# Linear Partial Differential Equations in Module of Copolynomials of Several
Variables over a Commutative Ring.

S.L.Gefter<EMAIL_ADDRESS>Department of Mathematics & Computer Sciences
V. N. Karazin Kharkiv National University  A.L.Piven<EMAIL_ADDRESS>Department of Mathematics & Computer Sciences
V. N. Karazin Kharkiv National University

###### Abstract

We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from the
ring of polynomials $K[x_{1},...,x_{n}]$ into the commutative ring $K$. We
prove an existence and uniqueness theorem for a linear differential equation
of infinite order which can be considered as an algebraic version of the
classical Malgrange-Ehrenpreis theorem for the existence of the fundamental
solution of a linear differential operator with constant coefficients. We find
the fundamental solutions of linear differential operators of infinite order
and show that the unique solution of the corresponding inhomogeneous equation
can be represented as a convolution of the fundamental solution of this
operator and the right-hand side. We also prove the existence and uniqueness
theorem of the Cauchy problem for some linear differential equations in the
module of formal power series with copolynomial coefficients.

Keywords: copolynomial, fundamental solution, convolution, $\delta$-function,
differential operator of infinite order, Cauchy problem, Laplace transform
2020 Mathematics Subject Classification. 35R50, 34A35, 13B25

This article is accepted for publishing in Journal of Mathematical Analysis
and Applications

##  1 Introduction

The Poisson formula

$u(t,x)=\frac{1}{2a\sqrt{\pi
t}}\int\limits_{-\infty}^{\infty}e^{-\frac{y^{2}}{4a^{2}t}}Q(x-y)dy$

for the solution of the Cauchy problem for the one-dimensional heat equation

$\frac{\partial u(t,x)}{\partial t}=a^{2}\frac{\partial^{2}u(t,x)}{\partial
x^{2}},\quad u(0,x)=Q(x)$

is very interesting. Externally, this formula is ”very transcendental”. In
other direction, if the initial condition $Q(x)$ is a polynomial of degree $m$
with integer coefficients and $a\in\mathbb{Z}$, then the considered Cauchy
problem has the unique polynomial solution with integer coefficients

$u(t,x)=\sum_{k=0}^{[m/2]}a^{2k}\frac{Q^{(2k)}(x)}{k!}t^{k}.$

The form of this solution shows that it is defined over the ring $\mathbb{Z}$,
i.e. to find it we need only to add and to multiply (unlike to the Poisson
formula, where coefficient $a^{2}$ be in a denominator of an exponent of
power). Thus, in some sense the Poisson formula has an arithmetic origin. The
analogue of the Poisson formula in the case of the non-invertible operator
coefficient $a^{2}$ and the convergent power series $Q(x)$ was considered in
[9]. In the present paper we consider a purely algebraic version of the
Poisson formula (see Example 6.11) and other similar formulas in the case when
the initial condition $Q$ is a copolynomial over the commutative integral
domain $K$, that is a $K$-linear functional on the ring $K[x_{1},...,x_{n}]$
of polynomials of $n$ variables. General properties of copolynomials of $n$
variables were considered in Section 2. Recently the case $n=1$ was studied in
[6, 7, 14] partially. In these papers copolynomials were called a formal
generalized functions (see also [8]). Note that given properties of
copolynomials connected with the convolution are consequences of general
constructions of the theory of Hopf algebras (see, for example, [20, 26]). In
Section 3 differential operators of infinite order on the module of
copolynomials are studied. In Section 4 with the help of the Laplace transform
a connection between copolynomials and formal power series is established (see
Propositions 4.2, 4.5 and Theorem 4.3). The main results of the present paper
are contained in Sections 5 and 6. Theorem 5.1 and Corollary 5.2 can be
considered as an algebraic version of the classical Malgrange-Ehrenpreis
theorem for the existence of the fundamental solution of a linear differential
operator with constant coefficients (see, for example, [16, Theorem 7.3.10],
[17, Section 10.2])). Moreover, it was shown in Theorem 5.1 and Corollary 5.4
that the unique solution of the inhomogeneous equation ${\mathcal{F}}u=T$ with
a linear differential operator ${\mathcal{F}}$ of infinite order can be
represented as a convolution of the fundamental solution of this operator and
the right-hand side $T$ from the module of copolynomials. We note that unlike
the classical theory (see, for example, [17]), the solution of the
inhomogeneous equation ${\mathcal{F}}u=T$, and in particular, its fundamental
solution in the module of copolynomials are defined uniquely. In Section 6 the
concept of a fundamental solution of the Cauchy problem for the equation
$\frac{\partial u}{\partial t}={\mathcal{F}}u$ is introduced and is studied.
The main result of this section is Theorem 6.9 which states that under the
fulfillment of additional restrictions on the ring $K$ the Cauchy problem
$\frac{\partial u}{\partial t}=({\mathcal{F}}u)(t,x),\ u(0,x)=Q(x)$ with a
copolynomial $Q(x)$ has a unique solution and furthermore this solution is a
convolution of the fundamental solution of the Cauchy problem and the initial
condition. In Sections 5 and 6 we present meaningful examples which illustrate
the constructed theory.

