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Electronic Transactions on Numerical Analysis.
Volume 30, pp. 128-143, 2008.
Copyright 2008, Kent State University.
ISSN 1068-9613.
Kent State University
http://etna.math.kent.edu
THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS
OF ,
AND ITS RELATIONSHIP TO DISCREPANCY THEORY
RICHARD S. VARGA , AMOS J. CARPENTER , AND BRYAN W. LEWIS
Dedicated to Edward B. Saff on his 64th birthday, January 2, 2008.
With by
! " Abstract.
% denoting the -th partial sum of% , let its zeros be denoted
"
$# &% for any positive integer . If ' and '( are any angles with )+*,' *-'.(/*102 , let 35476 498 be the
associated sector, in the z-plane, defined by
! "
If K,L 1924 that
%
!
%CBEDFHG B
"
354:6 94 8; =<?>@A'
'.( #JI
"
"
5# &%NM 34:6 498O represents the number of zeros of P in the sector 354:6 498 , then Szegő showed in
SQ RST
VUXW
K1LZY.
"
"
%
\[ &% M 34:6 498O
( _
`
^]
]
0 2
a
( are defined in the text. The associated discrepancy function is defined by
%
%
bVRScHd
"
"
(`_
e
' '(f& gK-hY. \[ &%Ni 34:6 498j _ lkm]
]
I
a
02
n
%
%
One of our new results shows, for any ' with )=*E' *E2 , that
D c
%
%
bVRScod
QSs G
?tvu
P' 02 _ ' 5prq
a
a
a
%
q
'
where
is
a
positive
constant,
depending
only
on
.
Also
new
in
this
%
%
% paper is a study of the cyclical nature of
bVRScHd
' ' ( , as a function of , when )=*E' % *E2 and ' ( w02 _ ' . An upper bound for the approximate cycle
length, in a this case, is determined in terms of . All this can be viewed in our Interactive Supplement, which shows
] of the partial sums of f and their associated discrepancies.
the dynamical motion of the (normalized) zeros
where
]
and
]
Key words. partial sums of , Szegő curve, discrepancy function
AMS subject classifications. 30C15, 30E15
yf
|
|
}
1. Introduction. Let xzy|{H}\~
denotez the -th partial sum of for each
yf
`
positive integer , and denote the zeros of xy|{H}\~ by V} y$
. Let 6 8 be the sector in
the complex plane > , defined by
^.
(1.1)
6 8 V}w>
}
z yf
`J§
for angles and such that E ¡£¢¤ . Let
¥¦{:z} y
6 8 ~ represent the
number of zeros of x y {¨}N~ that lie in the sector 6 8 . A beautiful result of Szegő [7] states
that
©«ª¬
z yf
``§
¥£{7V} y
6 8 ~
X²
(1.2)
±°
°
y®­¡¯
¢¤
where if ³´¯
¼
(1.3)
is the closed curve (called the Szegő curve) in the unit disk which is given by
f·
³¯µV}lr>$¶ }
¶v¸¹»º ¶ }&¶
¸®
Received April 12, 2007. Accepted for publication January 31, 2008. Published online on May 29, 2008.
Recommended
by D. Lubinsky.
Institute for Computational Mathematics, Kent State University, Kent, OH 44242, U.S.A. E-mail:
varga@math.kent.edu.
Department of Mathematics and Actuarial Science, Butler University, Indianapolis, IN 46208, U.S.A. E-mail:
acarpent@butler.edu.
Rocketcalc, 100 West Crain Avenue, Kent, OH 44240, U.S.A. E-mail: blewis@rocketcalc.com.
128
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THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS OF 129
6
then there are unique positive values ½¯r{» ~ and ½V¯g{» ~ , in {¨5¸V~ , such that ½V¯g{¨ ~ z¾ and
8
V¾
ª
Á
½z¯r{¨ ~
are points of ³E¯ of (1.3). The angles
in (1.2) are then defined by
°¿
(1.4)
S¦ª ²1½ ¯ {¨ ~eÀ
Âlø®¢eÄ
¿
¿
¿
° ¿
The discrepancy
function Å À.Æ y {» :~ , for the zeros in 6 8 , is defined as
ª
V yf
`JÊ
X²
(1.5)
Å À.Æ y {¨ :~ÇS¥ÉÈ®V} y ª
6 8Ë ²1-Ì°
°
Ä
¢¤
Í
It follows from (1.2) that the function Å ÀÆy|{¨ ~ would behave, at worst, like Îe{»J~ , as
ª in this paper is to
© show in Theorem
©©
Á\Ù©Ú¡
©
Ø result that
rÏÑÐ . Our first aim
2.5ª«Øthe
sharper
Õ
®
(1.6)
Å À.Æ y {» :~Ò£Ó º CÔ¹»º
EÀ:Ö×´Æ
where Ó is a positive constant depending on and . It is of interest to note that Szegő [7]
showed that the associated discrepancy function, but now in the Û -plane under the mapping
f·
Û££}
ª
is bounded as a function of .
¯
`
Our second aim is to
.
closely study the cyclical
nature of the sequence zÅ À.Æ y {» :~f y
Specifically, for r ¤ and É¢¤Ü²Ý , the approximate
cyclic
length,
of
what
we
call the short-term pattern, is determined as a function of . (Long-term patterns are also
°
described in Section 3.)
Our third aim in this paper is to illustrate the dynamical motion of the zeros of x y {»m}N~ , as
Þ
VÞ accompanying this paper. This allows the reader to
varies,
with our Interactive
Supplement
? ß
input , in the range ¤
, where ¡Sࢤg², , and to input , the degree of
¤
xzy|{»m}N~ , in the range ¸
¢ . The reader’s computer then graphs the zeros of xy|{¨m}\~
inª the } -plane. On increasing , one sees the actual “fanning out” of the zeros of xy|{¨m}\~ ,
into the upper and lower half-planes of the } -plane. In addition, the discrepancy function,
. each -th step. This calculation is
Å ÀÆy|{» ~ , is then displayed to four decimal digits, at
`
based on the stored zeros of all polynomials zx y {»m}N~ y
, whose zeros were all determined
to ¢ decimal digits.
