PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 53, Number 1, November 1975
FACTORS AND ROOTS OF THE VAN DER POL POLYNOMIALS
F. T. HOWARD
ABSTRACT.
The van der Pol polynomials
V (a) are defined
by means
of
oo
xlexa\jox(ex
+ 1) - 12(e* - l)]-1
= 1
V (a)xrt/n\.
n=0
In this
that
paper
new properties
neither
V', (a)
nor
K,
of these
polynomials
,,(a)/(a
~ 1/2)
"
are derived.
has
rational
n = 2 • 3™, m > 0, or n = 3m + 3 , m > t > 0, or n - m(p -3),
ber,
3m < p, then
rational
the
field.
rational
V (a)
It is also
field.
and
V
,,(a)/(a
shown
Finally,
that
-
1/2)
are both
if n = 2 , then
possible
factors
n
van
and that
p a prime
irreducible
V (a)
of the
It is shown
roots,
if
num-
over
is irreducible
der
Pol
the
over
polynomials
are discussed.
la Introduction.
fined
by means
The van
der
of the generating
Pol
numbers
V , V.,
can be de-
function
2
U ;
V2> •••
OO
6x{e*+ 1) - 12U* - 1) Zo "nl
This
definition
is apparently
bers
in a problem
three
variables.
bers
are closely
due to B. van der Pol [l2j,
concerning
The
present
related
defined
in terms
subject
of a number
others.
(The
early
the smoothing
and unsmoothing
writer
out [7J that
of investigations
history
(1.2)
defined
a
= (.-If"1
of functions
of
which
has
[S>L Carlitz
can be found
been
is
the
[2], [4], and
in [l4,
p.
5 • 2ln~lV 2n/(2n)\.
the van der Pol polynomial
1)-
num-
the van der Pol num-
and which
fe],
function
these
(v), a function
function
by Kishore
of the Rayleigh
-—-=£v(*)*
6x(ex+
function
of the Bessel
502].) In fact, for « > 1, ^(3/2)
In [7] the writer
pointed
to the Rayleigh
of the zeros
who used
12(e*-l)
„=Q "
V (a) by means
of
"!
It follows that Vn (0) = Vn , and
(1.3)
v'>>=zrW""r>
Received by the editors September
AMS (MOS) subject classifications
27, 1974.
(1970).
Primary 10A40; Secondary
12D05,
12D10.
Key words and phrases,
van der Pol numbers
and polynomials,
tion, Bernoulli
and Euler polynomials,
irreducible
over the rational
stein's
irreducibility
criterion,
Eisenstein
polynomial.
Rayleigh
funcfield,
Eisen-
Copyright © 1975. American Mathematical
1
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Society
2
F.T.HOWARD
(1.4)
V ia) = (-l)nV
n
In [6] it was proved
0 and
1, and
0 and
1„ It was
that
x = 0, x = x/i, and
shown
that
der Pol polynomials
have
Bernoulli
and Euler
polynomials.
example.
The first
al factors
(a)
properties
• • • + c,a
then
(e)
to those
n
V (a)/(a
- XA)has
c Q, . . . , c
for
in V5.
of the roots
we prove
and ration-
the following:
roots.
no rational
are integers,
c , . . ., c,
2], and [ll],
are listed
- l/2) has
~ ^
that the van
of the well-known
In particular
V 2n+l(a)/(a
on the lines
[10, Chapter
no rational
between
roots.
a factor
of the form
c
+
k > 0, c, / 0 (mod 5).
^as a factor
are integers,
of the form cQ + c^a +
then
if c_ ^ 0 (mod 5), we
c. = 0 (mod 5) for all i > 0.
ff n = 2 • 3m, m > 0, or n = 3m + 3', m > t > 0, or n = m(p - 3) where
p is a prime
and
3m < p, then
over the rational
Preliminaries.
terms,
we examine
&
lemma.
the lemma
for
V +,(a)/(«
V (a) is irreducible
The van der Pol numbers
of 2, 3, and 5 dividing
lowest
V (a) and
- Vi) are irreducible
field.