Note that differential operators of infinite order on different spaces were
studied in numerous works (see, for example, [22, 19, 2, 23, 24, 25, 10, 11]).
In the classical scalar case, the series with respect to the derivatives of
the $\delta$-function are intensively studied because of their applications to
differential and functional-differential equations and the theory of
orthogonal polynomials (see, for example, [15, 3]).

##  2 Preliminary

Let $K$ be an arbitrary commutative integral domain with identity and let
$K[x_{1},...,x_{n}]$ be a ring of polynomials with coefficients in $K$.

######  Definition 2.1.

By a copolynomial over the ring $K$ we mean a $K$-linear functional defined on
the ring $K[x_{1},...,x_{n}]$, i.e. a homomorphism from the module
$K[x_{1},...,x_{m}]$ into the ring $K$.

We denote the module of copolynomials over $K$ by
$K[x_{1},..,x_{n}]^{\prime}$. Thus $T\in K[x_{1},...,x_{n}]^{\prime}$ if and
only if $T:K[x_{1},...,x_{n}]\to K$ and $T$ has the property of $K$-linearity:
$T(ap+bq)=aT(p)+bT(q)$ for all $p,q\in K[x_{1},...,x_{n}]$ and $a,b\in K$. If
$T\in K[x_{1},..,x_{n}]^{\prime}$ and $p\in K[x_{1},...,x_{n}]$, the for the
value of $T$ on $p$ we use the notation $(T,p)$. We also write a copolynomial
$T\in K[x_{1},...,x_{n}]^{\prime}$ in the form $T(x)$, where
$x=(x_{1},...,x_{n})$ is regarded as the argument of polynomials $p(x)\in
K[x_{1},...,x_{n}]$ subjected to the action of the $K$-linear mapping $T$. In
this case, the result of action of $T$ upon $p$ can be represented in the form
$(T(x),p(x))$.

Let $\mathbb{N}_{0}$ be the set of nonnegative integers. For a multi-index
$\alpha=(\alpha_{1},...,\alpha_{n})\in\mathbb{N}_{0}^{n}$ we put [22, Chapter
1, §1–2]

$D^{\alpha}=\frac{\partial^{|\alpha|}}{\partial x_{1}^{\alpha_{1}}\partial
x_{2}^{\alpha_{2}}\cdots\partial
x_{n}^{\alpha_{n}}},\quad|\alpha|=\sum_{j=1}^{n}\alpha_{j},$
$x^{\alpha}=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots
x_{n}^{\alpha_{n}},\quad\alpha!=\alpha_{1}!\alpha_{2}!\cdots\alpha_{n}!.$

For multi-indexes $\alpha,\beta\in\mathbb{N}_{0}^{n}$ the relation
$\alpha\leq\beta$ means that $\alpha_{j}\leq\beta_{j}$ for all $j=1,...,n$. If
$\alpha\leq\beta$ , then we will use the notation
$\left(\begin{array}[]{c}\beta\\\
\alpha\end{array}\right)=\prod\limits_{j=1}^{n}\left(\begin{array}[]{c}\beta_{j}\\\
\alpha_{j}\end{array}\right)$.

Let $p(x)=\sum\limits_{|\alpha|\leq m}a_{\alpha}x^{\alpha}\in
K[x_{1},...,x_{n}]$. If $h=(h_{1},...,h_{n})$, then the polynomial $p(x+h)\in
K[x_{1},...,x_{n}][h_{1},...,h_{n}]$ can be represented in the form

$p(x+h)=\sum_{|\alpha|\leq m}p_{\alpha}(x)h^{\alpha},$

where $p_{\alpha}(x)\in K[x_{1},...,x_{n}]$. Since in the case of a field with
zero characteristic $p_{\alpha}(x)=\frac{D^{\alpha}p(x)}{\alpha!}$, we also
assume that, by definition,
$\frac{D^{\alpha}p(x)}{\alpha!}=p_{\alpha}(x),\quad|\alpha|\leq m$ is true for
any commutative ring $K$. For $m<|\alpha|$ we assume that
$\frac{D^{\alpha}p(x)}{\alpha!}=0$.