2. Background and statement of results. To study the behavior of the zeros of the partial sums x y ®{¨}N&~ of V , it is convenientª to study instead the Vnormalized
partial sums x y {¨m}\~á
yf
`
yA
`
y
{¨m}\~
, whose zeros, henceforth denoted
by
z
}
,
have
the same arguments
as the zeros of x y {H}\~ . This leaves Å À.Æ y {» ~ in (1.5) unchanged. An application of the
Eneström-Kakeya
(see [4, p. 137, Exercise 2], or [6, p.88, Problem
z Theorem
22]) shows
yf
`
that all zeros V} y
of x y {»m}N~ lie in the unit disk âãäV}åv>æ?¶ }&¶
¸ for any
compactness
considerations,
there
are
necessarily
accumulation
points
in â for
èç¸ . From
é ¯
` z yA
`
, and Szegő [7] established that each such accumulation point must lie on the
z} y
y
curve ³ ¯ of (1.3), and, conversely, that each point of ³ ¯ is an accumulation point of these
zeros of xVy|{»m}N~ . Buckholtz [1] later proved that all zeros of all xVy|{¨m}\~ lie outside of ³´¯ for
any 1çภ, and that
ª ÙVê
ÁNÚ
(2.1)
Å À
where
ª Ùzê
Å À
V}
z
yë
yf
ìVí
³´¯lî
¢
ï
¬
z}
z
yë
yf
` í
³´¯lîJ
Õ
Ô¹»º
èçฮ
ª Ùzê
ð
fñ&zñ H{ Å À
y
}
z í
y ³´¯lî»~f
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130
ª Ùzê
R. S. VARGA,
¬òªÁ A. J. CARPENTER, AND B. W. LEWIS
z í
z
where Å À } y ³¯lîáS
and
ó®ôõ ¶ } y²Ý}&¶Ä It was shown in [2] that the exponent of
¸ ¢ ß for in (2.1) is the best possible, and that the constant ¢ can be reduced in (2.1) to
Ä ö öXö÷\ø .1
In [2], the following curve was defined for each positive integer :
f·
f·
ï
y
¶ } ¶ ýy ¢¤|gþ þ
ûü
þ
· þ
(2.2)
³ y úù }w>àÿ¶ }ë¶ . ¸® and ·|
y
¶
}ë¶\çåÆAº\À
y
where ý y , defined by
ý y
· ï
y ¢ ¤|
y
is the exact error in Stirling’s formula. For calculations of ýVy when is very large, the
following asymptotic series (cf. Henrici [3]) for ýy can be useful:
·
·m
ß ·
¸
Þ
ýy Ò ¸ ²
À`rÏ Ð¦
¸z¢
¢ ÷¸
and
©
º
·
·|
ý y Ò
¸V¢
²
ß
ö
·
²
z¸ ¢ö
·
·
z¸ ö®
²A
¸ ¸
®
ÀJrÏÑЦÄ
¸ , each ³´y curve gives a much better approximation to
For any fixed with \
}
~
where the zeros of xzª y|ÙV{¨m
lie,
in
that
from [2, Theorem 4],
ê
Å À
(2.3)
V}
V
yë
yf
`
í
³y\î&
Ì
¸
Í
ÀJÜÏ
Ц
Svz}r>à$¶ }/² ¸N¶N!\ . The exponent of ¢ for in (2.3) was shown in [2] to be
where
best possible.
As defined in (2.2), the curve ³y is not a closed curve, so we make the following modifications of (2.2). First, as will be explained in the proof of Proposition 2.1 in Section 5,
the curve ³ y of (2.2) can· be extended, for each äçú¸ , to the boundary of â in two
#"$ and %¾ "$ , where '
unique points, ¾!
& y æ¤ for each ±ç ¸ . Then, the circular
+è
z¾
#
)
arc (
*
² & y
,& .y - is annexed to the extended ³ y curve, thereby producing the
following closed curve ³ / y in â :
f·
Á f·
¶ y ¦ýy ï ¢¤| þ þ
+è
8
}w >ë¶ }
#¾ ) *
^®
(2.4) ³ / y S1
0
(
² & y
,& .y - Ä
þ þ
y 476
¶ }&¶
¸®
Å & y
2
}
¢¤w3
² & 5
The curves ³
/ y , for rø®÷ , and Ð , are given below in Figure 2.1.
We remark that it can be shown that & y can be expressed as the convergent series
Þ
¸®Äø : 9 ø
5Ä ¢ ®< 9 ø;ø <¸ 9 À`rÏÑÐ Ä
(2.5)
&y
=
ÔÌ
Í
With the above definitions, we state the following proposition, whose proof is sketched
in Section 5.
P ROPOSITION 2.1. For each positive integer , the following are valid:
bzRScod
%
for discrepancy numbers
' ' ( , which are given to four decimal digits in the Interactive Supa
plement, we truncate, in the text, the displayed fractional
part of noninteger numbers to six decimal digits.
