Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 44, pp. 1–25.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ENTIRE SOLUTIONS FOR THE HEAT EQUATION

VASSILIS G. PAPANICOLAOU, EVA KALLITSI, GEORGE SMYRLIS

Abstract. We consider the solutions of the heat equation

∂tF = ∂2zF

which are entire in z and t (caloric functions). We examine the relation of

the z-order and z-type of an entire caloric function F (t, z), viewed as function

of z, to its t-order and t-type respectively, if it is viewed as function of t.
Also, regarding the zeros zk(t) of an entire caloric function F (t, z), viewed as

function of z, we show that the points (t, z) at which

F (t, z) = ∂zF (t, z) = 0

form a discrete set in C2 and, then, we derive the t-evolution equations of

zk(t). These are differential equations that hold for all but countably many ts
in C.

1. Introduction

In this article we study certain properties of the solutions F (t, z) of the standard
heat equation ut = uzz which are entire in z and t. We believe that such solutions
deserve an independent study, since they exhibit some theoretically interesting phe-
nomena, while, at the same time, they are computationally friendly.

Suppose that the function F (t, z) is entire in z for every t ∈ C and entire in t for
every z ∈ C. Then, a consequence of the celebrated theorem of Hartogs (see [11])
is that, for any given pair (t0, z0) of complex numbers the function F (t, z) equals
to its Taylor expansion about (t0, z0), namely

F (t, z) =
∑
j,k≥0

cjk(t− t0)j(z − z0)k, where cjk =
∂jt ∂

k
zF (t0, z0)

j!k!
(1.1)

and the series converges absolutely for any t, z ∈ C. A further consequence of the
absolute convergence is that F (t, z) can be expanded as

F (t, z) =
∑
k≥0

ak(t)(z − z0)k, as well as F (t, z) =
∑
j≥0

bj(z)(t− t0)j , (1.2)

where ak(t), k ≥ 0, and bj(z), j ≥ 0, are entire functions.
Let us now assume that F (t, z) is also caloric, namely it satisfies the heat equa-

tion
∂tF (t, z) = ∂2zF (t, z) (1.3)

2010 Mathematics Subject Classification. 35KO5, 32W30.
Key words and phrases. Entire solution; heat equation; entire caloric functions; order;

dynamics of the zeros.
c©2021 Texas State University.

Submitted October 20, 2020. Published May 25, 2021.

1



2 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

with “initial condition”

f(z) := F (0, z) (1.4)

(notice that if (1.3) is satisfied in an open subset of C2, then, by analytic contin-
uation of ∂tF (t, z) and ∂2zF (t, z) we have that (1.3) is automatically satisfied for
every (t, z) ∈ C2). Since the operators ∂t and ∂z commute, by differentiating (1.3)
with respect to t repeatedly we obtain

∂jtF (t, z) = ∂2jz F (t, z), j ≥ 0. (1.5)

Also, a rather trivial observation is that if F (t, z) satisfies (1.5), so does

F̃ (t, z) = F (t+ t′, z + z′).

Using (1.5) in (1.1) yields

F (t0 + t, z0 + z) =
∑
j,k≥0

∂2j+kz F (t0, z0)

j!k!
tjzk, t, z ∈ C. (1.6)

It is convenient to write (1.6) as

F (t0 + t, z0 + z) =

∞∑
m=0

∂mz F (t0, z0)

m!
Pm(t, z), (1.7)

where

Pm(t, z) :=
∑

2j+k=m

m!

j!k!
tjzk =

bm/2c∑
j=0

m!

j!(m− 2j)!
tjzm−2j , m ≥ 0. (1.8)

The quantity Pm(t, z) is called the m-th caloric polynomial and it is clear from (1.8)
that it is parabolically m-homogeneous, namely

Pm(λ2t, λz) = λmPm(t, z). (1.9)

The first six caloric polynomials are

P0(t, z) ≡ 1, P1(t, z) = z, P2(t, z) = z2 + 2t, P3(t, z) = z(z2 + 6t),

P4(t, z) = z4 + 12tz2 + 12t2, P5(t, z) = z(z4 + 20tz2 + 60t2).

Let us review some important properties of the caloric polynomials. For each
m ≥ 0 the polynomial Pm(t, z) satisfies the heat equation with initial condition

Pm(0, z) = zm, (1.10)

hence, the entire solution F (t, z) of (1.3), with f(z) = F (0, z) = 6
∑M
m=0 amz

m, is

the polynomial F (t, z) =
∑M
m=0 amPm(t, z).

Also, from the standard integral formula, involving the heat kernel, which gives
the solution of the heat equation in terms of the initial condition, we have

Pm(t, z) =

∫ ∞
−∞

1

2
√
πt
e−(z−ξ)

2/4tξmdξ for <(t) > 0. (1.11)

Since

m!

j!(m− 2j)!
=

m!

(2j)!(m− 2j)!
(j + 1)(j + 2) · · · 2j =

(
m

2j

)
(j + 1)(j + 2) · · · 2j

it follows that the coefficients of the caloric polynomials are positive integers.



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 3

It is easy to see from (1.8) that a crude bound of Pm(t, z) is

|Pm(t, z)| ≤ m!(bm/2c+ 1)

(κm)!(m− 2κm)!
max

0≤j≤bm/2c
|t|j |z|m−2j , (1.12)

where

κm =
⌊4m− 1−

√
8m+ 17

8

⌋
+ 1. (1.13)

If we differentiate (1.8) with respect to z we obtain

∂zPm(t, z) =
∑

2j+k=m

m!

j!(k − 1)!
tjzk−1

= m
∑

2j+l=m−1

(m− 1)!

j! l!
tjzl = mPm−1(t, z)

(1.14)

for m ≥ 1 (another way to see that ∂zPm(t, z) = mPm−1(t, z) is by observing that,
since Pm(t, z) is the solution of the heat equation (1.3) with Pm(0, z) = zm, the
derivative ∂zPm(t, z) is the solution of (1.3) with initial condition mzm−1).

The function
Eλ(t, z) := eλ

2t+λz (1.15)

is entire in (t, z) and satisfies the heat equation (1.3) for any value of the com-
plex parameter λ (actually, even for the case of a square matrix λ with constant
elements). Thus, we can apply (1.7) to Eλ(t, z) (for t0 = z0 = 0) and obtain

Eλ(t, z) = eλ
2t+λz =

∞∑
m=0

∂mz Eλ(0, 0)

m!
Pm(t, z) =

∞∑
m=0

λm

m!
Pm(t, z). (1.16)

In other words, eλ
2t+λz is the generating function of the caloric polynomials.

Formula (1.8) also implies

Pm(−1, 2z) = m!

bm/2c∑
j=0

(−1)j

j!(m− 2j)!
(2z)m−2j = Hm(z), m ≥ 0, (1.17)

where Hm(z) is the (physicists’) Hermite polynomial of order m, i.e. of degree m
(see, e.g., [19]). Thus, the parabolic homogeneity (1.9) yields

Pm(t, z) =
(
i
√
t
)m
Hm

( z

2i
√
t

)
(1.18)

and equation (1.7) can be written as (for t0 = z0 = 0)

F (t, z) =

∞∑
m=0

∂mz F (0, 0)

m!

(
i
√
t
)m
Hm

( z

2i
√
t

)
=

∞∑
m=0

f (m)(0)

m!

(
i
√
t
)m
Hm

( z

2i
√
t

)
,

(1.19)

where the second equality follows from (1.4). Thus, if F (t, z) and G(t, z) are two
entire solutions of (1.3) with F (0, z) ≡ G(0, z) (or F (t0, z) ≡ G(t0, z) for some fixed
t0 ∈ C), then (1.19) tells us that they have to be identical, namely F (t, z) ≡ G(t, z).
Equivalently, if for some fixed t0 we have F (t0, z) ≡ 0, then F (t, z) ≡ 0. On the
other hand, the relation F (t, z0) ≡ 0, for some z0 does not imply that F is identically
0 (e.g., if f(z) 6≡ 0 is odd, then F (t, z) is a solution of (1.3) which is odd in z, and
hence F (t, 0) ≡ 0).



4 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

There are many known facts about the zeros of the Hermite polynomials. For
instance, it is well known [19] that Hm(z) is an even (odd) function if and only if
m is even (odd). Furthermore, the zeros of Hm(z) are real and simple. It, then,
follows from (1.18) that [15] if m = 2l, the polynomial Pm is of the form

Pm(t, z) = (z2 + ρm,1t) · · · (z2 + ρm,lt), with 0 < ρm,1 < · · · < ρm,l, (1.20)

while if m = 2l + 1, then Pm is of the form

Pm(t, z) = z(z2 + ρm,1t) · · · (z2 + ρm,lt), with 0 < ρm,1 < · · · < ρm,l. (1.21)

From (1.20) and (1.21) we have that if t ∈ C\{0}, then the zeros of Pm(t, z), viewed
as a polynomial of z, are simple (the case t = 0 is exceptional since Pm(0, z) = zm).
Furthermore, by (1.14) and Rolle’s Theorem we obtain the interlacing properties

0 < ρm,1 < ρm−1,1 < ρm,2 < · · · < ρm−1,l−1 < ρm,l, if m = 2l, (1.22)

while

0 < ρm−1,1 < ρm,1 < ρm−1,2 < · · · < ρm−1,l < ρm,l, if m = 2l + 1. (1.23)

Let us also notice that (1.20) and (1.21) tell us that, if m > 0 is even, then the zeros
of Pm(t, z) (viewed as a function of z) are real if and only if t ∈ R− := (−∞, 0]
and the same is true for the non-zero zeros of Pm(t, z) in the case where m is odd
(z = 0 is always a zero of Pm(t, z), if m is odd).