We now introduce the notion of shift for a copolynomial [8, 7]. For $T\in
K[x_{1},...,x_{n}]^{\prime}$ and fixed $h=(h_{1},...,h_{n})\in K^{n}$ we
define a copolynomial $T(x+h)$ by the formula

$(T(x+h),p)=(T,p(x-h)),\quad p\in K[x_{1},...,x_{n}].$

######  Definition 2.2.

The partial derivative $\frac{\partial T}{\partial x_{j}}$ of a copolynomial
$T\in K[x_{1},...,x_{n}]^{\prime}$ with respect to the variable $x_{j}\
(j=1,...,n)$ is defined as in the classical case by the formula

$\left(\frac{\partial T}{\partial x_{j}},p\right)=-\left(T,\frac{\partial
p}{\partial x_{j}}\right),\quad p\in K[x_{1},...,x_{n}].$ (2.1)

By using (2.1), we arrive at the following expression for the derivative
$D^{\alpha}T$:

$(D^{\alpha}T,p)=(-1)^{|\alpha|}(T,D^{\alpha}p),\quad p\in
K[x_{1},...,x_{n}].$

Therefore

$(D^{\alpha}T,p)=0,\ \text{where}\ p\in K[x_{1},...,x_{n}]\ \text{and}\
|\alpha|>{\rm deg}p.$

By virtue of the equality

$\left(\frac{D^{\alpha}T}{\alpha!},p\right)=(-1)^{|\alpha|}\left(T,\frac{D^{\alpha}p}{\alpha!}\right),\quad
p\in K[x_{1},...,x_{n}],$ (2.2)

the copolynomials $\frac{D^{\alpha}T}{\alpha!}$ are well defined for any $T\in
K[x_{1},...,x_{n}]^{\prime}$ and $\alpha\in\mathbb{N}_{0}^{n}$.

######  Example 2.3.

The copolynomial $\delta$-function is given by the formula

$(\delta,p)=p(0),\quad p\in K[x_{1},...,x_{n}].$

Therefore

$(D^{\alpha}\delta,p)=(-1)^{|\alpha|}(\delta,D^{\alpha}p)=(-1)^{|\alpha|}D^{\alpha}p(0),\quad\alpha\in\mathbb{N}_{0}^{n}.$

######  Example 2.4.

Let $K=\mathbb{R}$ and let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a
Lebesgue-integrable function such that

$\int\limits_{\mathbb{R}^{n}}|x^{\alpha}f(x)|dx<+\infty,\quad\alpha\in\mathbb{N}_{0}^{n}$
(2.3)

Then $f$ generates the regular copolynomial $T_{f}$:

$(T_{f},p)=\int\limits_{\mathbb{R}^{n}}p(x)f(x)dx,\quad
p\in\mathbb{R}[x_{1},...,x_{n}].$

Note that, in this case, unlike the classical theory, all copolynomials are
regular [3, Theorem 7.3.4], although a nonzero function $f$ can generate the
zero copolynomial [7, Example 2.2], [8, Remark 1]. Moreover, if $f\in
C^{1}(\mathbb{R}^{n})$ and the conditions (2.3) are satisfied for
$\frac{\partial f}{\partial x_{j}}\ (j=1,...,n)$, then it can be shown that

$\left(\frac{\partial T_{f}}{\partial
x_{j}},p\right)=\int\limits_{\mathbb{R}^{n}}p(x)\frac{\partial f}{\partial
x_{j}}dx,\quad p\in\mathbb{R}[x_{1},...,x_{n}],\ j=1,...,n.$

######  Remark 2.5.

The notion of a copolynomial differs from that of a formal distribution used
in the theory of vertex operator algebras (see [4, 18]), although there are
some natural connections between these notions.

We now consider the problem of convergence in the space
$K[x_{1},...,x_{n}]^{\prime}$. In the ring K, we consider the discrete
topology. Further, in the module of copolynomials
$K[x_{1},...,x_{n}]^{\prime}$, we consider the topology of pointwise
convergence. It is easy to show that the last topology is generated by the
following metric:

$d(T_{1},T_{2})=\sum_{|\alpha|=0}^{\infty}\frac{d_{0}((T_{1},x^{\alpha}),(T_{2},x^{\alpha}))}{2^{|\alpha|}},$

where $d_{0}$ is the discrete metric on $K$. The convergence of a sequence
$\\{T_{k}\\}_{k=1}^{\infty}$ to $T$ in $K[x_{1},...,x_{n}]^{\prime}$ means
that for every polynomial $p\in K[x_{1},...,x_{n}]$ there exists a number
$k_{0}\in\mathbb{N}$ such that

$(T_{k},p)=(T,p),\quad k=k_{0},k_{0}+1,k_{0}+2,....$

The series $\sum\limits_{k=0}^{\infty}T_{k}$ converges in
$K[x_{1},...,x_{n}]^{\prime}$ if a sequence of its partial sums
$\sum\limits_{k=0}^{N}T_{k}$ converges in $K[x_{1},...,x_{n}]^{\prime}$.

The following lemma shows the possibility of the decomposition of an arbitrary
copolynomial in series about the system $\frac{D^{\alpha}\delta}{\alpha!},\
\alpha\in\mathbb{N}_{0}^{n}$.