1 Except
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THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS OF &
1
131
³/
&
â
0.5
0
³´
/¯
0
-0.5
³/
-1
-1
-0.5
0
F IG . 2.1. The curve
?>
, for /A@
0.5
aCB
, and
1
u .
i) The curve ³
/ y is a simple
closed curve in â which is star-shaped with respect to }+¦ ;
¢¤´²D&$y , there is a unique number ½Vym{¨~ , with @½zym{»\~
¸ ,
ii) For each with &$y
such that }+å½zym{¨~ V¾ is a point of ³ò
/f· y , and satisfies
~y
}${H}
#¾ E.$GF I H
ï
(2.6)
ê
ýy ¢¤|Õ{:¸Ç²1}N~
ªÁ
where J y {»~ is defined,
& y .¢¤ Ù K
² Á & y î , by
ê on the interval
ª«Á
·|
í
½ y {»~eÀ
(2.7)
J y {»\~;S /²1½ y {¨~eÀ Lî ÝM
Ì
¸Ç²è½zym{»\~eÆAº\ÀmÍ
ê
iii) N
J y O{ & y ~= ¢¤ for each çภ;
iv) For each fixed @çø , J y {»~ is a strictly increasing function of on & y .¢¤wK
² & y î , and
z
z
T S<U
for each integer with ¸
, there is a unique point } P y Q
½ P y V¾<R $ on the
V
² &$y , such that
curve ³
/ y , with &$yw P y ¢¤w3
V
(2.8)
J y { P y ~£¢¤ í
z yf
`
z
vz}òr>£
denotes the (exact) zeros of x y {»m}N~ , then, for any } y not in
v) If z} y
¶ }¡² ¸N¶\8
\ , where is fixed with l!
¸ ,
V
z
¸
y ¶®W Ì
YXex1rÏÑÐ
Í
where the constant, implicit in { y 8 ~ , depends only on .
(2.9)
¶}
y ²V} P
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132
R. S. VARGA, A. J. CARPENTER, AND B. W. LEWIS
z yf
` §
¥
V} yë
6 8 , one would
Next, it would appear that to precisely
z determine
yA
`
need to know many very precise zeros z} yë
of xzy|{¨m}\~ . This,z for large, would
be a
yf
` §
daunting task! However, we give a very accurate estimate of ¥
z} y
6 8 , which
avoids finding any zeros of x y {»m}N~ . This estimate is stated below in Proposition 2.2, after
some preliminary definitions are introduced.
Fixing any with ^± ¤ , consider any positive integer such that & y
,
where we see from (2.5) that this inequality holds for all sufficiently large.
Then, from
y
y
part ii) of Proposition
2.1,
there
is
a
unique
number
,
with
½
¨
{
~
´
½
¨
{
X
~
à
¸ , such that
6
}èSÔ½ y {» ~ z¾ is a point of ³ / y which satisfies (2.6). With J y {» ~ defined in (2.7), and,
Ø ÙØ ÙJªÁ\ÙØ Ø
ÁNÚ Ø ©
with the following notation2:
ZTZ\[]T] ®
[ Õ [
S
À
`¹»º
it follows, from the strictly increasing nature of J/y from part iv) of Proposition 2.1, that
ZTZ
]:]
Jy?{» ~ ¢¤ ç^5Ä
(2.10)
This brings us to the statement of
P ROPOSITION
Given
any with ¤ , the number of zeros of x y {¨m}\~ in the
. 2.2.
sector ²Ç
}
is approximately
ZTZ
]:]
(2.11)
¢ J y {» ~ ¢¤
so that, by symmetry,
(2.12)
¥ÔÈ®z}
z
y
yA
` Ê
6 :^
·
Ä
6 Ë w²-¢
ZTZ
J
y {» ~
¢¤
]:]
Ä
The proof of Proposition 2.2 is given in Section 5.
To illustrate
¤ ¢ , and we
now the result
of (2.12) of Proposition 2.2, suppose that
choose -
, and , . Then, from part ii) of Proposition 2.1, ½ y {»¤ ¢®~ and J y {¨¤ ¢~
are numerically determined to be
Þ
ß
ß
I_
:_
Ä ß M
ß ¸ ö
and J : {»¤ ¢~ ¢¤gv¸ 5 Ä 5¸ö ¸ Þ and
(2.13)
0 ½½ I {¨{¨¤¤ ¢®¢®~~£
£Ä and J {»¤ ¢~ ¢¤gv¸ Ä XNø Ä
From (2.12), this gives
:_
z I_ f
` §
9 9
ZTZ :_ ]T] Ä
^ ^ ~
²,¢ J {»¤ ¢~ ¢¤ ²-¢{:¸~C£ö¢ and
û ü ¥ {9z}
:
ù
z I f
` §
9 9
ZTZ : ]T] Ä
¥ {9z}
^ ^ ~
²,¢ J
{»¤ ¢~ ¢¤
²-¢{:¸ ~C£ö¸Ä
(2.14)
`
Because we have all the zeros of Vx y {¨m}\~f y
to an accuracy of ¢®® decimal digits, it turns
out
that
the
final
numbers
of
(2.14),
i.e.,
and
ö
¢
ö5¸ , are
:_ : exactly
åß the number of the zeros of
x { ®}N~ and x { }\~ , respectively, in the sector ¤ ¢
¤ ¢ , without having directly
¤
and
determined any zeros of x y {¨m}\~ . In addition, in the symmetric sector case of
ß
ß
¢
¤
¤
¸
¤
¸
, it follows from (1.4) that
and
²
, so that
°
°
¢
¢
¢
Dz
¸
°
°
¸ ¤ £Ä ö¸VøÇ
¢¤
¢
B
2
Note that this is not the floor function
`badc , which is defined as the greatest integer a
.
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THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS OF which givesª us from (1.5) and (2.14) that
Á
ª
ß
Þ
:_ ¤ ¤
ß®ß
(2.15)
Å ÀÆ
l¸®ÄS÷®¢ ¢
e
ÅÅ À.Æ
Ì
¢
¢ Í
I
Ì
ß
¤
¢
¤
¢ Í
²X5Ä
133
¢ö®öÄ
The two numerical discrepancies of (2.15) agree with the rounded numbers, in these cases,
of the Interactive Supplement of this paper.
We remark X
that
in (2.11) to estimate the number of zeros of x y {¨m}\~ ,
åusing
.
the expression
in the sector ²Ç
} , is generally very accurate, but it is evident that this estimate
can be faulty when J y {» ~ ¢¤ is exceedingly close to anª integer, and this can change the
estimate in (2.12) by f?¢ . This will be considered in more detail in Section 5.