The rest of this article is organized as follows. In Section 2 we recall some general
results regarding the order and the type of an entire function. These results are
used in Section 3, where we study the relation of the orders ρz and ρt, as well as
the types τz and τt, of an entire caloric function F (t, z), viewed as function of z and
t respectively. The main results of Section 3 are Theorems 3.4 and 3.5. Finally,
in Section 4 we first show that the multiple zeros of F (t, z) are isolated (Theorem
4.1). This enables us to get the dynamics of the zeros of F (t, z), namely

z′k(t) = 2
∑
j 6=k

1

zk(t)− zj(t)
(1.24)

for all but countably many t ∈ C. The above equations were first derived by Csor-
das, Smith, and Varga [5] for the case where t is restricted in a real semiaxis (see
[16, 18]), since one must use Theorem 4.1 of the present paper in order to analyt-
ically extend (1.24) to t ∈ C. The equations (1.24) are a kind of “ characteristics”
for the heat equation and they also remind the equations which arise in the solution
of the inverse spectral problem for the Hill operator (see [20]).

2. Order and type of an entire function

Let
g(z) =

∑
n≥0

anz
n, z ∈ C, (2.1)

be an entire function and

M(r) = Mg(r) := sup
|z|≤r

|g(z)| = max
|z|=r

|g(z)|, r > 0, (2.2)

its maximum modulus.
We recall that the order of g(z) is the quantity [7]

ρ = ρ(g) := lim sup
r→∞

ln lnM(r)

ln r
. (2.3)



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 5

In other words, the order ρ of g(z) is the smallest exponent ρ′ ≥ 0 such that for
any given ε > 0 there is a r0 = r0(ε) > 0 for which

|g(z)| ≤ exp
(
|z|ρ

′+ε
)

whenever r = |z| ≥ r0. (2.4)

Clearly, 0 ≤ ρ ≤ ∞.
Let us also recall [7] that if 0 < ρ <∞, the quantity

τ = τ(g) := lim sup
r→∞

lnM(r)

rρ
(2.5)

is the type of g(z). In other words, τ is the smallest (extended) number τ ′ ≥ 0 such
that for any given ε > 0 there is a r0 = r0(ε) > 0 for which

|g(z)| ≤ exp
(

(τ ′ + ε)|z|ρ
)

whenever r = |z| ≥ r0. (2.6)

Clearly, 0 ≤ τ ≤ ∞. If τ = 0, we say that g(z) is of minimal type, whereas if
τ = ∞, we say that g(z) is of maximal type. In the extreme cases where ρ = 0 or
ρ =∞ the type is not defined.

A well-known fact of complex analysis is [7] that the order ρ and the type τ of
g(z) are given by the formulas

ρ = lim sup
n

n lnn

− ln |an|
(2.7)

and (in the case where 0 < ρ <∞)

τ =
1

eρ
lim sup

n
n|an|ρ/n (2.8)

respectively, where an, n = 0, 1, . . ., are the coefficients of the power series of g(z)
as seen in (2.1).

Let us now set

an(z) :=
g(n)(z)

n!
n = 0, 1, . . . (2.9)

(so that an(0) = an). Then, in view of (2.3), formulas (2.7), (2.8), and (2.9) yield

ρ = lim sup
n

n lnn

− ln |an(z)|
(2.10)

and (in the case where 0 < ρ <∞)

τ =
1

eρ
lim sup

n
n|an(z)|ρ/n =

eρ−1

ρ
lim sup

n
n1−ρ

∣∣∣g(n)(z)∣∣∣ρ/n . (2.11)

Sometimes, it is convenient to write (2.10) in the equivalent form (using that
limn |an(z)| = 0 and, hence, − ln |an(z)| is eventually positive)

e−1/ρ = lim sup
n
|an(z)| 1

n lnn (2.12)

or, in view of (2.9) and the fact that limn(n!)
1

n lnn = e, as

θ := e1−(1/ρ) = lim sup
n

∣∣g(n)(z)∣∣ 1
n lnn . (2.13)

Notice that θ = θ(ρ) is smooth and strictly increasing for ρ ∈ [0,+∞], with θ(0) :=
θ(0+) = 0 and θ(+∞) = e.



6 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

2.1. Additional properties of the order. For the purposes of the present work
we need to consider the cases where the lim sup appearing in (2.10) is taken over
the subsequences a2k(z) and a2k+1(z), that is

ρ0(z) := lim sup
k

2k ln(2k)

− ln |a2k(z)|
= 2 lim sup

k

k ln k

− ln |a2k(z)|
, z ∈ C, (2.14)

and

ρ1(z) := lim sup
k

(2k + 1) ln(2k + 1)

− ln |a2k+1(z)|
= 2 lim sup

k

k ln k

− ln |a2k+1(z)|
, z ∈ C, (2.15)

respectively. Clearly, for every z ∈ C we have

ρ = max{ρ0(z), ρ1(z)}. (2.16)

It is convenient to introduce the quantities

θ0(z) := exp
(

1− 1

ρ0(z)

)
= lim sup

k

∣∣g(2k)(z)∣∣ 1
2k ln k , z ∈ C, (2.17)

and

θ1(z) := exp
(

1− 1

ρ1(z)

)
= lim sup

k

∣∣g(2k+1)(z)
∣∣ 1
2k ln k , z ∈ C. (2.18)

The following theorem tells us that ρ0(z) and ρ1(z) are essentially the same as
the order ρ of g(z).

Theorem 2.1. For an entire function g(z) let θ, θ0(z), and θ1(z) be as in (2.13),
(2.17), and (2.18) respectively. Then

θ0(z) = θ and θ1(z) = θ for a.e. z ∈ C. (2.19)

For a proof of the above theorem we refer to [11].

2.2. Additional properties of the type. We, now, turn our attention to the
type of g(z) for the case 0 < ρ <∞. In view of (2.11) we set (in the spirit of (2.14)
and (2.15))

τ0(z) :=
1

eρ
lim sup

k
2k |a2k(z)|ρ/2k

=
2

eρ
lim sup

k
k |a2k(z)|ρ/2k

=
(e/2)ρ−1

ρ
lim sup

k
k1−ρ

∣∣g(2k)(z)∣∣ρ/2k, z ∈ C

(2.20)

and

τ1(z) :=
1

eρ
lim sup

k
(2k + 1) |a2k+1(z)|

ρ
2k+1

=
2

eρ
lim sup

k
k |a2k+1(z)|

ρ
2k

=
(e/2)ρ−1

ρ
lim sup

k
k1−ρ

∣∣g(2k+1)(z)
∣∣ρ/2k, z ∈ C,

(2.21)

where an(z) is given by (2.9). From (2.11) it is obvious that

τ = max{τ0(z), τ1(z)}. (2.22)

The following theorem gives a property of the type of g(z) analogous of the
property of the order of g(z) given in Theorem 2.1.



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 7

Theorem 2.2. Let τ be the type of the entire function g(z), while τ0(z) and τ1(z)
be as in (2.20) and (2.21) respectively.

(i) If τ <∞, then

τ0(z) = τ and τ1(z) = τ for a.a. z ∈ C. (2.23)

(ii) If τ =∞, then there exists a dense Gδ subset U∞ of C such that

τ0(z) =∞ and τ1(z) =∞ for all z ∈ U∞. (2.24)

For a proof of the above theorem we refer to [11]. In the case where τ =∞, the
question whether τ0(z) =∞ and τ0(z) =∞ for a.a. z ∈ C remains open.

2.3. Canonical products. Here we give a brief review of some basic facts regard-
ing canonical products. Suppose z1, z2, . . . is a finite or infinite sequence of non-zero
complex numbers such that

σ := inf
{
s ≥ 0 :

∑
k≥1

1

|zk|s
<∞

}
<∞. (2.25)

Then, the canonical product (associated with {zk}k≥1) is the quantity

Π(z) :=
∏
k≥1

ep
( z
zk

)
, (2.26)

where

e0(z) := 1− z, ep(z) := (1− z) exp
(z

1
+
z2

2
+ · · ·+ zp

p

)
, p > 0, (2.27)

and p is related to σ as follows:

(i) If σ is not an integer, then p = bσc.
(ii) If σ is an integer and ∑

k≥1

1

|zk|σ
=∞, (2.28)

then p = σ.
(iii) If σ is an integer and ∑

k≥1

1

|zk|σ
<∞, (2.29)

then p = max{σ − 1, 0}.
It is a well-known fact in complex analysis [7] that the canonical product of

(2.26) is entire in z of order σ and, furthermore, in the case σ > 0, this entire
function is of minimal type if

∑
k≥1 |zk|−σ <∞.