######  Lemma 2.6.

Let $T\in K[x_{1},...,x_{n}]^{\prime}$. Then

$T=\sum_{|\alpha|=0}^{\infty}(-1)^{|\alpha|}(T,x^{\alpha})\frac{D^{\alpha}\delta}{\alpha!}.$

###### Proof.

For any multi-index $\beta\in\mathbb{N}_{0}^{n}$ we have

$\sum_{|\alpha|=0}^{\infty}(-1)^{|\alpha|}(T,x^{\alpha})\left(\frac{D^{\alpha}\delta}{\alpha!},x^{\beta}\right)=(-1)^{|\beta|}(T,x^{\beta})\left(\frac{D^{\beta}\delta}{\beta!},x^{\beta}\right)=(T,x^{\beta}).$

∎

######  Definition 2.7.

If $p\in K[x_{1},...,x_{n}]$ and $T\in K[x_{1},...,x_{n}]^{\prime}$ is a
copolynomial, then their convolution $T*p$ is defined naturally as follows:

$(T*p)(x)=(T(y),p(x-y))=\sum\limits_{|\alpha|\leq
m}(-1)^{|\alpha|}(T,y^{\alpha})\frac{D^{\alpha}p(x)}{\alpha!},$

where $m={\rm deg}p$. Thus $T*p$ is a polynomial with coefficients in $K$,
i.e. $T*p\in K[x_{1},...,x_{n}]$.

######  Remark 2.8.

By Definition 2.7 we have $\delta*p=p$.

######  Definition 2.9.

The tensor product $T_{1}\otimes T_{2}\in
K[x_{1},...,x_{n},y_{1},...,y_{m}]^{\prime}$ of copolynomials $T_{1}\in
K[x_{1},....,x_{n}]^{\prime}$ and $T_{2}\in K[y_{1},....,y_{m}]^{\prime}$ is
defined by the equality:

$(T_{1}\otimes
T_{2},x^{\alpha}y^{\beta})=(T_{1},x^{\alpha})(T_{2},y^{\beta}),\quad\alpha\in\mathbb{N}_{0}^{n},\beta\in\mathbb{N}_{0}^{m}.$

Further with the help of $K$-linearity the result of the action $(T_{1}\otimes
T_{2},p)$ of the copolynomial $T_{1}\otimes T_{2}$ to an arbitrary polynomial
$p\in K[x_{1},...,x_{n},y_{1},...,y_{m}]$ is defined.

######  Definition 2.10.

Let $T_{1},T_{2}\in K[x_{1},...,x_{n}]^{\prime}$. We remind the definition of
their convolution [20], [26, Section 2.1]. If $p\in K[x_{1},...,x_{n}]$ and
$p(x+y)=\sum\limits_{|\alpha|\leq m}\frac{D^{\alpha}p(x)}{\alpha!}y^{\alpha}$,
then

$(T_{1}*T_{2},p)=(T_{1}\otimes T_{2},p(x+y))=\sum_{|\alpha|\leq
m}\left(T_{1}(x),\frac{D^{\alpha}p(x)}{\alpha!}\right)(T_{2}(y),y^{\alpha}).$
(2.4)

Note that $T_{1}*T_{2}\in K[x_{1},...,x_{n}]^{\prime}$.

The following assertion establishes the commutativity and associativity for a
convolution of copolynomials.

######  Proposition 2.11.

Let $T_{1},T_{2},T_{3}\in K[x_{1},...,x_{n}]^{\prime}$. Then

$T_{1}*T_{2}=T_{2}*T_{1}$ $(T_{1}*T_{2})*T_{3}=T_{1}*(T_{2}*T_{3}).$

######  Example 2.12.

Let $T\in K[x_{1},...,x_{n}]^{\prime}$. We find the convolution $\delta*T$.
For $p\in K[x_{1},...,x_{n}]$, ${\rm deg}p=m$ we obtain

$(\delta*T,p)=\sum_{|\alpha|\leq
m}\left(\delta,\frac{D^{\alpha}p(x)}{\alpha!}\right)(T,y^{\alpha})=\sum_{|\alpha|\leq
m}\left(\frac{D^{\alpha}p}{\alpha!}\right)(0)(T,y^{\alpha})=(T,p).$

Hence $\delta*T=T$.

######  Corollary 2.13.

The module $K[x_{1},...,x_{n}]^{\prime}$ under the convolution operation is an
associative commutative algebra with identity over the ring $K$.

The following theorem establishes the property of the continuity for a
convolution.

######  Theorem 2.14.

Let $T_{k}\in K[x_{1},...,x_{n}]^{\prime},\ k\in\mathbb{N}$ and $T_{k}\to 0,\
k\to\infty$ in the topology of $K[x_{1},...,x_{n}]^{\prime}$. Then $T_{k}*S\to
0,\quad k\to\infty$ in the topology of $K[x_{1},...,x_{n}]^{\prime}$ for every
copolynomial $S\in K[x_{1},...,x_{n}]^{\prime}$.