Our next result gives an equivalent representation for Å À.Æ y {» .¢¤l² ~ , where its proof
is given in Section 5.
P ROPOSITION 2.3. Given any with ^ µ¤ , assume that Ü ¢¤1²^ , the
symmetric case. Then,
for any positive integer ,
ª
:
Z
Z
ª«Á
y {¨ ~
y {» ~ ]:]
J
J
Å À.ÆAym{¨ : ~á
²-Ù ¢ Á
0 ¤
4
¢
¤
·m
½ y {» ~eÀ
ª«Á ¸
¤ ê
(2.16)
² 0
Ì
¤
¸Ç²1½ym{¨ ~eƺÀe Í 4
0 /À ¤ ½ y {¨ ~Õ²1½ ¯ {¨ ~ î 4 Ä
We remark that each of the three quantities in braces, in (2.16), can be seen to be positive.
For example, the first term in braces in (2.16) canÁNÚ be seen,
ª ÙIi using (2.10), to satisfy
ÁNÚ
Jy|{» ~ ²-¢ Z:Z Jy|{¨ ~ ]:] ¢e h g
ò ^¤C
çฮÄ
(2.17) 0
4
¤
¢¤
Next, we have the result of Proposition
2.4, whose proof is again given in Section 5.
ªÁ
©
ªÁ
P ROPOSITION
ê 2.4. Given any fixed with ò^ ¤ ,
?À
º {H¢¤|J~
½ ¯ {» ~eÀ
½yJ{» ~Õ²1½z¯Ü{» ~9î|Ò
À`ÜÏ Ð Ä
(2.18)
Ì
¯
¤
¢¤
¸Ç²1½ {¨ ~eƺÀe Í
Then, because of the properties
of the terms
in braces in (2.16), we have
ª
with ò^ © ¤ , assume ¢¤w²1 . Then,
T HEOREM 2.5. Given any
(2.19)
Å À.Æ y {» :~Ò¦Ó º Cj
Xex1rÏ Ð¦
Ó kå is dependent only on .
where l
¤
To numerically illustrate here the result of (2.19) we have, in the case
, that, as
¢
ªÁ
©
shown in Section 5,
/À
¤
¤
¤
¸
º {H¢¤|J~
½y È Ë ² q Ò
(2.20)
¤
m ½y È ¢ Ë ²è½V¯ È ¢ oË n ¤ p
¢
¢ ¤
as ÜÏÿÐ . This means that the last term in braces in (2.16) tends slowly to /Ð as ÜÏÑÐ ,
while the other two terms in braces in (2.16) can be seen to be bounded. More concretely, we
¤
have that for
,
¢
ß
¤
¸
(2.21)
½ y È Ë ² q v¸V¢eÄ ÷öX÷\ø÷E¹»º Üv¸z Ä
¤ p
¢
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134
R. S. VARGA, A. J. CARPENTER, AND B. W. LEWIS
ª
of the
3. The interesting oscillations of rs7x ty|{» : ~ in the symmetric case. One
¯
`
most intriging results, from this research, is that actual calculations of zÅ À.ÆVy?{» : ~f y
,
in the symmetric case, produce patterns of two distinct types, which likely could not have
been conjectured purely from theoretical results. For the
ª symmetric case, these patterns can
be classified as
ª
short-term patterns of increases of the Å À.Æ y {¨ :z~f and
(3.1)
0 long-term patterns of increases or decreases of the Å À.Æ y {» ~fÄ
ª
Both of these patterns can be immediately seen from our Interactive Supplement,
which was
screen,
written in Java by our third author. On setting ¦¤ ¢ , one sees, at the bottom of the
ß
Å
À
V
Æ
m
y
»
{
¤
e
¢
¤
¢~ ,
a short-term pattern of a sequence of four or five successive
increases
in
ß
where the increases at each step are approximately Ä \¢ , followed by a long-term pattern,
in which the short-term patterns are successively slightly increasing or slightly
decreasing
from step to step. This can also be seen to be the case in other choices of , as well. We
remark that these short-term and long-term patterns are valid only for symmetric sectors.
Our next theoretical result here
has to do with the short-term patterns.
T HEOREM 3.1. Given
any
with ^ ¤ , assume that u ¦¢¤ ²ò , the symmetric
sector case, and let be determined from (1.4). Then, the length P of each short-term pattern
°
is at most
uP
(3.2)
v¸vxw
¢¤
oy
° [|{
[
for all sufficiently large, where the floor function z
is defined
.
as ^ } the greatest integer
^} . It follows
Þ result of (3.2), consider theÞ case
Þ
ÞÞ
As an example ofthe
of
and
ß
I^
~;àÄS÷ ¢\;ø , and
Ä ø®
ø . In this case, ~ 6 Ô¸V÷eÄS÷®¢¢ ÷
from (1.4) that ½V¯r{»¤
,
°
so that from (3.2),
uP
v¸xw
¢¤
v¸xw
¢¤
o y v¸öÄ
°
In this case, the short term pattern
consists of at most ¸zö steps. This can
from our
^} , to be correct. Similarly, for be^ seen,
^
Interactive
Supplement,
with
and
,
ß
ß ~ ^
I
½ ¯ {»¤ ¢~
£Ä öNø \ø , and
ø®ÄS¢\¢ ¸zö , so that 6 £÷eÄS¢®¢ ¢ ¸ , so that
°
uP
oy £öÄ
°
In this case, the short term pattern consists of at most 6 steps. Again, this can be verified from
our Interactive Supplement.
4. Extensions. With satisfying £ ¤ , and with 䢤@² , we have
considered only symmetric
sectors in the previous
sections, and we now extend these results
to general sectors 6 8 , of (1.1), where 1à wÔ¢¤ . Note however that since the
zeros, of the real polynomial x y {¨m}\~ , occur in conjugate complex pairs, we
may assume,
without
loss
of
generality,
that
.
Then,
either
satisfies
¦
¤
¤
Ý ¢¤ , or
@ ^
¤ .