Remark 2.3. In view of (2.27) formula (2.26) can be written as

Π(z) =
∏
k≥1

(
1− z

zk

)
exp

( z
zk

+
z2

2z2k
+ · · ·+ zp

pzpk

)
. (2.30)

Obviously,

Π(0) = 1. (2.31)



8 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

Also, in case where p ≥ 1,

Π′(z)

Π(z)
=
∑
k≥1

( 1

z − zk
+

1

zk
+

z

z2k
+ · · ·+ zp−1

zpk

)
. (2.32)

Thus
Π′(0)

Π(0)
= 0, which implies Π′(0) = 0 (2.33)

and, more generally,

dr−1

dzr−1
[
Π′(z)

Π(z)
]
∣∣
z=0

= 0 for r = 1, 2, . . . , p,

which implies

Π(r)(z) = 0 for r = 1, 2, . . . , p. (2.34)

Finally, let us notice that if

g(z) := eA1z+A2z
2+···+AmzmΠ(z), (2.35)

where A1, A2, . . . , Am are complex constants and Π(z) is as in (2.30) with p ≥ 1,
then

g(0) = 1 and g′(0) = A1 (2.36)

(if p ≥ 2, then g′′(0) = 2A2 +A2
1).

2.4. Some notation and terminology. From now on we will use the following
notation/terminology for typographical convenience.

Let g(z) be an entire function of order ρ ∈ (0,∞) and type τ .

(i) If τ = 0 we will say that the exact order of g(z) is ρ−.
(ii) If 0 < τ <∞ we will say that the exact order of g(z) is ρ.

(iii) If τ =∞ we will say that the exact order of g(z) is ρ+.

When ρ = 0 or ρ =∞ we can consider the notion of exact order as equivalent to the
order. For instance, the statement that the exact order of g(z) is in [0, 2−] means
that the order is between 0 and 2 (included), and in the case where it is equal to 2
the type of g(z) is 0.

3. Order and type considerations for entire caloric functions

3.1. t- and z-power series of a caloric function. For t0 = 0 the second equality
in (1.2) can be written in the (Taylor) form

F (t, z) =
∑
j≥0

∂jtF (0, z)

j!
tj . (3.1)

If F (t, z) satisfies the heat equation (1.3), we can use (1.5) in (3.1) and get the
following expansion in powers of t,

F (t, z) =
∑
j≥0

∂2jz F (0, z)

j!
tj =

∑
j≥0

f (2j)(z)

j!
tj , (3.2)

where (recall (1.4)),

f(z) = F (0, z) =
∑
k≥0

ckz
k. (3.3)



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 9

Let us also expand F (t, z) in powers of z. For z0 = 0 the first equality in (1.2)
can be written in the (Taylor) form

F (t, z) =
∑
k≥0

∂kzF (t, 0)

k!
zk =

∑
k≥0

∂2kz F (t, 0)

(2k)!
z2k +

∑
k≥0

∂2kz [∂zF (t, 0)]

(2k + 1)!
z2k+1. (3.4)

If F (t, z) is caloric, so is ∂zF (t, z). Thus, we can use (1.5) in (3.4) and obtain the
expansion

F (t, z) =
∑
k≥0

∂kt F (t, 0)

(2k)!
z2k +

∑
k≥0

∂kt [∂zF (t, 0)]

(2k + 1)!
z2k+1, (3.5)

or

F (t, z) =
∑
k≥0

φ(k)(t)

(2k)!
z2k +

∑
k≥0

ψ(k)(t)

(2k + 1)!
z2k+1, (3.6)

where we have set

φ(t) := F (t, 0) and ψ(t) := ∂zF (t, 0). (3.7)

We can say that formula (3.6) is the counterpart of (3.2).

Remark 3.1. If F (t, z) satisfies the heat equation (1.3), so does F (t,−z). Conse-
quently, the z-even and z-odd parts of F (t, z), namely

Fe(t, z) :=
F (t, z) + F (t,−z)

2
and Fo(t, z) :=

F (t, z)− F (t,−z)
2

(3.8)

respectively, also satisfy (1.3), and, moreover, from (3.6) we obtain immediately
that

Fe(t, z) =
∑
k≥0

φ(k)(t)

(2k)!
z2k and Fo(t, z) =

∑
k≥0

ψ(k)(t)

(2k + 1)!
z2k+1. (3.9)

Finally, let us notice that if φ(t) and ψ(t) are two arbitrary entire functions,
then, for any fixed t ∈ C the radii of convergence of the two z-power series in
(3.6) are infinite (and the same is true for the t-derivatives of those z-power series),
hence both series are entire in (t, z) and satisfy the heat equation (1.3). Thus, every
entire solution F (t, z) of (1.3) is determined uniquely by a pair of (arbitrary) entire
functions φ(t) and ψ(t) via (3.6)-(3.7). As we will see in the next subsection, the
situation is quite different if we want to determine an entire caloric solution F (t, z)
from the initial condition f(z) via (3.2), since in this case the entire function f(z)
cannot be arbitrary.

3.2. Order of the initial condition f(z). Recall that in view of (2.7) the order
ρ of the entire function f(z) can be expressed as

ρ = lim sup
k

k ln k

− ln |ck|
, (3.10)

where, in view of (3.3),

ck =
f (k)(0)

k!
, k ≥ 0. (3.11)

The following theorem is not new (actually it probably goes back to Kovalevskaya,
see [8]), but we include its proof here for the sake of completeness.



10 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

Theorem 3.2. Suppose f(z) is the initial condition of an entire solution F (t, z) of
the heat equation (1.3). Then the order ρ of f(z) satisfies 0 ≤ ρ ≤ 2. Furthermore,
in the extreme case ρ = 2 the type τ of f(z) must be minimal, i.e. zero. Thus,
under the terminology introduced in Subsection 2.4, the exact order of f(z) is in
[0, 2−].

Conversely, if f(z) is an entire function whose exact order is in [0, 2−], then
F (t, z) given by (3.2) is entire in (t, z) and satisfies the heat equation (1.3) with
F (0, z) = f(z).

Proof. We need to consider the cases k = 2j and k = 2j + 1 separately. Observe
that ρ = max{ρ0, ρ1} where

ρ0 := lim sup
j

2j ln(2j)

− ln |c2j |
= 2 lim sup

j

j ln j

− ln |c2j |
, (3.12)

ρ1 := lim sup
j

(2j + 1) ln(2j + 1)

− ln |c2j+1|
= 2 lim sup

j

j ln j

− ln |c2j+1|
. (3.13)

Let us first estimate ρ0. For z = 0 , in view of (3.11), formula (3.2) becomes

F (t, 0) =
∑
j≥0

tj

j!
f (2j)(0) =

∑
j≥0

(2j)!c2j
j!

tj . (3.14)

Since the second power series in (3.14) converges for all t ∈ C, i.e. has an infinite
radius of convergence, we must have

lim sup
j

∣∣ (2j)!c2j
j!

∣∣1/j = 0
(

hence, lim
j

∣∣ (2j)!c2j
j!

∣∣1/j = 0
)

(3.15)

equivalently, there is a sequence εj → 0 such that

|c2j |1/j < εj
[ j!

(2j)!

]1/j
. (3.16)

By applying Stirling’s asymptotic formula for n! to (3.16) we obtain that there is a
sequence εj → 0 (not necessarily the same εj appearing in (3.16)) such that

|c2j |1/j <
εj
j
. (3.17)

Now, (3.12) tells us that 2/ρ0 is the supremum of all exponents r such that

|c2j |1/j <
1

jr
for every sufficiently large j. (3.18)

Therefore, by comparing (3.17) and (3.18) we obtain that

2/ρ0 ≥ 1, i.e. ρ0 ≤ 2. (3.19)

To determine ρ1 we differentiate (3.2) with respect to z and arrive at

∂zF (t, z) =
∑
j≥0

tj

j!
f (2j+1)(z). (3.20)

Then, in exactly the same way as in the case of ρ0 we conclude that there is a
sequence εj → 0 such that

|c2j+1|1/j <
εj
j
. (3.21)

Hence
ρ1 ≤ 2 (3.22)



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 11

and, therefore, the order of f(z) satisfies ρ ≤ 2.
Next, let us estimate the type τ of f(z) in the case where ρ = 2. Recall that if

ρ = 2, then

τ =
1

2e
lim sup

k
k|ck|2/k. (3.23)

As in the case of ρ we, again, need to consider the cases k = 2j and k = 2j+ 1 odd
separately. We, thus, observe that τ = max{τ0, τ1} where

τ0 :=
1

e
lim sup

j
j|c2j |1/j and τ1 :=

1

e
lim sup

j
j|c2j+1|1/j . (3.24)

Then, by (3.17) and (3.21) we obtain immediately that τ0 = τ1 = 0 and, conse-
quently τ = 0. In other words, if f(z) is of order 2, then it is of minimal type.