###### Proof.

Indeed, by Formula (2.4), for every polynomial $p\in K[x_{1},...,x_{n}]$ of
degree $m$ we have

$(T_{k}*S,p)=\sum_{|\alpha|\leq
m}(T_{k}(y),y^{\alpha})\left(S(x),\frac{D^{\alpha}p(x)}{\alpha!}\right)\to 0,\
k\to\infty$

∎

######  Corollary 2.15.

Assume that $T_{k}\in K[x_{1},...,x_{n}]^{\prime},\ k\in\mathbb{N}$ and the
series $\sum\limits_{k=1}^{\infty}T_{k}$ converges in the topology of
$K[x_{1},...,x_{n}]^{\prime}$. Then for every copolynomial $S\in
K[x_{1},...,x_{n}]^{\prime}$ the series $\sum\limits_{k=1}^{\infty}(T_{k}*S)$
converges in the same topology and

$\sum_{k=1}^{\infty}(T_{k}*S)=\left(\sum_{k=1}^{\infty}T_{k}\right)*S.$

##  3 Linear differential operators of infinite order on the module of
copolynomials

We now consider the following linear differential operator of infinite order
on $K[x_{1},...,x_{n}]^{\prime}$:

${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha},$

where $a_{\alpha}\in K$. This operator acts upon a copolynomial $T\in
K[x_{1},...,x_{n}]^{\prime}$ by the following rule: if $p\in
K[x_{1},...,x_{n}]$ and $m={\rm deg}p$, then

$({\mathcal{F}}T,p)=\left(\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}T,p\right)=\sum_{|\alpha|\leq
m}(-1)^{|\alpha|}a_{\alpha}(T,D^{\alpha}p)=\sum_{|\alpha|\leq
m}a_{\alpha}(D^{\alpha}T,p).$

Thus, the differential operator ${\mathcal{F}}:K[x_{1},...,x_{n}]^{\prime}\to
K[x_{1},...,x_{n}]^{\prime}$ is well defined and for any polynomial $p$ of
degree at most $m$ the equality

$({\mathcal{F}}T,p)=\sum_{|\alpha|\leq m}a_{\alpha}(D^{\alpha}T,p).$ (3.1)

is true.

######  Lemma 3.1.

The differential operator ${\mathcal{F}}:K[x_{1},...,x_{n}]^{\prime}\to
K[x_{1},...,x_{n}]^{\prime}$ is a continuous $K$-linear mapping.

###### Proof.

Assume that a sequence of copolynomials $\\{T_{k}\\}_{k=0}^{\infty}$ converges
to $T$ in $K[x_{1},...,x_{n}]^{\prime}$. Then there exists
$k_{0}=k_{0}(p)\in\mathbb{N}$ such that the equality $(T_{k},p)=(T,p)$ is true
for all $k\geq k_{0}(p)$. Let $m={\rm deg}p$ and
$s(p)=\max\\{k_{0}(D^{\alpha}p):|\alpha|\leq m\\}$. Then, by using (3.1), we
obtain

$({\mathcal{F}}T_{k},p)=\left(\sum_{|\alpha|\leq
m}a_{\alpha}D^{\alpha}T_{k},p\right)=\sum_{|\alpha|\leq
m}a_{\alpha}(-1)^{\alpha}(T_{k},D^{\alpha}p)$ $=\sum_{|\alpha|\leq
m}a_{\alpha}(-1)^{\alpha}(T,D^{\alpha}p)=\left(\sum_{|\alpha|\leq
m}a_{\alpha}D^{\alpha}T,p\right)=({\mathcal{F}}T,p),\quad k\geq s(p),$

i.e., the sequence $\\{{\mathcal{F}}T_{k}\\}_{k=0}^{\infty}$ converges to
${\mathcal{F}}T$. The lemma is proved. ∎

The following assertion shows that the convolution operation and the
differential operator of infinite order commute. We also show that every such
differential operator is a convolution operator.

######  Theorem 3.2.

Let $T_{1},T_{2}\in K[x_{1},...,x_{n}]^{\prime}$ and let
${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ be a
differential operator of infinite order on $K[x_{1},...,x_{n}]^{\prime}$ with
coefficients $a_{\alpha}\in K$. Then

${\mathcal{F}}(T_{1}*T_{2})=({\mathcal{F}}T_{1})*T_{2}.$

Therefore ${\mathcal{F}}(T)={\mathcal{F}}(\delta)*T$ for all $T\in
K[x_{1},...,x_{n}]^{\prime}$.