With / S£¢¤g ²è and / /S£¢¤g²1V , we consider
the
following three cases.
Case 1. ^ ¤ , ¤
? ¢¤ , and /
¤ , which is shown in Figure 4.1.
It is then geometrically evident, from Figure 4.1, that
V yf
` §
z yA
` §
%
¥ È z} y&
6 6 Ë ¥ È z} yë
8 8 Ë
z yf
ì Ê
¥ È V} y
6 8 Ë
¢
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THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS OF 1
/
135
0.5
0
0
-0.5
/
-1
-1
-0.5
F IG . 4.1. Case 1:
0
%
)=* '
*2 , 2
B
0.5
1
' ( * 02 , and )=* '
%
*K' > (
and, on using the definition
of (1.5), it can
ª
ª be verified that
ª
(4.1)
Å ÀÆ y {»
~
m
Å À.Æ y {¨
G / .
~ ÝÅ À.Æ y {/ :~
n
B
2 .
¢eÄ
Case 2. @¤ , ¤è^?墤 , and / / @¤ , which is shown in Figure 4.2.
Similarly, we obtain ª
ª
ª
Å À.Æy|{¨ : ~
Å ÀÆy|{ / :
~ @Å ÀÆym{»
/ ~ n ¢e
(4.2)
m
The final case to be considered
is
¤ , which isª shown in Figureª 4.3, and it similarly follows that
Case 3. ò^ ^
ª
y
n
(4.3)
Å À.Æ y {¨ :z~
Å ÀÆ y {» / ~Õ²,
Å
À
Æ
»
{
/
~
¢e
ª
m
Thus, we have shown how the general function Å À.Æ y {¨ Vz~ can be expressed in terms of
symmetric sectors. This will be used below to extend the result of Theorem 2.5, on symmetric
sectors, to general sectors. Its proof is given
in Section 5.
ª
T HEOREM 4.1. Given any
angles and © with ò^ ? ¢¤ , then,
(4.4)
Å À.ÆAym{» : ~Ò¦Ó
where Ólkå is dependent only on
and .
º
CjXex1rÏ
Ц
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136
R. S. VARGA, A. J. CARPENTER, AND B. W. LEWIS
1
V/
0.5
0
0
-0.5
/
-1
-1
-0.5
F IG . 4.2. Case 2:
)=* '
%
0
0.5
1
*´2 , 2+*E'(* 02 , and )á* '> (* '
1
%
*´2 .
0.5
0
0
-0.5
/
/
-1
-1
-0.5
F IG . 4.3. Case 3:
0
)=*´'
%
0.5
* ' (
B
2 .
1
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THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS OF 137
5. Proofs. Proof of Proposition 2.1. For each positive integer , let &ëy be the largest
positive number such that ¾%"$ is a point of ³
from (2.4),
/ y , i.e.,
·
:
T9
:9
ï
y F
"$ H ¦ýy ¢¤|Õ{o¢²,¢ÆAºÀ&5y~ ¢ýy ï ¤|{:¸Ç²1ƺÀo&$y5~ Ä
(5.1)
With yw£¢{7¸;²,ÆAºÀ&5y5~ , the squaring of the expression in (5.1) gives
d $ £¢¤|ý y y|Ä
(5.2)
Then, the largest of the two positive solutions of (5.2), called y , can be expressed as the
convergent expression
Þ
Þ
Þ
ß
ß Þ
5Ä ÷
¸
5Ä ¢÷ ¢®
(5.3)
y £¢Ä
M
²
Ì
`
À rÏ Ð¦Ä
Í
From (5.2) and y S£¢{:¸Ç²1ƺÀo& y ~ , it can then be verified that
Þ
·|h
y
¸ÄSø ø
ÄS¢
®I9 ø®ø;
I¸ 9 ÀJÜÏ
ï
(5.4) & y ƺ¸Ç²
ÔÌ
¢2
Í
Ц
which was stated in (2.5). Then, improving slightly on the discussion in [2, Section 3], it
follows that, for any with &ëy-Ã袤ܲ&$y , there is a unique positive ½Vy|{»~l ¸ such
that ½ym{¨~ V¾ is a point on ³ò
/ y . Thus, having annexed the circular arc V}+ ¾ ²*&$y
,&5yë to form the curve ³/ y , then ³/ y is a simple closed curve in â which is star-shaped with
respect to }è± , giving part i) of Proposition 2.1. Part ii) of Proposition 2.1 then follows
Þ
from equations (3.12) and (3.13) of [2].
It can be verified from (5.4) and (2.7) that
J O{ & ~ Ä \¢®¢ ¢®¢ radians, and that J?y|C{ &5y5~
is strictly decreasing in to ¤ ¢¡¸Ä ÷\ø ø ö , as ÜÏ Ð . Thus,
ß
(5.5)
J y C{ & y ~=墤r£öÄS
¢ ¸ ÷
for each èçฮ
from which part iii) of Proposition 2.1 follows.
·
·that J y {»~ is a strictly increasing
Next, for any fixed Ýç¸ , it·mwas
stated
in [2, p. ·|118]
y
y
function of on the interval ƺy ¢¤g²1ƺÀ
y1 , where the end-points of this
interval come from the definition of the curve ³ y in (2.2). Recalling thatê the curve ³ / y of (2.4)
is just an extension of the curve ³ y to the boundary of â , the proof from [2] similarly shows
that J y {»~ is a strictly increasing function of on the longer interval & y .¢¤ =
² & y î , where
& y is the largest number such that #¾ "$ is a point of ³ / y , and this gives iv) of Proposition 2.1.
Finally, the proof of v) of Proposition 2.1 again comes directly from [2, p. 118], completing the proof.