(f) If n = 2 , & > 0, then
following
it appears
\i\,
roots
a = M, between
Thus
our investigation
V (a) has
or ^2n+l(fl^fl
, k > 0, where
must have
when
[l],
including
roots
we continue
where
(d) If V2n(a)
roots,
van der Pol polynomials
V2 (a) nor
c .a + • • •+ c,ak
its
similar
See
two real
no nonreal
plane.
of the van der Pol polynomials.
(c) Neither
powers
real
eight
If n is even
'
exactly
paper
(b) If n is odd then
2.
three
V n (a) has
x = 1 in the complex
In the present
(1 - a).
for n > 0, V 2 (a) has
V2 + {(a) nas exactly
also
n
[6], Ui.
the roots
and factors
Since
2V, In
V.,In . Similar
are rational
the denominator
are known
over the rational
and the exact
of V , when
Because
this
V
information
+ l)V
congruences
b
is reduced
will
of V (a), we incorporate
., = -(in+1
field.
it into
the
for n > 1 [7], we shall
In
obviously ' hold
for
to
be useful
state
V,,2n +.,.l
Lemma 2.1. (a) // n > 1 then 2V2n = 1 (mod 4).
(b) If n> 1 then for b = 0, 1, 2, 3V2n = & (mod 3) «'/ 2n = b + 1 (mod 3).
(c)
// n > 0 //>en 5"V2n/(2«)!
(d) // p is a prime,
= 3" (mod 5).
p > 3, /Ack /or k > 0, pnVn(
3)/[n(p
- 3)]! = (-12)"
(mod p).
When proving
of Eisenstein's
Lemma
2.2.
coefficients
p) for all
irreducibility
theorems,
we shall
use
the following
version
criterion.
If f(x) - c
and if there
i > 0, c
^0
+ c.x
is a prime
+ • • •+c
p such
(mod p ), then
f(x)
x"
that
is a polynomial
c.
with
integer
^ 0 (mod p), c. = 0 (mod
is irreducible
over
the rational
field.
Suppose
f(x)
is a polynomial
with
rational
coefficients.
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If there
is an
FACTORS AND ROOTS OF VAN DER POL POLYNOMIALS
integer
ses
h such
that
of Lemma
ly an Eisenstein
Lemma
hf(x)
2.2, then
2.3.
coefficients
// f(x) = c.
+ c x -t-• • • + c x"
Proof,
is a prime
if f(u/b)
and satisfies
f{x) an Eisenstein
is irreducible
i > 0, then
we must have
coefficients
call
polynomial
and if there
p) for all
has integer
we shall
ft is easy
= 0 where
u and
b are integers,
to see that if f(x)
satisfies
the above
induction
integer.
number
reduced
on i, we can show that each
the denominator
Assume
an integer,
r,
(u, b) = 1,
paper
of p dividing
2.4.
Let
hypotheses,
then
r. is integral
for
we use the notation
Now,
(mod p); that
using
is, p does
cn = br.
0
at f(x)
is an
0
, - ur^, and since
looking
p
terms.
i = 0 since
c, = br,
(mod p). Now,
)
lowest
c. (mod p), which is impossible
this
n > 3 with nonzero
to its
This is true
I
must be integral
Throughout
Lemma
of r..
true for i = k — 1; then
that bx - u divides
est power
integer
b £ 0 (mod p). We can write
r. is a rational
not divide
with
c„ ^ 0 (mod p), c. = 0 (mod
f(x) = (bx - u)(r0 + r,x + • • • + rn_ xxn~
each
field.
is a polynomial
that
Clear-
b = 0 (mod p).
u £ 0 (mod p). Now suppose
where
the hypothe-
polynomial.
over the rational
p such
3
c,
is
(mod p), we see
(see [l3, pp. 74—77]).
h tI
\ \ c to mean
p
h
is the high-
c.
f(x) -c+c.x+.-.+
integer
c x"
coefficients.