###### Proof.

Let $p\in K[x_{1},...,x_{n}]$, $m={\rm deg}p$ and let
$\beta\in\mathbb{N}_{0}^{n}$ be an arbitrary multi-index. By the definition of
the convolution

$(D^{\beta}(T_{1}*T_{2}),p)=(-1)^{|\beta|}(T_{1}*T_{2},D^{\beta}p)=(-1)^{|\beta|}\sum_{|\alpha|\leq
m}\left(T_{1},\frac{D^{\alpha+\beta}p}{\alpha!}\right)(T_{2},y^{\alpha})$
$=\sum_{|\alpha|\leq
m}\left(D^{\beta}T_{1},\frac{D^{\alpha}p}{\alpha!}\right)(T_{2},y^{\alpha})=((D^{\beta}T_{1})*T_{2},p).$

Therefore

$D^{\beta}(T_{1}*T_{2})=(D^{\beta}T_{1})*T_{2}.$ (3.2)

By (3.1), for every $T\in K[x_{1},...,x_{n}]^{\prime}$ the series
$\sum\limits_{|\beta|=0}^{\infty}a_{\beta}D^{\beta}T$ converges in the
topology of $K[x_{1},....,x_{n}]^{\prime}$. Therefore, by Corollary 2.15 and
the equality (3.2),

$({\mathcal{F}}(T_{1}*T_{2}))=\sum_{|\beta|=0}^{\infty}a_{\beta}D^{\beta}(T_{1}*T_{2})=\sum_{|\beta|=0}^{\infty}a_{\beta}((D^{\beta}T_{1})*T_{2})=({\mathcal{F}}T_{1})*T_{2}.$
(3.3)

Now, substituting $T_{1}=\delta$, $T_{2}=T$ into (3.3), we get
${\mathcal{F}}(T)={\mathcal{F}}(\delta)*T$. The theorem is proved. ∎

##  4  The Laplace transform in the module of copolynomials

Let $z=(z_{1},...,z_{n})$ and let
$K\left[\left[z_{1},...,z_{n},\frac{1}{z_{1}},...,\frac{1}{z_{n}}\right]\right]$
be the module of formal Laurent series with coefficients in $K$. For the
multi-index $\alpha=(\alpha_{1},...,\alpha_{n})\in\mathbb{Z}^{n}$ we put
$z^{\alpha}=z_{1}^{\alpha_{1}}z_{2}^{\alpha_{2}}\cdots z_{n}^{\alpha_{n}}$.
For $g\in
K\left[\left[z_{1},...,z_{n},\frac{1}{z_{1}},...,\frac{1}{z_{n}}\right]\right]$,
$g(z)=\sum\limits_{\alpha\in\mathbb{Z}^{n}}g_{\alpha}z^{\alpha}$ we naturally
define the formal residue:

$Res(g(z))=g_{(-1,...,-1)}.$

Now we define a Laplace transformation of a copolynomial $T\in
K[x_{1},...,x_{n}]^{\prime}$.

######  Definition 4.1.

Let $T\in K[x_{1},...,x_{n}]^{\prime}$. Assume that the ring $K$ contains the
field of rational numbers $\mathbb{Q}$. Consider the following formal power
series from $K[[z_{1},...,z_{n}]]$:

$L(T)(z)=\widetilde{T}(z)=\sum\limits_{|\alpha|=0}^{\infty}\frac{(T,x^{\alpha})}{\alpha!}z^{\alpha}.$

A power series $\widetilde{T}(z)$ will be called the Laplace transform of the
copolynomial $T$.

We may write informally as follows:: $\widetilde{T}(z)=(T,e^{<z,x>})$.
Obviously the mapping $L:K[x_{1},...,x_{n}]^{\prime}\to K[[z_{1},...,z_{n}]]$,
$L(T)=\widetilde{T}$ is a continuous isomorphism of $K$-modules, if we
consider the standart Krull topology on $K[[z_{1},...,z_{n}]]$ [12, Section 1,
§3, Section 4], i.e. the topology of coefficient-wise stabilization.

######  Proposition 4.2.

(the inversion formula or the Parseval identity). Let $K\supset\mathbb{Q}$,
$T\in K[x_{1},...,x_{n}]^{\prime}$ and $p(x)=\sum\limits_{|\alpha|\leq
m}c_{\alpha}x^{\alpha}\in K[x_{1},...,x_{n}]$. Then

$(T(x),p(x))=Res(\widetilde{T}(z)\widetilde{p}(z)),$

where $\widetilde{p}(z)=\sum\limits_{|\alpha|\leq
m}\frac{\alpha!c_{\alpha}}{z^{\alpha+\iota}}$ is the Laplace transform of the
polynomial $p(x)$.

###### Proof.

It is sufficient to consider the case $p(x)=x^{\beta}$ for some multi-index
$\beta\in\mathbb{N}_{0}^{n}$. We have
$\widetilde{p}(z)=\frac{\beta!}{z^{\beta+\iota}}$. Therefore

$\widetilde{T}(z)\widetilde{p}(z)=\sum_{|\alpha|=0}^{\infty}\frac{(T,x^{\alpha})}{\alpha!}z^{\alpha}\frac{\beta!}{z^{\beta+\iota}}$

and $Res(\widetilde{T}(z)\widetilde{p}(z))=(T,x^{\beta})$. ∎

The following theorem asserts that the Laplace transform sends the convolution
of copolynomials to the product of their Laplace transforms.