The second part of the theorem is easy (see also Remark 3.7 below). �

Using the Hadamard Factorization Theorem [7] we obtain immediately the fol-
lowing corollary of Theorem 3.2.

Corollary 3.3. Suppose f(z) 6≡ 0 is the initial condition of an entire solution
F (t, z) of the heat equation (1.3). Then f(z) has the form

f(z) = eλz+βzd Π(z), (3.25)

where λ and β are complex constants, d is a nonnegative integer and Π(z) is a
canonical product whose exact order is in [0, 2−].

Finally note that from Subsection 2.1 it follows that the canonical product Π(z)
of (3.25) must be of the form

Π(z) =
∏
k≥1

ep

( z
ak

)
, where p = 0 or 1 or 2. (3.26)

The case p = 2 can happen only if the exact order of Π(z) is 2− and
∑
k≥1 |zk|−2 =

∞.

3.3. Order and type of a caloric function F (t, z) when t or z is fixed.
Suppose we freeze t ∈ C and we consider

ρz := ordz F (t, z) and τz := typez F (t, z), (3.27)

namely the order and the type of F (t, ·), which for convenience we call the z-order
and z-type of F (t, z). One might think that these quantities depend on t. However,

by applying [4, Theorem 1.1] for the choice f(D) = etD
2

(in the notation of [4,

Theorem 1.1]) and noticing that F (t, z) = etD
2

F (0, z) with D = ∂z, we obtain that
ρz and τz are, actually, independent of t (actually, this also follows from (3.6)). We
can, therefore, choose t = 0 and conclude that

ρz = ρ and τz = τ, (3.28)

where ρ and τ are the order and type of f(z) = F (0, z) respectively. And then,
Theorem 3.2 implies immediately that the exact z-order of F (t, z) is in [0, 2−].

The following open question envisages a refinement of formula (3.28).

Open Question. Let z1(t), z2(t), . . . be the zeros of F (t, z) (viewed as an entire
function of z). Is it true that for any t1, t2 ∈ C and any α > 0 we have that∑

k≥1

1

|zk(t1)|α
<∞ if and only if

∑
k≥1

1

|zk(t2)|α
<∞?



12 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

More specifically, if the zeros are infinitely many and arranged so that |zk(t)| ≤
|zk+1(t) for all k ≥ 1, how close are the asymptotics of the sequences {zk(t1)}k≥1
and {zk(t2)}k≥1 as k →∞?

Regarding the order and the type of F (t, z) viewed as a function of t things
are more complicated, since these quantities may depend on z. For instance it is
clear from (3.6)-(3.7) that there are nontrivial caloric functions F (t, z) such that,
for a given z0 the quantity F (t, z0) can vanish for all t ∈ C (e.g., just think of

F (t, z) = e−λ
2t sin(λz), for z0 = kπ/λ, k ∈ Z).

The goal of this subsection is to clarify the notions of t-order and t-type of
an entire caloric function F (t, z) and to relate them to its z-order and its z-type
respectively.

Theorem 3.4. Let F (t, z) be an entire solution of the heat equation (1.3). For a
fixed z ∈ C we consider the orders of F (t, z) and ∂zF (t, z) viewed as functions of
t, namely

ρt,0(z) := ordt F (t, z), ρt,1(z) := ordt ∂zF (t, z), (3.29)

and we set

ρt := max{ρt,0(z), ρt,1(z)}. (3.30)

Then:

(i) The quantity ρt is independent of z and is related to the z-order ρz (= ρ)
of F (t, z) via the formula

ρt =
ρ

2− ρ
(3.31)

(in particular, if ρ = 2, then ρt = ∞, while ρt is finite in the case where
ρ ∈ [0, 2), and ρt = 0 if and only if ρ = 0).

(ii) We have

ρt,0(z) = ρt and ρt,1(z) = ρt for a.a. z ∈ C. (3.32)

Proof. Let us set

ck(z) :=
f (k)(z)

k!
, k ≥ 0, z ∈ C (3.33)

(so that, in view of (3.11), ck(0) = ck). Then, by substituting (3.33) in (3.2) we
obtain

F (t, z) =
∑
j≥0

(2j)!

j!
c2j(z)t

j (3.34)

∂zF (t, z) =
∑
j≥0

(2j)!

j!
c′2j(z)t

j =
∑
j≥0

(2j + 1)!

j!
c2j+1(z)tj . (3.35)

Then, by (3.34) and (2.10) we obtain

ρt,0(z) = ordt F (t, z) = lim sup
j

j ln j

− ln |c2j(z)(2j)!/j!|
, (3.36)

which implies

ρt,0(z) = lim sup
j

1
− ln |c2j(z)|

j ln j − ln((2j)!)
j ln j + ln(j!)

j ln j

. (3.37)



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 13

Now, by Stirling’s formula we have that ln ((2j)!) /(j ln j)→ 2 and ln (j!) /(j ln j)→
1 as j →∞. Thus (3.37) becomes

ρt,0(z) = lim sup
j

1
− ln |c2j(z)|

j ln j − 1
= lim sup

j

j ln j
− ln |c2j(z)|

1− j ln j
− ln |c2j(z)|

=
ρ0(z)

2− ρ0(z)
, (3.38)

where

ρ0(z) := 2 lim sup
j

j ln j

− ln |c2j(z)|
. (3.39)

Likewise, from (3.35) we have

ρt,1(z) = ordt ∂zF (t, z) = lim sup
j

j ln j

− ln |c2j+1(z)(2j + 1)!/j!|
(3.40)

and in the same way as above we obtain

ρt,1(z) =
ρ1(z)

2− ρ1(z)
, (3.41)

where

ρ1(z) := 2 lim sup
j

j ln j

− ln |c2j+1(z)|
. (3.42)

But in view of (3.33), the quantities ρ0(z) and ρ1(z) of (3.39) and (3.42) re-
spectively, associated with f(z), are the analogs of the quantities introduced in
(2.14) and (2.15), associated to g(z). Hence, by Theorem 2.1 we obtain that
ρ0(z) = ρ1(z) = ρ for a.a. z ∈ C and, therefore, formulas (3.38) and (3.41),
together with (3.30), imply (3.31) and (3.32). �

We will refer to ρt of (3.30) as the caloric t-order of F (t, z). Also, let us set

E0 := {z ∈ C : ρt,0(z) = ρt} and E1 := {z ∈ C : ρt,1(z) = ρt}, (3.43)

so that, in view of (3.30), E0 ∪ E1 = C. Theorem 3.4 tells us that both E0 and E1
have full measure. Consequently,

E := E0 ∩ E1 is a subset of C of full measure. (3.44)

In general it might happen that ρt,0(z0) < ρt for some z0 and ρt,1(z1) < ρt for

some z1 6= z0. For instance, if F (t, z) = e−λ
2t sin(λz), then ρt = 1. However,

F (t, z0) ≡ 0 for z0 = kπ/λ, k ∈ Z, while ∂zF (t, z1) ≡ 0 for z1 = [k + (1/2)]π/λ,
k ∈ Z, and, consequently, ρt,0(z0) = ρt,1(z1) = 0 for those values of z0 and z1.

Next, we present the analog of Theorem 3.4 regarding the t-type of F (t, z).

Theorem 3.5. Let F (t, z) be an entire solution of the heat equation (1.3) whose
z-order ρz (= ρ) satisfies 0 < ρz < 2 (or, equivalently, by (3.31) the caloric t-order
of F (t, z) satisfies 0 < ρt < ∞). Thinking of z as a parameter, we consider the
types of F (t, z) and ∂zF (t, z) viewed as functions of t, namely

τt,0(z) := typet F (t, z), z ∈ E0, (3.45)

τt,1(z) := typet ∂zF (t, z), z ∈ E1, (3.46)

where the sets E0 and E1 are the full measure sets defined by (3.43)-(3.29)-(3.30).
We also set

τt := max{τt,0(z), τt,1(z)}, z ∈ E = E0 ∩ E1. (3.47)

Then



14 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

(i) The quantity τt is independent of z ∈ E and is related to the z-order ρz
(= ρ) and the z-type τz (= τ) of F (t, z) via the formula

τt =
(

1− ρ

2

)
(2ρ)

ρ
2−ρ τ

2
2−ρ . (3.48)

In particular, τt = 0 if and only if τ = 0 and τt =∞ if and only if τ =∞.
(ii) If τt <∞, then

τt,0(z) = τt and τt,1(z) = τt for a.a. z ∈ C. (3.49)

Proof. (i) In view of (3.33) and (2.11) the type τ of f(z) = F (0, z) is given by

τ =
1

eρ
lim sup

k
k |ck(z)|ρz/k for any z ∈ C. (3.50)

If we set

τ0(z) :=
2

eρ
lim sup

j
j|c2j(z)|ρz/2j , τ1(z) :=

2

eρ
lim sup

j
j|c2j+1(z)|ρz/2j , (3.51)

then, obviously (3.50) implies

τ = max{τ0(z), τ1(z)} for any z ∈ C, (3.52)

while by Theorem 2.2 , if τt <∞, we have

τ0(z) = τ and τ1(z) = τ for a.a. z ∈ C. (3.53)

Next, let us set

τ̂t,0(z) :=
1

eρt
lim sup

j
j
∣∣ (2j)! c2j(z)

j!