Suppose
be a polynomial
f(x)
has
of degree
all the following
properties:
(a) There
is a prime
p such
that
cQ £ 0 (mod p), c{ = 0 (mod p) for all
i > 0.
(b)
// pki ||ii c i'., i > 0, then kn —i,>k
"■ ' > r
—
,, k.i >
k.., - 1,' k.i —
> k..,
- 1.
— i +l
i+2
n—2
(c) f(a) = 0 implies /(l - a) = 0.
T^ew we can conclude
that
f(x)
has no rational
roots.
Proof. We know from Lemma 2.3 that if f(u/b) = 0, then b = 0 (mod p).
By hypothesis
(c) we can write
f(x) = (bx - u)(bx - b + u)(r0 + r. + • • • + r _ 2xn~2)
where
each
r. is a rational
i
number
reduced
to its lowest
terms.
Since
c i. = u(b - u)r.i - b r.i-l , + b r.i-2 .,,
we can prove
show that
since
if p
|| c., then
cQ = u(b - "Vn.
u(b - u\.
p
by induction
on i that each
p
- b r{_x + b r._2>
|| numerator
of r..
|| numerator
Suppose
r. is integral
of r..
This
(mod p). Now we
is true for i = 0
it is true for all /' < z'. Then
and by part (b) of our hypotheses
Now since
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we have
c. =
we must have
4
F.T.HOWARD
c
T = b2r
n-i.
we have
A-b2r
n-4
n—i
a contradiction
We shall
make
, + u(b - u)r
n-2
of hypothesis
use
,
and
(b).
of the well-known
c
n-1
, = b2r
Therefore
facts
, - b2r
n- 3
f(u/b)
n-2'
,,
£ 0.
in the following
lemma
[5,
pp. 263, 271].
Lemma 2.5.
// p is a prime and
m = aQ + a xp + ■• ■ + ajin
r=bQ
(0 < a. < p),
+ b1p + --- + bnpn
(0<b.<p),
then
O'w(n)-C) <-°dw(b) // pk || m\ then k = (m - a
3. Rational
Lemma
rational
2.4,
roots
we shall
and factors
prove
that
- a^-«„)/(?
- l).
of the van der Pol polynomials.
neither
V2 (a) nor
V'
+l(a)/(a
By using
- Vi) has
roots.
Theorem 3.1. For n > 0,
5nV2nia)/i2n)\
5nV2n+1(a)/(2n+
Proof.
= 3"
(mod 5),
1)!= 3"_1 + 3" • a
(mod 5).
By (1.3) we have
5"V,
(a)
2n
(2n)!
» 5rV,
,„_,
= y_2z_L_fl
(2r)! ' (2« - 2r)l
r=0
2" -2r
(3-D
y
2r + 1 .
^
(2r+ 1)! ' (2b-
■>_fl2n-2r-l
2r-
1)!
Now by Lemma 2.5(b) we see that if 5k \\ (2b - 2r)l or 5k || (2b - 2r - 1)!,
then
k < n - r except
for the case
r - n.
Thus
by Lemma
5nV, (a)/(2n)! = 5nV, /(2b)! = 3"
472
The proof is similar
Corollary.
Zn
for V,
then
V (a)/(a
we have
(mod 5).
+\(a)'
For n > 0,
5"V2n + 1(a)/(2B + l)!(a - V) = 3"
Theorem
2.1(c)
3.2.
// n is even
— Vi) has
then
no rational
V (a) has
(mod 5).
no rational
roots.
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roots.
If n is odd
FACTORS AND ROOTS OF VAN DER POL POLYNOMIALS
Proof.
of Lemma
by the
When
corollary
V {a)/(a
3.3.
j (a) =c-
by Theorem
3.1 that
no rational
roots.