¤ , and let
Proof of Proposition 2.2. With Proposition 2.1, assume that &$y
6
}+ ½zy|{» ~ V¾ be the associated unique point of ³ò
/ y , i.e.,
·
y
þ } }
þ
þ ï
þ v¸®
þ
þ
þ ýy ¢¤|{7¸Ç²,}N~ þ
þ
þ
which means solving the following equation for ½y|{¨ ~ :
f
·
:
$ F 6 H 6 y : 9
½ y {» ~ ½ y {» ~
(5.6)
ï
ø®Ä
ýy ¢¤|l¸Ç²-¢½zy|{» ~eƺÀe ݽ y {» ~f
ZTZ
]:]
Then from (2.7), compute J y {» ~ , as well as J y {¨ ~ ¢¤ . This latter number then estimates the number of zeros of x y {»m}N~ in the sector
´^ , and, as x y {¨m}\~ , with positive
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138
R. S. VARGA, A. J. CARPENTER, AND B. W. LEWIS
ZTZ
]T]
|
y
»
{
~
esticoefficients, has no zeros on the positive real axis, the even number ¢ J¡
¢¤
number
ê
V
x
|
y
»
{
m
N
}
~
Ç
²
mates ZTthe
total
of
zeros
of
in
the
symmetric
sector
.
Thus,
Z
]T]
,²å¢ J y {» ~ ¢¤ estimates
the total number of zeros of x y {¨m}\~ in the complementary
symmetric sector .¢¤ ²1 î , which is stated in (2.12).
This leads us to the
Proof of Proposition
2.3. From (2.12) and (1.5), we have that
ª
ZTZ J y {» ~ ]T]
X²
(5.7)
Å ÀÆ y {» ¢¤w²è ~ w²-¢
²1¡Ì°
°
Ä
¢¤
¢¤
Í
Since ¢¤/ ²´ , it follows
from (1.3) and (1.4) that ½ ¯ {¨~ ½ ¯ {¨ ~ and £¢¤/²
.
°
°
X²
° øDz° . Substituting this in (5.7) gives
Thus, °
¢¤
¤ ª
Z:Z J y {¨ ~ ]:]
(5.8)
Å ÀÆ y {» .¢¤w²1 ~
° ²-¢
Ä
¤
¢¤
ª«Á
Next, we canê rewrite (2.7) asªÁ
Ù Á
ªÁ
·|
½y|{¨ ~eÀ
/À {»½ ¯ {» ~ì²è½ y {» ~:~&
J y {» ~å ²è½ ¯ {» ~À \î Ì
¸Ç²1½ y {» ~ÆAºÀ Í
ªÁ
which from (1.4) gives
ªÁ
Ù Á
·|
½y|{¨ ~eÀ
Jy|{» ~¦ ° - Ý?À {H½z¯g{¨ ~Õ²1½y|{¨ ~~&
Ì
¸Ç²è½ y {» ~eÆAº\Àe Í
ª«Á
or equivalently,
ª«Á
Ù Á
·|
½ y {¨ ~eÀ
Jy|{» ~C²è ²
W
²è/À {H½z¯g{¨ ~Õ²1½y|{¨ ~:~|Ä
(5.9)
Ì
°
y
¸Ç²è½ {» ~ÆAºÀ Í
Substituting the
above
expression
for
¤ in (5.8) then gives
ª
°
T
Z
Z
ªÁ
y|{» ~ ]T]
J
y|{» ~
J
²,Ù ¢ Á
Å À.Æ y {» .¢¤ ²1 ~
0 ¤
4
·| ¢¤
ªÁ ¸
½yJ{» ~eÀ
² 0
¤ ê
(5.10)
Ì
¤
¸Ç²è½ y {» ~eÆAº\Àe Í 4
0 /À ¤ ½zym{» ~C²è½V¯r{» ~9î 4
which gives the desired result of (2.16) of Proposition 2.3.
As previously remarked, the terms in the three braces of (5.10) are all positive. Moreover,
the first term in brackets satisfies, from the definition in (2.10), the inequalities of (2.17), for
any èçภ.
We next turn to the
Proof of Proposition 2.4. Given any fixed with @ ¤ , the quantity in the braces
ªÁ
of (2.18) satisfies
ê
Á\Ú
ì
/À
(5.11)
½ y {» ~C²è½ ¯ {» ~9
î k^ ¹»º
1çฮ
¤
ª«Á
since ½ y {» ~ k@½ ¯ {» ~ for any 1磸 . Next, setê
/À
y {¨ ~=
(5.12)
½ y {¨ ~Õ²1½ ¯ {¨ ~ î9Ä
¤
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THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS OF It follows from (5.6) that
f·
I
$GF 6 H 6 - y
½ym{¨ ª~ IÙ ( i ½ym{¨ ~
(5.13)
f·
:
g
6
õ F 6H
½ ¯ {» ~
ýy
ï
(
¢¤|
¸Ç²-¢½zy|{» ~eÆAº\Àe
139
-½
y {» ~
-
T9
¸Ä
Setting
½ y {¨ =
~ S¦½ ¯ {» ~= y {¨ ~A
(5.14)
so that y|{¨ ~
Ì|¸
(5.15)
k
for all 1磸 , then the first equation of (5.13) can be expressed as
y {¨ ~ 0òÌm¸ y {¨ ~ · $GF 6 H
: 6 y
4
½V¯r{» ~Í
½V¯r{» ~Í
: 9
ï
( ¸Ç²-¢½ y {» ~eƺÀe ݽ y {» ~ Ä
ýy ¢¤|
½ ¯ {» ~
©
On taking
© logarithms and dividing by , we have
© Ø
g
y {» ~ 3
º {o¢¤|J~
(5.16) º Ì|
¸ ² y {¨ ~eƺÀe
º
º
½z¯g{¨ ~Í
¢
Hence, for large,© we see that zy|{» ~ is small and positive,
© Ø so that
Ø Ù.Ø
y {» ~ y {» ~ º g º Å
º Ì ¸ ½ ¯ {¨ ~®Í
½ ¯ {¨ ~
©
This gives from (5.16) that
y {» ~
0
¸
²1ƺÀe
½V¯r{» ~
y|{¨
~Ò
º
(5.18)
H{ ¢¤|J~
¢
º
g
Ø
{o¢¤|J~
½ ¯ {» ~
Ì
¢
¸Ç²1½ ¯ {» ~eÆ ºÀe Í
Thus, from (5.12) and (5.14),©
º
©
º Å
Å
ªÁ
À
CÄ
¬
Ø
Ù.Ø ¬
ÀÄ
ÙØ ¬
ªÁ
À
C
©
which we can write as
(5.17)
4
Ø
y|{¨ ~Ò
º
À`rÏ
ЦÄ
ª«Á
o{ ¢¤|J~
½z¯r{¨ e
~ À
Ì
¢¤
¸Ç²1½ ¯ {» e
~ ƺÀe Í
À`gÏÑЦ
so that y {» ~ is unbounded as ÜÏ Ð .