######  Theorem 4.3.

Let $T_{1},T_{2}\in K[x_{1},...,x_{n}]^{\prime}$. Assume that the ring $K$
contains the field of rational numbers $\mathbb{Q}$. Then

$(\widetilde{T_{1}*T_{2}})(z)=\widetilde{T_{1}}(z)\widetilde{T_{2}}(z).$

###### Proof.

Since for any $\alpha\in\mathbb{N}_{0}^{n}$

$(T_{1}*T_{2},x^{\alpha})=\sum_{|\beta|\leq|\alpha|}\left(T_{1}(x),\frac{D^{\beta}x^{\alpha}}{\beta!}\right)(T_{2}(y),y^{\beta})$
$=\sum_{\beta\leq\alpha}\left(T_{1}(x),\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)x^{\alpha-\beta}\right)(T_{2}(y),y^{\beta}),$

we have

$(\widetilde{T_{1}*T_{2}})(z)=\sum\limits_{|\alpha|=0}^{\infty}\frac{(T_{1}*T_{2},x^{\alpha})}{\alpha!}z^{\alpha}$
$=\sum_{|\alpha|=0}^{\infty}\sum_{\beta\leq\alpha}\left(T_{1}(x),\frac{\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)}{\alpha!}x^{\alpha-\beta}\right)(T_{2}(y),y^{\beta})z^{\alpha}$
$=\sum_{|\alpha|=0}^{\infty}\sum_{\beta\leq\alpha}\left(T_{1}(x),\frac{x^{\alpha-\beta}}{(\alpha-\beta)!}\right)\left(T_{2}(y),\frac{y^{\beta}}{\beta!}\right)z^{\alpha}=\widetilde{T_{1}}(z)\widetilde{T_{2}}(z).$

∎

######  Example 4.4.

Suppose that $n=1$, $K$ is an arbitrary commutative integral domain, $a\in K$
and

$({\mathcal{E}}_{a},p)=\sum_{k=0}^{m}a^{k}p^{(k)}(0),$

where $p\in K[x]$, $m={\rm deg}p$. Then ${\mathcal{E}}_{a}\in K[x]^{\prime}$
and

${\mathcal{E}}_{a}=\sum_{j=0}^{\infty}{(-1)}^{j}a^{j}\delta^{(j)}.$

If $K=\mathbb{R}$ and $a>0$, then

$({\mathcal{E}}_{a},p)=\int\limits_{-\infty}^{\infty}p(x)f_{a}(x)dx,$

where

$f_{a}(x)=\left\\{{{\begin{array}[]{*{20}c}\frac{1}{a}e^{-\frac{x}{a}},\quad
x>0\\\ 0,\quad x<0.\\\ \end{array}}}\right.$

(see Example 4 in [8]). Let $K$ be again a commutative integral domain such
that $K\supset\mathbb{Q}$. If $a,b\in K$, then
$\widetilde{{\mathcal{E}}_{a}}(z)=\sum\limits_{j=0}^{\infty}a^{j}z^{j}$ and
$(a-b)\widetilde{{\mathcal{E}}_{a}}(z)\widetilde{{\mathcal{E}}_{b}}(z)=a\widetilde{{\mathcal{E}}_{a}}(z)-b\widetilde{{\mathcal{E}}_{b}}(z)$.
With the help of Theorem 4.3 we obtain
$L((a-b)({\mathcal{E}}_{a}*{\mathcal{E}}_{b}))=\widetilde{a{\mathcal{E}}_{a}-b{\mathcal{E}}_{b}}$
and

$(a-b)({\mathcal{E}}_{a}*{\mathcal{E}}_{b})=a{\mathcal{E}}_{a}-b{\mathcal{E}}_{b}.$
(4.1)

Now the convolution equation (4.1) can be checked for an arbitrary commutative
integral domain $K$ (see also Example 1.1 in [5], where it was considered the
similar equation for formal Laurent series). Indeed by Theorem 3.2 and
Corollary 2.15 we obtain

$(a-b)({\mathcal{E}}_{a}*{\mathcal{E}_{b}})=(a-b)\left(\sum_{j=0}^{\infty}{(-1)}^{j}a^{j}\delta^{(j)}\right)*\left(\sum_{k=0}^{\infty}{(-1)}^{k}b^{k}\delta^{(k)}\right)$
$=(a-b)\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}(-1)^{j+k}a^{j}b^{k}(\delta^{(j)}*\delta^{(k)})=(a-b)\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}(-1)^{j+k}a^{j}b^{k}\delta^{(j+k)}$
$=(a-b)\sum_{j=0}^{\infty}\sum_{l=j}^{\infty}(-1)^{l}a^{j}b^{l-j}\delta^{(l)}=(a-b)\sum_{l=0}^{\infty}(-1)^{l}\sum_{j=0}^{l}a^{j}b^{l-j}\delta^{(l)}$
$=\sum_{l=0}^{\infty}(-1)^{l}(a^{l+1}-b^{l+1})\delta^{(l)}=a{\mathcal{E}}_{a}-b{\mathcal{E}}_{b}.$