3.1 and by Lemma
all the hypotheses
If n is odd,
2.3 that
if u/b
b = 0 (mod 5). We can now proceed
2.4 to show
Theorem
we see
V (a) has
to Theorem
- Vi), then
of Lemma
form
n is even,
2.4 are met and
that
V (a)/(a
// either
+ c.«+•••+
where
is a root
of
roots.
- Vi) has
c., . . . , c,
we see
as we did in the proof
- Vi) has no rational
V2 (a) or V2 +1(a)/(«
c ,a
5
a factor
are integers,
of the
k > 0, c. ^
0 (mod 5), then c. = 0 (mod 5) for all i > 0.
Proof.
Suppose
5) for some
V 2 (a) does
m > 0. Then
have
such
a factor
f(a)
where
c
£ 0 (mod
we can write
(3.2)
5nV2n(a)/(2n)\
= f(a) ■ g(a)
where
(3.3)
each
5nV2
g(a) = rQ + r:a+
r.i a rational
number
(a)/(2n)l
is integral
reduced
• • ■ + r2n_ka2n~k,
to lowest
terms.
(mod 5), and since
Since
each
the coefficient
coefficient
of a'
r i.c 0 + r.i— ,c,
on i to rprove
l I +• • • + rnc
0 z'., we can use induction
(mod 5). Thus by Theorem 3-1 we see that f(a) divides
that
tegral
contradiction.
The proof for
Theorem
form
3.4.
f(a) =c»
Neither
+ c.«+•••+
V'
V,
in
+.(a)/(a
(a) nor
c, a
of
in (3.2) is
r.! is in-
1 (mod 5), a
— lA) is similar.
V,
2nTi
where
.,(a)/(a
- ]/2) has
c ., . . . , c,
a factor
are integers,
of the
k > 0, c,
^
0 (mod 5).
Proof.
Suppose
V2 (a) does
write (3.2) and (3.3).
we can use
induction
Thus
by Theorem
3.1 we see
criterion
and Euler
method
f{a)
/(«).
of <z
on i to prove
that
a factor
that
again
we can
in (3.2) is ci/i + Ck-1T' 1
r.
divides
Then
,_
is integral
(mod 5).
1 (mod 5), a contradiction.
+1(a)/(fl - Vi) is similar.
4. Some cases
ibility
such
Since the coefficient
+ ...
The proof for V.
have
of irreducibility.
to obtain
polynomials.
results
Carlitz
concerning
[3] used
With the aid of Lemma
on the van der Pol polynomials.
Eisenstein's
the irreducibility
2.1, we shall
It is perhaps
irreduc-
of the Bernoulli
use
useful
the same
to recall
that
Vn (a) is monic.
Theorem
V (a) and
4.1.
Let
V +,(a)/(a
n = m(p - 3) where
— Vi) are irreducible
p is a prime
over
and
the rational
field.
Proof. We have
" /(m - Dp + p - 3m\
pV„{a) = p'£
(
\
)Vra
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3m < p. Then
•
6
F.T.HOWARD
In 17J it is proved
that
[p, 2(p - 3)),...
if r is in any of the half open
. Vim- l)p, mip - 3)), then
p2Vr = 0
Thus
by Lemma
(mod p),
2.5 (a) and Lemma
V (a) is an Eisenstein
show
that
V +,(a)/(«
Vf is integral
[0, p — 3),
(mod p). Also
0 <r <m(p - 3).
2.1 (d) we have
pVn(a) ~ pVn 4 0
Thus
intervals
polynomial,
- Vi) is an Eisenstein
(mod p).
and in a similar
way we can
polynomial.
In that
case
also
we
first prove that
pmVn + l(a)/(n+
l)! = «(-12)m
+ 6(-12)m-1
(mod p),
so
pmVn+1(a)A(n+
and the theorem
l)\ia-V2)]^(-n)m
(mod p),
follows.