∣∣ρt/j , z ∈ C, (3.54)

τ̂t,1(z) :=
1

eρt
lim sup

j
j
∣∣ (2j + 1)!c2j+1(z)

j!

∣∣ρt/j , z ∈ C. (3.55)

Then, formulas (3.34), (3.35), (3.43), (3.45), and (3.46) imply

τ̂t,0(z) = τt,0(z) for all z ∈ E0, (3.56)

τ̂t,1(z) = τt,1(z) for all z ∈ E1. (3.57)

Application of Stirling’s asymptotic formula for the factorial to (3.54) gives

τ̂t,0(z) =
1

eρt
lim sup

j
j
(4j

e

)ρt(|c2j(z)|1/j)ρt
=

4ρt

ρt eρt+1

[
lim sup

j
j
(
|c2j(z)|1/j

) ρt
ρt+1

]ρt+1
, z ∈ C.

(3.58)

Since (3.31) can be written as
ρt

ρt + 1
=
ρ

2
, (3.59)

in view of (3.31), (3.59), and (3.51), formula (3.58) yields

τ̂t,0(z) =
4ρt

ρteρt+1

[
lim sup

j
j|c2j(z)|ρ/2j

]ρt+1

=
4ρt

ρt eρt+1

[eρτ0(z)

2

]ρt+1

=
(
1− ρ

2

)
(2ρ)

ρ
2−ρ τ0(z)

2
2−ρ , z ∈ C.

(3.60)



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 15

In the same way, starting from (3.55) we obtain

τ̂t,1(z) =
(
1− ρ

2

)
(2ρ)

ρ
2−ρ τ1(z)

2
2−ρ , z ∈ C. (3.61)

Therefore (3.48) follows by using (3.56) and (3.57) in (3.47), and then invoking
(3.60), (3.61) and (3.52). As for (3.49), it follows from (3.53), (3.56), (3.57), (3.60),
(3.61), and (3.47). �

We will refer to τt of (3.47) as the caloric t-type of F (t, z). The question whether
(3.49) remains valid in the case where τt =∞ remains open.

As an example let us observe that in the special case ρz = ρ = 1 formula (3.31)
implies that ρt = 1 and then (3.48) yields τt = τ2z = τ2 (e.g., this is the case of

the special solution Eλ(t, z) = eλ
2t+λz, where, clearly, ρz = ρt = 1, τz = |λ|, and

τt = |λ|2).

Remark 3.6. From the definition (3.30) of the caloric t-order ρt we know that the
t-orders ρt,0(z) = ordt F (t, z) and ρt,1(z) = ordt ∂zF (t, z) are ≤ ρt for every z ∈ C.
However, the t-type typet F (t, z) of F (t, z) or the t-type typet ∂zF (t, z) of ∂zF (t, z)
can become bigger than the caloric t-type τt of F (t, z) for some exceptional values
of z. For instance, suppose f(z) = f1(z)+f2(z), where the order of f1(z) is smaller
than the order of f2(z), while the type of f1(z) is bigger than the type of f2(z).
Furthermore, let us assume that f2(z) is an odd function. Then, we can easily
construct examples where typet F (t, 0) > τt.

Remark 3.7. Let f(z) be an entire function whose exact order is in [0, 2−]. Then

F (t, z) :=

∫ ∞
−∞

1

2
√
πt
e−(z−ξ)

2/4tf(ξ) dξ for <(t) > 0 (3.62)

satisfies the heat equation (1.3) for <(t) > 0. Actually,

F (t− t0, z) =

∫ ∞
−∞

1

2
√
π(t− t0)

e−(z−ξ)
2/4(t−t0)F (t0, ξ) dξ (3.63)

for <(t− t0) > 0. Also F (t, z) satisfies (1.3).
If t and z are real, with t > 0, then we can use the substitution η = (z − ξ)/

√
t

or η = (ξ − z)/
√
t in the integral of (3.62) and obtain

F (t, z) =
1

2
√
π

∫ ∞
−∞

e−η
2/4f

(
z + η

√
t
)
dη

=
1

2
√
π

∫ ∞
−∞

e−η
2/4f

(
z − η

√
t
)
dη .

(3.64)

Then, from our assumption for the order and type of f(z), the integral in the right-
hand side of (3.64) is entire in (t, z), satisfies the heat equation for every t, z ∈ C
(e.g., by analytic continuation) and it is clear from (3.64) that F (0, z) = f(z).

Example 3.8. (i) Suppose

f(z) = eaz
2

, where a ∈ C \ {0}, (3.65)

so that ρ = 2 and τ = |a| > 0 (thus the exact order of f(z) is 2). Here, if
F (t, z) is as in (3.62), and hence it satisfies the heat equation with initial condition
F (0, z) = f(z), then

F (t, z) =
1√

1− 4at
exp

( az2

1− 4at

)
. (3.66)



16 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

Obviously, this F (t, z) is not entire since it has a strong singularity at t = 1/4a
(independent of z), a combination of an essential singularity and a square-root
branch point (this singularity “disappears” as a → 0). If t starts at 0, makes a
loop around 1/4a and comes back to 0, then we obtain a different value of F (0, z).

Thus, if initially F (0, z) = eaz
2

, then, after one loop we obtain the other branch

of F (0, z), namely F (0, z) = −eaz2 . This counterexample is in agreement with the
previous discussion. Another observation is that for t 6= 1/4a the function F (t, z)
is entire in z and its z-order is ρz = ρ = 2, independent of t. However, its z-type is
τz = |a||1− 4at|−1 (thus depends on t). Finally, notice that F (t, z) is never 0.

(ii) As a variant of the above case, we consider

f(z) = cos(az2) =
eiaz

2

+ e−iaz
2

2
, where a ∈ C \ {0}, (3.67)

so that, again, ρ = 2 and τ = |a| > 0. Then F (t, z) of (3.62) becomes

F (t, z) =
1

2
√

1− 4iat
exp

( iaz2

1− 4iat

)
+

1

2
√

1 + 4iat
exp

( −iaz2
1 + 4iat

)
. (3.68)

Here F (t, z) has a strong singularities at t = ±1/4ia; furthermore, it has infinitely
many zeros. Actually, z is a zero of F (t, z) if and only if

z2 = (1 + 16a2t2)
[ 1

4ia
ln
(1− 4iat

1 + 4iat

)
+

π

2a

]
,

where ln(·) stands for the multivalued logarithmic function.

Example 3.9. Here we examine the case where the initial condition is

f(z) = zα, α ∈ C. (3.69)

For this f(z) the solution of the heat equation (1.3) cannot be entire in (t, z), unless,
of course, α = m, a nonnegative integer. We consider the function

F (t, z) := F (t, z;α) :=
iα

Γ(−α)
tα/2

∫ ∞
0

e−ξ
2+it−1/2zξ

ξα+1
dξ, <(α) < 0, (3.70)

where Γ(·) is the gamma function. Notice that F (t, z;α) is entire in z for any
complex t 6= 0 and analytic in t 6= 0 for any complex z. The singularity at t = 0
it is a combination of a branch point and an essential singularity. Also, it is not
hard to check that F (t, z) satisfies the heat equation for every z ∈ C, t ∈ C \ {0}.
Furthermore, if t → 0 in a way so that =(t−1/2z) ≥ 0, then F (t, z;α) approaches
(some branch of) zα. However, if t→ 0 in an arbitrary way, then limt→0 F (t, z;α)
may not be equal to zα. This is the sense in which the initial condition F (0, z;α) =
zα is satisfied.

The integral representation of F (t, z;α) given in (3.70) makes sense only for
<(α) < 0. We can give other (contour) integral representations of F (t, z;α) over
contours in the complex plane avoiding the positive semiaxis, which are valid for
any α ∈ C. But, instead of doing that, we continue the analysis of F (t, z;α) as
follows.