:
We remark that for very large, the accuracy of the approximation
of (5.18) is also very
large. We estimate that the result of (2.21), for ä¸ , is accurate to over ® decimal
digits!
ê 2.3 and 2.4. The
Proof of Theorem 2.5. This is an easy consequence of Propositions
the negative
first term in braces of (5.10) always
lies
in
the
interval
¨
{
V
¢
î
,
from
(2.17).
Next,
second term in (5.10), for ò^ ¤ , clearly always lies in the interval ² .î , since ¤ ,
by hypothesis, lies in {¨5¸V~ , and, because the argument of the next term in turn is always
positive, then this term can be no more than . Hence, as the third term in braces of (2.16)
tends to /Ð , as ÜÏ Ð , then (2.19) of Theorem 2.5 follows.
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140
R. S. VARGA, A. J. CARPENTER, AND B. W. LEWIS
Proof of Theorem 3.1. Given any with ¤ , assume that ¢¤-²å ,
so that the sector 6 8 of (1.1) is symmetric about the real axis. zTo
the number
estimate
yA
` of zeros of x y {»m}N~ in 6 8 , wez use the fact that the numbers } P y
, are, from
yf
ì
y
y
(2.9),
close
to
the
actual
zeros
V
}
of
x
»
{
m
N
}
~
.
In
particular,
consider
the
unique
points
z
z V¾ : R S<U $ y
f
`
} P yE1½ P y
of ³
/ y , for which (cf. (2.8))©©
z
C
(5.19)
J y { P y ~;¦¢¤ ¹»º
¸
C
so that, from (2.6),
z
}P
(5.20)
ý y
ï
f·
R S<U $ ~ y
z
¸Ä
¢¤|{:¸Ç²} P y ~
z
y|{} P
y
z
V
:S<U
Then, in place of the points } P y ½ P y V¾ R $
spaced (in angle) points, defined as
¶;Û P
y ¶®Ã¸® with
.
ÛP
z
f
` , we consider the following uniformly
y
for all ¸
CÄ
¸
z
We remark that asking
z if the approximate zero } P y of xzy|{»m}N~ is in the sector 6 8 is equivalent to asking if Û P y of (5.20) satisfies
y|{» ~
J
y|{¨ ~ å. z
J
ÛP y
¢¤ ²
Ä
(5.22)
£¸
¦¸
V
We further remark that, as Û P y can be expressed as
T 9
·|
F y H
f·
<
S
U
$
R
z
z
<
S
U
Y
$
R
ï
z
Û P yE
}P y
ý y ¢¤|{:¸Ç
² }P y ~
(5.21)
V
y
¢¤
we directly see, on letting Ï Ð , how the Szegő curve ³ ¯ of
z T(1.3)
AT
` plays a major role in
*
Û
P
the result of (1.2). We also show, in Figure 5.1, zthe
numbers
from (5.21).
yf
ì
Next, we order the approximate zeros } P y
of x y {»m}N~ by their increasing arguments, i.e.,
® f
.
.
(5.23)
}P y
}VP y
} P y y 墤CÄ
The “fanning out” of the exact zeros of x y {¨m}\~ , above and below the real axis as increases,
as can be seen in more detail in the Interactive Supplement, implies that, in the closed upper
half-plane,
. V
. z
¦¸
(5.24)
}P y k
} P y for all ¸
¢
and that, in the open lower half-plane, we have the reverse:
(5.25)
.
}P
z
y
®
}P
z
y
for all
¸
¢
CÄ
As the nonreal zeros of the real polynomial x y {»m}N~ occur in conjugate complex pairs, it is
sufficient to consider the motion (with respect Vto ) of the zeros, only in the upper half-plane
yf
`
of (5.24). Now, as the approximate zeros } P y
of x y {¨m}\~ , were derived (cf. (5.24))
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THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS OF 1
141
â
0.5
0
0
-0.5
³/
-1
-1
-0.5
0
0.5
1
!¡¢ " %\£ \% £ |
\% £
# % as ¤ ’s and the zeros of \
@T¥f as ¦ ’s.
F IG . 5.1.
z
z
T S¨U
as specific points } P y §½ P y z¾ R $ of the curve ³ / y , then the following analog of (5.24)
necessarily holds, i.e., from (5.21),
. z
. z
¦¸
¢¤
¢¤
Û P yE
k
Û P y
for all ¸
Ä
(5.26)
¦¸
©
@¢
¢
In addition, we see from (2.1) that
(5.27)
}P
V
y 6 8
J
if and only if
y »{ ~
¦¸
or, equivalently from (5.21),
(5.28)
}P
z
y r 6 8
if and only if
J
y »{ ~
¦¸
.
z
ÛP
¢¤
©¦¸
J
y
J
y {¨z~
©¦¸
y {»z~
Ä
¦¸
We note, for an odd positive integer, say g¢;ª ¸ , that from (5.21) we have
(5.29)
.