Theorem 4.2. // n = 3m + 3', m > t > 0, or if n = 2 • 3m, m > 0, then
V (a) and
Proof.
V
.,(a)/(a
Let
— Vi) are irreducible
over
the rational
n = 3m + 3' or n - 2 • 3m■ In either
case,
field.
by Lemma
2.5 (a)
and Lemma 2.1 (b) we have
3V (a) = 3 Y ( n V an~r = 3V =2
Thus
V (a) is an Eisenstein
polynomial.
3Vn+1 Aa) = 2a+2
and so
3V +,(a)/(a
Theorem
4.4.
(mod 3),
(mod 3).
Also
3Vrc +iAa)/(a - }/2)= 2 (mod 3)
— Vi) is an Eisenstein
polynomial.
If n = 2 , & > 0, «/>eB V (a)
z's irreducible
over
the
ration-
al field.
Proof.
By Lemma
2.5 (a) and Lemma
2.1 (a) we have
n (2k\
2Vnia) = 2 £
I
1 Van~T = 2Vn = 1 (mod 2),
r=0
since
/2A
l
We can now see
that
1= 0
V (a)
(mod 4)
is an Eisenstein
if r is odd.
polynomial
and the theorem
fol-
lows.
If we examine
V (a)
rc
for
0 < b < 100, the theorems
—
—
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of this
section
tell
FACTORS AND ROOTS OF VAN DER POL POLYNOMIALS
7
us that if n is even, n 4-.22, 46, 62, 66, 72, 74, 82, 92, 98, then Vn(a) is
irreducible
over the rational
field.
Also, if n is odd, n ^ 23, 47, 63, 67, 73,
75, 83, 93, 99, then V (a)/(a - Vi) is irreducible
over the rational
field.
5. The first 8 van der Pol polynomials.
V0(a)=l,
V,(a) = a--,
2
V 2Xa) = a - a -i—,
5
w
( \ = a-a3
K,(a)
3 2 +-a-,
3
3
2
1
5
20
V 4,(a) = a4 - 2a3 + -a2-a-,
5
5
,, / \
5
5 4
y Afl) = fl-a
5
2
.. 3
+ 2a ^-a-a
350'
1 2
1
2
70
,jv Xa)
l \ = a 6 - 3a
? 5 + 3a
-, 4 -a-a 3
6
,/f\
77^21574+—a-a-a
V Xa)
= a-a
'
2
5
+-,
32
1
140
3
1
+ —a
70 h-, 1050
70
I3
4
32
+—an-a-.
20
10
1
1
150
300
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On the Euler
and
Bernoulli
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Angew.
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L. Carlitz,
A sequence
of integers
related
to the Bessel
functions,
Proc.
Amer. Math. Soc. 14 (1963), 1-9. MR 29 #3425.
3.-,
Note
on irreducibility
Math. J. 19 (1952), 475-481.
4.-,
Recurrences
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and Euler
polynomials,
Duke
MR 14, 163.
for the Rayleigh
functions,
Duke Math. J. 34 (1967),
581-590. MR 35 #5670.
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History
of the theory of numbers.
Carnegie
Institution
of Washington,
Washington,
D. C,
6. F. T. Howard,
Properties
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1919.
of the van der Pol numbers
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Reine Angew. Math. 260 (1973), 35-46. MR 47 #6603.
7.-,
The van der Pol numbers
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numbers,
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533. MR 27 # 1633.
9. -,
The Rayleigh
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uber
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R. Spira,
The nonvanishing
of the Bernoulli
polynomials
Proc. Amer. Math. Soc. 17 (1966), 1466-1467. MR 34 #2967.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
in the
critical
strip,
8
ics
F. T. HOWARD
12. B. van
in Physical
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DEPARTMENT OF MATHEMATICS, WAKE FOREST UNIVERSITY, WINSTON-SALEM,
NORTH CAROLINA 27109
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