We observe that F (t, z) = F (t, z;α) in (3.70) can be written as

F (t, z) =
iαtα/2

Γ(−α)
h
( it−1/2z

2

)
, where h(x) := h(x;α) :=

∫ ∞
0

e−ξ
2+2xξ

ξα+1
dξ, (3.71)



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 17

where the function h(x) of (3.71) satisfies the Hermite equation, namely

u′′(x)− 2xu′(x) = −2αu(x), (3.72)

with

h(0) =
1

2
Γ
(
− α

2

)
and h′(0) = Γ

(1− α
2

)
. (3.73)

Clearly, every solution of (3.72) is entire in x. Actually, the general solution of
(3.72) can be expressed as

u(x) =

∞∑
n=0

anx
k, (3.74)

where the coefficients ak satisfy the recursion

an+2

an
=

2(n− α)

(n+ 1)(n+ 2)
. (3.75)

We can single out two linearly independent solutions of (3.72) in a convenient way.
By taking a0 = 1 and a1 = 0 we obtain the solution ue(x) = ue(x;α) which is
even in x and satisfies ue(0) = 1, while by taking a0 = 0 and a1 = 1 we obtain the
solution uo(x) = uo(x;α) which is odd in x and satisfies u′o(0) = 1. Thus,

ue(x) = 1 +

∞∑
k=1

(−1)k
2kα(α− 2) · · ·

(
α− 2(k − 1)

)
(2k)!

x2k (3.76)

and

uo(x) = x+

∞∑
k=1

(−1)k
2k(α− 1)(α− 3) · · · (α− 2k + 1)

(2k + 1)!
x2k+1. (3.77)

Evidently (because of the analytic dependence on the parameter α), ue(x;α) and
uo(x;α) are also entire in α. Furthermore, in view of (3.73),

h(x) = h(x;α) =
1

2
Γ
(
− α

2

)
ue(x;α) + Γ

(1− α
2

)
uo(x;α). (3.78)

An additional consequence of formula (3.78) is that, by analytic continuation in α,
we obtain the meromorphic extension of h(x;α), which we also denote by h(x;α),
for all complex α.

Using (3.78) in (3.71) yields

F (t, z;α) =
iαtα/2

Γ(−α)

[1

2
Γ
(
− α

2

)
ue
( it−1/2z

2
;α
)

+ Γ
(1− α

2

)
uo
( it−1/2z

2
;α
)]
. (3.79)

With the help of the well-known Legendre’s duplication formula for the Gamma
function, namely the formula

√
πΓ(2z) = 22z−1Γ(z)Γ

(
z +

1

2

)
,

equation (3.79) simplifies as

F (t, z;α)

=
√
π(2i)αtα/2

[ 1

Γ( 1−α
2 )

ue

( it−1/2z
2

;α
)

+
2

Γ(−α/2)
uo

( it−1/2z
2

;α
)]
.

(3.80)

It follows that for t 6= 0 the function F (t, z;α) is entire in z and α, while for
any z and α it is analytic in t, except for t = 0, where it may have a strong
singularity (branch point combined with an essential singularity). Thus, if t makes



18 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

a loop around 0 in the complex t-plane, then we may arrive at a different value
of F (t, z;α). Notice, however, that if m ≥ 0 is an integer, F (t, z;m) becomes the
m-th caloric polynomial, i.e.

F (t, z;m) = Pm(t, z). (3.81)

Generically, the z-order of F (t, z;α) is 2, while its z-type depends on t.
The case α = −1 is of particular interest. The solutions ue(x;α) and uo(x;α)

become respectively

ue(x;−1) = ex
2

and uo(x;−1) = ex
2

∫ x

0

e−ξ
2

dξ (3.82)

for every x ∈ C. Then, with the help of (3.82) and (3.80) we can construct the
solution of the heat equation

F (t, z) =
i√
t
e−z

2/4t

∫ −iz/2√t
−∞

e−ζ
2

dζ

=
i√
t
e−z

2/4t
(√π

2
+

∫ −iz/2√t
0

e−ζ
2

dζ
)
,

(3.83)

where the first contour integral is taken over a contour which approaches the neg-
ative real axis at −∞. Formula (3.83) implies that

F (t, z)→ 1

z
as t→ 0 in certain directions. (3.84)

From F (t, z) of (3.83) we can also obtain the solution of the heat equation with
“initial condition” f(z) = ln z as

∫ z
1
F (t, ζ)dζ (the quotations here remind us that

the initial condition is satisfied in a certain sense). Furthermore, the solutions of the
heat equation with “initial conditions” f(z) = z−m, m ∈ N, can be also obtained
from F (t, z) by differentiating it m− 1 times with respect to z (or (m− 1)/2 times
with respect to t, in m is odd).

4. Zeros of entire caloric functions

We start with a result stating that the multiple zeros of a (nontrivial) entire
caloric function F (t, z), viewed as a function of z, cannot accumulate in C2.

Theorem 4.1. Suppose F (t, z) 6≡ 0 is entire in (t, z) and satisfies the heat equation
(1.3). If

F (t∗, z∗) = ∂zF (t∗, z∗) = 0 for some (t∗, z∗) ∈ C2, (4.1)

then there is a (C2-open) neighborhood U of (t∗, z∗) such that

|F (t, z)|+ |∂zF (t, z)| > 0 for every (t, z) ∈ U \ {(t∗, z∗)}. (4.2)

Proof. Without loss of generality and for typographical convenience we take t∗ =
z∗ = 0. If the statement of the theorem is false, then there exists a sequence of
points (tn, zn) 6= (0, 0), n = 1, 2, . . ., such that (tn, zn)→ (0, 0) and

F (tn, zn) = ∂zF (tn, zn) = 0 for every n ≥ 1. (4.3)

Let µ be the smallest value of m such that ∂mz F (0, 0) 6= 0 (since F (t, z) 6≡ 0,
formula (1.7) guarantees that µ exists; of course, due to the assumption (4.1) we



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 19

have µ ≥ 2). Then, by taking t0 = z0 = 0 in (1.7) we obtain

F (t, z) =
∂µz F (0, 0)

µ!
Pµ(t, z) +

∞∑
m=µ+1

∂mz F (0, 0)

m!
Pm(t, z), (4.4)

which also implies

∂zF (t, z) =
∂µz F (0, 0)

µ!
∂zPµ(t, z) +

∞∑
m=µ+1

∂mz F (0, 0)

m!
∂zPm(t, z). (4.5)

In view of (1.8), formulas (4.4) and (4.5) imply that, given an open ball B ⊂ C2

centered at (0, 0) there is an C > 0 (i.e. depending only on B) such that for every
(t, z) ∈ B we have∣∣F (t, z)− ∂µz F (0, 0)

µ!
Pµ(t, z)

∣∣ ≤ C max
{
|z|µ+1, |z|µ−1|t|, . . . , |t|b(µ+2)/2c} (4.6)

and∣∣∂zF (t, z)− ∂µz F (0, 0)

µ!
∂zPµ(t, z)

∣∣ ≤ C max
{
|z|µ, |z|µ−2|t|, . . . , |t|b(µ+1)/2c}. (4.7)

Now, given B there is an n0 such that (tn, zn) ∈ B for all n ≥ n0. Hence, by using
(4.3) in (4.6) and (4.7) we obtain

|Pµ(tn, zn)| ≤ C ′max
{
|zn|µ+1, |zn|µ−1|tn|, . . . , |tn|b(µ+2)/2c}, (4.8)

|∂zPµ(tn, zn)| ≤ C ′max
{
|zn|µ, |zn|µ−2|tn|, . . . , |tn|b(µ+1)/2c} (4.9)

for every n ≥ n0, where for typographical convenience we have set

C ′ :=
µ!C

|∂µz F (0, 0)|
. (4.10)

Let us consider the case where µ = 2l. Then, substituting (1.20) in (4.8) yields

|z2n + ρµ,1tn| · · · |z2n + ρµ,ltn|

≤ C ′max
{
|zn|2l+1, |zn|2l−1|tn|, . . . , |zn||tn|l, |tn|l+1

} (4.11)

for every n ≥ n0, while, in view of (1.14), substituting (1.21) in (4.9) yields

|zn||z2n + ρµ−1,1tn| · · · |z2n + ρµ−1,l−1tn|

≤ C ′

µ
max

{
|zn|2l, |zn|2l−2|tn|, . . . , |zn|2|tn|l−1, |tn|l

} (4.12)

for every n ≥ n0.
If zn = 0 (hence tn 6= 0), then (4.11) becomes |ρµ,1 · · · ρµ,l tln| ≤ C ′|tn|l+1, which,

in view of (1.22) and the assumption tn → 0, cannot be satisfied for any sufficiently
large n. Thus, without loss of generality we can assume zn 6= 0.

