O
Ú K8mM
cG
B w:K E wðM
Y
@ÚÛ@PRQ9STGDQAcG=@P=Q}SG3U
is a pseudo-multivariate function with
¹
cÛ©jᦩ歩¾©>©j©(©*@Ú¾¯
J I
@³TG&H÷
²¾N
f
Ú ²
@¼¾óôjÝ!Þ8ß,@BG
@Ú5¯pQAcGT¯&U
BDCFBDC
Y
cG
B w:²¾m
f
HÜÝFÞ8ß
CF¯\G
Ñ
f
B
@{dC!|9G
Y
BDz
Then we can use mpa
to compute the
general order multiY
®
õÅ@B\C!E;G
?
C!¯&z
variate Padé approximant to
on the index set for any given positive integers ®
For some pseudo-multivariate series, we might not be able to have the explicit functions
for the sum of their projection series ofQÂone
variable. In this case, we can use Maple
to write
¯øQyc
@
C!¯\G
the partial sum up to the degree of ®
in the variable, and compute the ®
Padé
@
CFBDC
CF¯\G
@{dC:|}G
®
approximant to the partial sum and then use mpa ×
to compute the
general
?
order multivariate Padé approximant to the multivariate series on the index set
for any
C!¯&z
given positive integers ®
E XAMPLE 2.6. The function
K E M
B
J I
eo@<B\C!E]G^H
@PRQ9SG3U5Q@<PRQ9STG ã
K<L M!N=O
is a pseudo-multivariate function with
¹
J I
@³TG&H
²¾N=O
³ ²
¯&UQÒ¯
ã
Qc
z
Qc
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308
P. ZHOU
¯
First,
we write a short procedure Sn(n,x) to write the sum of the first
¹
@³TGª
b
b
b
b
b
b
b
Sn:=proc(n,x)
local i,s;
s:=0;
for i from 0 to n do:
s:=s+xˆi/(factorial(i)+iˆ3+1); od;
return (s);
end proc:
@
Then running mpa Sn
@BDCFE;G#
Padé approximant
terms of the series in
@BDCFE;GhH
Y
@<ì;C!BGVCFB\C:á;CÚ*G
@B\C!E;G
to
á#c>ñTÚ¾ì(Ú;c>ë*â(á*ñ(á;cjá*B ã
Qoä5ëTæ*Ú*æ*á¾â]ä5á*ì(á(Ú*q¾B ã E¦QÜc>á(ì*q*ì(ë*ñä5ë*ñ#c>qäB ã E w
cä¾á¾âTáTä5ñ#c>ë(ñ(æ¾ñ(á¾á¾B w E
Y
Q c>á(ì*q(ì*ë*ñä5ë(ñ;c>qäE ã B w
¨
Y
Y
Qä5ëTæ*Ú(æ¾á¾â]ä5á(ì*á(Ú*q5E ã B
Q¦Ú]cq*ìTÚ¾q*q#c>qTæ¾á;cë¾á(Ú]c
@{dC!|}G
gives the the
general order multivariate
?C
on the index set
where
í@BDCFE;G
Y
Y
æTä5ë(æ*ë*q(á¾â#cæ¾q(á(æ*ä5B w
Qc>q¾âTá*qTæ]c>ì(á*ëä5á¾âTá5B w E w
cñTä¾Ú*ì;cë*ñ¾âTá;cá*ñ*ë(ñ5B#EÛQÆñä*ä¾ñ¾â(ë*âTÚ;cæ¾q(ñTä¾ä5â¾B
cä¾á¾â(áä5ñ#c>ë*ñTæ¾ñ(á*á5BàE w
á;cñ(Ú*ì(Ú]cë¾âTá*ñ*á#c>á*E ã
æ(ä¾ë(æ*ë*q*á*â#cæ¾q*áTæ(äE w
Y
Q
Å
ñTä(ä5ñ¾âTë¾âÚ]cæ*q*ñä*äâ(EàC
and
@B\CFE;GH¸c*cÚ*Ú
Ð
Y
cë*áä5ì;cëªQÆâ(ñ*â#c>ì(q¾B/QÆñTÚ¾ñ*â#cjB
wVÓ
{dC:|~C!?
Ð
Y
c>ë(áTä5ì#c>ëQÒâTñ¾â#cì*q*EQÅñ(Ú*ñ¾âàcjE w3Ó
@
@
Qø¯
Q9c(CFBGVCFBDC
C
¯\G
with the same
as given in Example 2.1. We can use mpa Sn ®
® ,
@{dC!|}G
e@BDCFE;G
to compute the
general order multivariate Padé approximant to
on the index
?
C!¯&z
set for any given positive integers ®
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[5] A. C UYT, K. D RIVER , AND D. L UBINSKY , Kronecker type theorems, normality and continuity of the multivariate Padé operator, Numer. Math., 73 (1996), pp. 311–327.
[6] A. C UYT, K. D RIVER , J. TAN , AND B. V ERDONK , Exploring multivariate Padé approximants for multiple
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[7] A. C UYT, J. TAN , AND P. Z HOU , General order multivariate Padé approximants for pseudo-multivariate
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[8] D. S. L UBINSKY AND E. B. S AFF, Convergence of Padé approximants of partial theta functions and the
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[9] L. J. S LATER , Generalized Hypergeometric Functions, Cambridge University Press, 1966.
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17.
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