ÛP «
I «
¢¤{¬ª¦¸z~
;¢ ª ¢ ¦¤CÄ
This means that xzfI « {:{o¢;ª¦¸z~:}\~ , which has exactly one (negative) real zero, corresponds
to the point ÛP « I« , Vwhich
is also real and negative, from (5.21).
Next, suppose that } P y is exactly on the boundary of the symmetric sector 6 8 in the
upper half-plane, i.e.,
® V
¢¤
J y {» ~ where ¸
¦¸
Û P y´
Ä
(5.30)
¸
¦¸
¢
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142
If satisfies ¸
(5.31)
R. S. VARGA, A. J. CARPENTER, AND B. W. LEWIS
y
·m
, then
.
ÛP
z
®
y
ÛP
f
¢¤{ ¦¸z~
¦¸
y
¤C
f
which implies that } P y is
also
f in the upper half-plane of 6 8 . It is then evident, from
(5.31), that the numbers MÛ P «l «®­ y are all in the upper half-plane, with strictly decreasing
arguments, as ª increases. (This is the analog, in the Û -plane, of the “fanning out” of the
zeros of x y {»m}N~ , in the upper half-plane.)
Û P « «®­ y , is the smallest
M
Next, what we seek,
from
this
fanning
out
of
the
numbers
f
f
f
u
} P y
} P y ¯ are all in the upper half-plane
nonnegative
integer
f so that
} P y&G
of 6 8 , while } P y ¯
is out of this sector. This implies from (5.30) that
y {¨ ~
¢¤{ ¸V~
J
¢¤
¢¤{ ¦¸z~
u ¸ ç Ì
u ¢
(5.32)
k
¸
¦¸ Í
so that
u
(5.33)
¦
¸
u ¦¸®Ä
From (5.30), we can write these inequalities as
(5.34)
u
J
¢ ¤
y {» ~ {¨¦¸z~
u ¸Ä
Next, it follows from©(2.7)
and (1.4) that
ªÁ
ª«¬
Jyë{¨ ~ ¦ ²è½V¯g{» ~eÀ l
(5.35)
y®­¡¯
°
¦¸
where ½ ¯ {¨ ~ lies on ³ ¯ . Then, as (5.30) implies that
(5.36)
¦
¸
¢ ¤
y {¨ ~ »{ © ¸V~
J
we see, assuming that is large, from (5.35) and (2.7) that
Jy|{¨ ~
Ä
°
±
¸ °
Moreover, (5.36), coupled with the last inequality of (5.33), gives approximately that
(5.37)
u
¢¤
u ¸Ä
°
u
Thus, (5.37) says that ¢¤
is a good approximation of the positive in (5.33), when
°
is large, but, as the Interactive Supplement shows, it can also be quite
good for small
f as
u such
y
¯ ²
well. Furthermore,
this
implies
that
the
maximum
nonnegative
integer
that
}
P
z
6 8 when } P y , is given approximately by
(5.38)
¸
w ¢¤ |y
°
which is independent of , and this is the desired result of (3.2) of Theorem 3.1.
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THE DYNAMICAL MOTION OF THE ZEROS OF THE PARTIAL SUMS OF 143
z
We return to the assumption
z that } P y is exactly on the boundary off á 6 8 in the
upper
half-plane. Suppose now that } P y is outside the sector 6 8 , while } P y is in 6 8 . This
gives the inequalities
. f
. V
y&{¨ ~
J
ÛP y k
k
ÛP y Ä
¸
©
y ´ is in ª 6 8 , while
Then
the smallest nonnegative integer
f on seeking
u ³ suchu that } P y ´ is not, it similarly follows that ³
}P , where satisfies (5.33).
Proof of Theorem 4.1. The results of (4.1) - (4.3) show in these cases how Å ÀÆ y {» :z ~
can be expressed
in terms of discrepancies
for symmetric sectors. In particular, for any
ªÁ
with @¤ ª , with ?£¢¤ © ²1 , it follows from Propositions
2.3 and 2.4 that
º {o¢¤|J~
½ ¯ {» ~À
q
(5.39)
Å ÀÆ y {» ~Ò
À`rÏ Ð¦Ä
¢¤
p ¸Ç²è½V¯r{» ~eÆAº\Àe
ªÁ
But, it can be verified that the function
(5.40)
ص$ÁØ
½z¯Ü{»~eÀ
gÅ
¸Ç²è½ ¯ {»\~eÆAº\À
Á
Å+º
{H¢¤J~f
ª
©
is strictly
decreasing,
we see, in all cases, that
from +1 to -1, in , so that, from (4.1) - (4.3),
Å ÀÆ y {» VV~Ò£Ó º , where ¶
Ó kå is dependent only on and .
Acknowledgment. The authors thank Professor Thomas Kriecherbauer for his sage
comments and correction.
REFERENCES
[1] J. D. B UCKHOLTZ , A characterization of the exponential series, Part II, Amer. Math. Monthly, 73 (1966),
pp. 121–123.
[2] A. J. C ARPENTER , R. S. VARGA , AND J. WALDVOGEL , Asymptotics for the zeros of the partial sums of f .
I., Rocky Mountain J. Math., 21 (1991), pp. 99–120.
[3] P. H ENRICI , Applied and Computational Complex Analysis, Vol. 2, John Wiley and Sons, New York, 1977.
[4] M. M ARDEN , Geometry of Polynomials, Math. Surveys, No. 3, Amer. Math. Soc., Providence, RI, 1966.
[5] D. J. N EWMAN AND T. J. R IVLIN , The zeros of the partial sums of the exponential function, J. Approx. Theory,
5 (1972), pp. 405–412.
[6] G. P ÓLYA AND G. S ZEG Ő , Aufgaben und Lehrsätze aus der Analysis, Erster Band, Springer, Berlin, 1954.
Problem 22 of Section III, p. 88.
[7] G. S ZEG Ő , Über eine Eigenschaft der Exponentialreihe, Sitzungsber. Berlin Math. Ges., 23 (1924), pp. 50–64.