We set

λn :=
tn
z2n
. (4.13)

Then, (4.11) and (4.12) become respectively

|1 + ρµ,1λn| · · · |1 + ρµ,lλn| ≤ C ′|zn|max
{

1, |λn|, . . . , |λn|l, |λn|l+1|zn|
}

(4.14)

and

|1 + ρµ−1,1λn| · · · |1 + ρµ−1,l−1λn| ≤
C ′

µ
|zn|max

{
1, |λn|, . . . , |λn|l

}
(4.15)



20 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

for sufficiently large n.
If |λn| becomes arbitrarily large, then, in view of (1.22), formula (4.14) should

imply that there is a constant C ′′ > 0 such that

|λn|l ≤ C ′′|zn|max
{
|λn|l, |λn|l+1|zn|

}
= C ′′max

{
|λn|l|zn|, |λn|l|tn|

}
, (4.16)

i.e. 1 ≤ C ′′max{|zn|, |tn|}, which is, obviously, impossible since zn, tn → 0. There-
fore, the sequence λn must be bounded and, hence, by (4.14) and (4.15) there must
exist a constant M > 0 such that

|1 + ρµ,1λn| · · · |1 + ρµ,lλn| ≤M |zn|. (4.17)

|1 + ρµ−1,1λn| · · · |1 + ρµ−1,l−1λn| ≤M |zn| (4.18)

for every sufficiently large n. Let λnk be a convergent subsequence of the sequence
λn, with limλnk = λ ∈ C. However, if take limits in (4.17) and (4.18) as nk → ∞
we obtain

|1 + ρµ,1λ| · · · |1 + ρµ,lλ| = 0, (4.19)

|1 + ρµ−1,1λ| · · · |1 + ρµ−1,l−1λ| = 0 (4.20)

which contradict (1.22).
The remaining case is µ = 2l+ 1. Here, by substituting (1.21) in (4.8) we obtain

|zn||z2n + ρµ,1 tn| · · · |z2n + ρµ,l tn| ≤ C ′max
{
|zn|2l+2, |zn|2l|tn|, . . . , |tn|l+1

}
(4.21)

for every n ≥ n0, while, in view of (1.14), substituting (1.20) in (4.9) yields

|z2n + ρµ−1,1tn| · · · |z2n + ρµ−1,ltn|

≤ C ′

µ
max

{
|zn|2l+1, |zn|2l−1|tn|, . . . , |zn||tn|l, |tn|l+1

} (4.22)

for every n ≥ n0.
By proceeding in the same manner as in the case µ = 2l, we again arrive at a

contradiction. Therefore, our assumption of the existence of the sequence (tn, zn)
is false. �

An immediate consequence of Theorem 4.1 is that, if F (t, z) 6≡ 0 is entire in (t, z)
and satisfies the heat equation, then the set

MF := {(t, z) ∈ C2 : F (t, z) = ∂zF (t, z) = 0} (4.23)

is discrete in C2. For example, if F is a caloric polynomial, then by (1.20) and
(1.21) we have

MPm = {(0, 0)}, m ≥ 2, while MP0 =MP1 = ∅. (4.24)

One peculiar consequence of Theorem 4.1 (together with the fact [11] that zeros
of entire functions of two or more complex variables are never isolated) is that if
an entire function A(t, z) can be written as

A(t, z) = A1(t, z)2A2(t, z), (4.25)

where A1(t, z) and A2(t, z) are entire and A1(t0, z0) = 0 for some point (t0, z0) ∈ C2,
then A(t, z) cannot satisfy the heat equation (1.3).

Finally, let us notice that the analog to Theorem 4.1 in the case where F (t, z)
is viewed as a function of t does not hold. For example if we consider the caloric

function F (t, z) = e−λ
2t sin(λz), then F (t, 0) ≡ 0, hence ∂jtF (t, 0) ≡ 0 for every

j ∈ N.



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 21

4.1. Dynamics of the zeros. The following lemma is well known (see [5, Lemma
2.3]). Since its proof is very short, we include it here for the sake of completeness.

Lemma 4.2. Let g(z) be analytic in a domain D of C. If z0 ∈ D is such that
g(z0) 6= 0 and we set G(z) := (z − z0)g(z), then

G′′(z0)

G′(z0)
= 2

g′(z0)

g(z0)
.

Proof. For z ∈ D We have

G′′(z)

G′(z)
=

(z − z0)g′′(z) + 2g′(z)

(z − z0)g′(z) + g(z)

and the statement follows by setting z = z0. �

Corollary 4.3. Suppose G(z) is an entire function with zeros z0, z1, z2, . . . , where
z0 is a simple zero of G(z). Furthermore, let us also assume (essentially without
loss of generality) that G(0) 6= 0.

(i) If Σk≥0|zk|−1 <∞ and

G(z) = CeAz
∏
k≥0

(
1− z

zk

)
, (4.26)

where A and C 6= 0 are complex constants, then

G′′(z0)

G′(z0)
= 2A+ 2

∑
k≥1

1

z0 − zk
. (4.27)

(ii) If Σk≥0|zk|−2 <∞ and

G(z) = CeAz
∏
k≥0

(
1− z

zk

)
ez/zk , (4.28)

where, again, A and C 6= 0 are complex constants, then

G′′(z0)

G′(z0)
= 2A+ 2

∑
k≥1

( 1

z0 − zk
+

1

zk

)
. (4.29)

Proof. Set

g(z) :=
G(z)

z − z0
(4.30)

(thus G(0) 6= 0 implies g(0) 6= 0). Then, by Lemma 4.2 we have

G′′(z0)

G′(z0)
= 2

g′(z0)

g(z0)
. (4.31)

Using (4.26) in (4.30) yields

g(z) = ceAz
∏
k≥1

(
1− z

zk

)
, (4.32)

where c 6= 0. Thus, the Mittag-Leffler expansion of g′(z)/g(z) is [7]

g′(z)

g(z)
= A+

∑
k≥1

1

z − zk
.

Therefore, formula (4.27) follows by setting z = z0 in the above formula and sub-
stituting in (4.31).

The proof of formula (4.29) is very similar. �



22 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

Notice that formula (4.29) differs from (4.27) only if Σk≥0|zk|−1 = ∞. Let us
now consider the set

Γ := {F (t, z) = 0} :=
{

(t, z) ∈ C2 : F (t, z) = 0
}
. (4.33)

If Γ is empty, that is if F (t, z) is never 0, then by the Hadamard Factorization
Theorem and by Theorems 3.2 and 3.4 of the previous section it follows that (recall
(1.15))

F (t, z) = cEλ(t, z) = ceλ
2t+λz, (4.34)

for some constants c, λ ∈ C, with c 6= 0.
Suppose now that F (t, z) is a nontrivial entire solution of the heat equation,

which is not of the form (4.34) and, hence, it vanishes for some values of t and z.
Then Γ is a nonempty set in C2 (and, of course, Γ 6= C2). If F (t, z) is irreducible,
namely it cannot be written as

F (t, z) = A1(t, z)A2(t, z), (4.35)

where both A1(t, z) and A2(t, z) are entire and assume the value 0, then we can
say that Γ is a “curve” in C2. Otherwise Γ is a union of such component-curves (as
we have seen, a consequence of Theorem 4.1 is that multiple components do not
exist). For example, if F is a caloric polynomial, then, in view of (1.20) and (1.21)
we have

Γ = ∪lj=1{z2 + ρm,jt = 0}, if m = 2l, (4.36)

Γ = {z = 0} ∪ ∪lj=1{z2 + ρm,jt = 0}, if m = 2l + 1. (4.37)

Let z1(t), z2(t), . . . be the zeros of F (t, z). These zeros can be seen as branches
of a global analytic function, say Z(T ) defined on a Riemann surface which can
be identified with Γ. In general, this will be an infinitely sheeted surface. The
ramification points of Γ are the points (t∗, z∗) satisfying (4.1) and Theorem 4.1
assures us that they form a discrete set in C2. Thus, for every zero zk(t) we
have that ∂zF (t, zk(t)) 6= 0 for a.a. t ∈ C, where here “a.a.” means “almost all,”
namely all except for a discrete subset of C. Then, by differentiating (implicitly)
F (t, zk(t)) = 0 with respect to t we obtain

∂tF (t, zk(t)) + ∂zF (t, zk(t))z′k(t) = 0 (4.38)

or, in view of (1.3),

z′k(t) = − ∂tF (t, zk(t))

∂zF (t, zk(t))
= −∂

2
zF (t, zk(t))

∂zF (t, zk(t))
for a.a. t ∈ C. (4.39)

Thus, if for F (t, z) we have that ρz < 1, then we can apply Corollary 4.3 to (4.39)
and obtain

z′k(t) = −2
∑
j 6=k

1

zk(t)− zj(t)
for a.a. t ∈ C. (4.40)

The above derivation of (4.40) is an imitation of the derivation of (1.24) as presented
in [5].

As an application, let us consider the system of ordinary differential equations

z′k(t) = −2
∑
j 6=k

1

zk(t)− zj(t)
, 1 ≤ k ≤ N, (4.41)

with initial condition
zk(0) = ak 1 ≤ k ≤ N, (4.42)



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 23

where a1, . . . , aN are distinct non-zero complex numbers. To solve this system, we
form the polynomial

f(z) :=
(

1− z

a1

)
· · ·
(

1− z

aN

)
= 1 +A1z + · · ·+ANz

N . (4.43)

Then, the solution z1(t), . . . , zN (t) of the system is the set of zeros of the polynomial
in (t, z) given by

F (t, z) := 1 +

N∑
k=1

AkPk(t, z), (4.44)

where Pk(t, z) is the k-th caloric polynomial.
We expect that this application extends to the infinite case (i.e. N =∞) under

the restriction that
∑
k |ak|−1 <∞.

4.1.1. Even caloric functions. Suppose that the initial condition f(z) is even, i.e. it
satisfies f(−z) = f(z), Then, by formula (3.2) we have that the solution F (t, z) of
the heat equation also satisfies F (t,−z) = F (t, z). Hence F (t, z) = Φ(t, z2), where
Φ(t, µ) is entire in (t, µ) and if ρz is the z-order of F (t, z), then the order of Φ(t, µ)
with respect to µ (the µ-order) is ρz/2. Furthermore, Φ(t, µ) satisfies the heat-type
equation

∂tΦ(t, µ) = 4µ∂2µΦ(t, µ) + 2∂µΦ(t, µ). (4.45)

Let ±z1(t),±z2(t), . . . be the zeros of F (t, z). Then the zeros of Φ(t, µ) are µ1(t) =
z1(t)2, µ2(t) = z2(t)2, . . ., and by imitating the derivation of (4.39) we now have, in
view of (4.45)

µ′k(t) = − ∂tΦ(t, µk(t))

∂µΦ(t, µk(t))
= −4µk(t)

∂2µΦ(t, µk(t))

∂µΦ(t, µk(t))
− 2 for a.a. t ∈ C. (4.46)

If ρz < 2, then the µ-order of Φ(t, µ) is less than 1. We can, therefore apply
Corollary 4.3 to (4.46) and obtain

µ′k(t) = −2− 8µk(t)
∑
j 6=k

1

µk(t)− µj(t)
for a.a. t ∈ C. (4.47)

4.1.2. Odd caloric functions. Now, suppose that the initial condition f(z) satisfies
f(−z) = −f(z), i.e. is odd. Then, by formula (3.2) we have that the solution F (t, z)
of the heat equation also satisfies F (t,−z) = −F (t, z). Hence F (t, z) = zΨ(t, z2),
where Ψ(t, µ) is entire in (t, µ) and if ρz is the z-order of F (t, z), then the µ-order
of Ψ(t, µ) is ρz/2. Furthermore, Ψ(t, µ) satisfies the heat-type equation

∂tΨ(t, µ) = 4µ∂2µΨ(t, µ) + 6∂µΨ(t, µ). (4.48)

Let z0(t) ≡ 0,±z1(t),±z2(t), . . . be the zeros of F (t, z). Then the zeros of Ψ(t, µ)
are µ1(t) = z1(t)2, µ2(t) = z2(t)2, . . ., and by imitating the derivation of (4.39) we
now have, in view of (4.48)

µ′k(t) = − ∂tΨ(t, µk(t))

∂µΨ(t, µk(t))
= −4µk(t)

∂2µΨ(t, µk(t))

∂µΨ(t, µk(t))
− 6 for a.a. t ∈ C. (4.49)

If ρz < 2, then the µ-order of Ψ(t, µ) is less than 1. We can, therefore apply
Corollary 3.3 to (4.49) and obtain

µ′k(t) = −6− 8µk(t)
∑
j 6=k

1

µk(t)− µj(t)
. (4.50)



24 V. G. PAPANICOLAOU, E. KALLITSI, G. SMYRLIS EJDE-2021/44

4.1.3. General examples. Let us start with an observation whose validity is easily
checked.

Observation. Suppose F (t, z) satisfies the heat equation (1.3), with F (0, z) =
f(z). Then

G(t, z) := F (t, z + 2λt)Eλ(t, z) = F (t, z + 2λt) eλ
2t+λz, (4.51)

where λ is a complex constant, also satisfies (1.3), with G(0, z) = eλzf(z) (actually,
we can even take λ to be a square matrix with constant elements).

As an example, let us take f(z) = zm, where m is a positive integer. Then
F (t, z) = Pm(t, z), the m-th caloric polynomial, and

G(t, z) = eλ
2t+λzPm(t, z + 2λt), (4.52)

is the solution of the heat equation with initial condition G(0, z) = zmeλz.
(i) Let F (t, z) and G(t, z) be entire solutions of the heat equation (1.3) with

initial conditions

f(z) =
∏
k

(
1− z

ak

)
and g(z) = eλz

∏
k

(
1− z

ak

)
(4.53)

respectively, where the order of the product Πk[1 − (z/ak)] is σ < 1 (in other
words, there is an α < 1 such that Σk|ak|−α < ∞). Then, as we have seen the
z-order of F (t, z) is σ. It follows that if z1(t), z2(t), . . . are the zeros of F (t, z), then
Σ′k|zk(t)|−α <∞ for some α < 1 (the prime on the sum indicates that we omit the
zk(t)’s which are equal to 0).

The relation of G(t, z) and F (t, z) is given by (4.51). Thus, if w1(t), w2(t), . . .
are the zeros of G(t, z) (viewed as a function of z), then (4.51) implies that

wk(t) = zk(t)− 2λt, k ≥ 1, (4.54)

and it follows that Σ′k|wk(t)|−α <∞.
Finally, since (4.52) implies w′k(t) = z′k(t) − 2λ, while zk(t), k ≥ 1, satisfies

(4.40), we have

w′k(t) = −2λ− 2
∑
j 6=k

1

wk(t)− wj(t)
for a.a. t ∈ C. (4.55)

(ii) Let F (t, z) be an entire solution of the heat equation (1.3) with initial con-
dition

f(z) = eλzzd
∏
k

(
1− z

ak

)
ez/ak , (4.56)

where, d ≥ 0 is an integer and the order ρ of f(z) is < 2. Then, by applying
Corollary 4.3(ii) we can get that the zeros z1(t), z2(t), . . . of F (t, z) satisfy

z′k(t) = −2λ− 2
∑
j 6=k

[ 1

zk(t)− zj(t)
+

1

zj(t)

]
for a.a. t ∈ C.

References

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I: Asymptotic Methods and Perturbation Theory, Springer-Verlag, New York, 1999.

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Evaluation of Integrals, Cambridge University Press, Edinburgh, 2004.

[3] N. C. de Bruijn; The roots of trigonometric integrals, Duke J. Math., 17 (1950), 197–226.



EJDE-2021/44 ENTIRE SOLUTIONS FOR THE HEAT EQUATION 25

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Appl., 290 (2004), 534–541.

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Λ, and the Riemann hypothesis, Constr. Approx., 10 (1994), 107–129.

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mations, 17, No. 1 (2011), 1-110.
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first paper, Expositiones Mathematicae, 12 (1994), 79–95.
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Mathématique, 91 (2003), 369–387.

[10] H. Ki, Y. O. Kim, J. Lee; On the de Bruijn-Newman constant, Advances in Mathematics,22
(2009), 281–306.

[11] S. G. Krantz; Function Theory of Several Complex Variables, Second Edition, AMS Chelsea
Publishing, 2001.

[12] E. H. Lieb, M. Loss; Analysis, Second Edition, Graduate Studies in Mathematics, Volume

14, American Mathematical Society, Providence, RI, 2001.
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246–251.

[14] V. G. Papanicolaou, E. Kallitsi, G. Smyrlis; Analytic Solutions of the Heat Equation,
arXiv:1906.02233v1 [math.CV] (June 2019).

[15] P. Poláčik, V. Šverák; Zeros of complex caloric functions and singularities of complex viscous
Burgers equation, J. reine angew. Math., 616 (2008), 205–217.

[16] D. H. J. Polymath; Effective Approximation of Heat Flow Evolution of the Riemann ξ

Function, and a New Upper Bound for the de Bruijn-Newman constant, arXiv:1904.12438
[math.NT], (29 Apr 2019).

[17] G. F. B. Riemann; Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On

the Number of Prime Numbers less than a Given Quantity), Monatsberichte der Berliner
Akademie (November 1859), 671-680.

[18] B. Rodgers, T. Tao; The de Bruijn-Newman constant is nonnegative, arXiv:1801.05914v1

[math.NT]—, (18 Jan 2018).
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4th ed., Vol. 23, Providence,RI, 1975.

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Applied Mathematics 30 (1977) 321–337.

Vassilis G. Papanicolaou

National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece
Email address: papanico@math.ntua.gr

Eva Kallitsi
National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece

Email address: evpapa@hotmail.com

George Smyrlis
National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece

Email address: gsmyrlis@math.ntua.gr


	1. Introduction
	2. Order and type of an entire function
	2.1. Additional properties of the order
	2.2. Additional properties of the type
	2.3. Canonical products
	2.4. Some notation and terminology

	3. Order and type considerations for entire caloric functions
	3.1. t- and z-power series of a caloric function
	3.2. Order of the initial condition f(z)
	3.3. Order and type of a caloric function F(t, z) when t or z is fixed

	4. Zeros of entire caloric functions
	4.1. Dynamics of the zeros

	References