Now we established a connection between Laplace transform of the copolynomial
${\mathcal{F}}T=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}T$, where
$T\in K[x_{1},...,x_{n}]^{\prime}$, and the symbol
$\varphi(z)=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}z^{\alpha}$ of the
differential operator ${\mathcal{F}}$.

######  Proposition 4.5.

Let $K\supset\mathbb{Q}$ and let
${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ be a
linear differential operator of infinite order on
$K[x_{1},...,x_{n}]^{\prime}$ with coefficients $a_{\alpha}\in K$. Then for
every $T\in K[x_{1},...,x_{n}]^{\prime}$ the equality

$\widetilde{{\mathcal{F}}T}(z)=\varphi(-z)\widetilde{T}(z)$ (4.2)

holds.

###### Proof.

By the definition of a Laplace transform for any multi-index
$\alpha\in\mathbb{N}_{0}^{n}$ we have

$\widetilde{D^{\alpha}T}(z)=\sum_{|\beta|=0}^{\infty}\frac{(D^{\alpha}T,x^{\beta})}{\beta!}z^{\beta}=\sum_{\beta\geq\alpha}(-1)^{|\alpha|}\frac{(T,x^{\beta-\alpha})}{(\beta-\alpha)!}z^{\beta}$
$=(-z)^{\alpha}\sum_{|\beta|=0}^{\infty}\frac{(T,x^{\beta})}{{\beta}!}z^{\beta}=(-z)^{\alpha}\widetilde{T}(z).$

Multiplying this equality by $a_{\alpha}$ and summing on all
$\alpha\in\mathbb{N}_{0}^{n}$, we obtain Equality (4.2). ∎

##  5 Fundamental solution of a linear differential operator of infinite order

Let $T\in K[x_{1},...,x_{n}]^{\prime}$ be a copolynomial and let
${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ be a
linear differential operator of infinite order on
$K[x_{1},...,x_{n}]^{\prime}$ with coefficients $a_{\alpha}\in K$. Consider
the following differential equation

${\mathcal{F}}u=T.$ (5.1)

We prove an existence and uniqueness theorem for Equation (5.1) and continuous
dependence for the unique solution of this equation on $T$.

######  Theorem 5.1.

Let $a_{0}$ be an invertible element of the ring $K$. Then the linear
differential operator ${\mathcal{F}}$ of infinite order is bijective and its
inverse operator ${\mathcal{F}}^{-1}$ is a continuous mapping. Moreover, if
$I$ denotes the identity mapping, then

${\mathcal{F}}^{-1}=a_{0}^{-1}\sum_{k=0}^{\infty}(I-a_{0}^{-1}{\mathcal{F}})^{k},$
(5.2)

where the series in the right-hand side of (5.2) converges in the topology of
$K[x_{1},...,x_{n}]^{\prime}$. In particular, for any copolynomial $T\in
K[x_{1},...,x_{n}]^{\prime}$ there exists a unique solution $u\in
K[x_{1},...,x_{n}]^{\prime}$ of Equation (5.1). This solution admits
representations

$u={\mathcal{F}}^{-1}(T)={\mathcal{F}}^{-1}(\delta)*T$

and continuously depends on $T$ in the topology of
$K[x_{1},...,x_{n}]^{\prime}$.

###### Proof.

We have the following representation of the operator ${\mathcal{F}}$:

${\mathcal{F}}=a_{0}\left(I-\sum_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right),$ (5.3)

where ${\mathcal{G}}_{j}\ (j=1,...,n)$ are some linear differential operators.
For every $k\in\mathbb{N}$ we have

$\left(\sum_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{k}=\sum_{|\alpha|=k}\frac{k!}{\alpha!}D^{\alpha}{\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}.$

Now for every copolynomial $T\in K[x_{1},...,x_{n}]^{\prime}$ and polynomial
$p\in K[x_{1},...,x_{n}]$ of degree $m$ we have

$\sum_{k=0}^{\infty}\left(\left(\sum_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{k}T,p\right)=\left(\sum_{k=0}^{\infty}\sum_{|\alpha|=k}\frac{k!}{\alpha!}D^{\alpha}{\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}T,p\right)$
$=\sum_{|\alpha|=0}^{\infty}\left(\frac{|\alpha|!}{\alpha!}D^{\alpha}{\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}T,p\right)=\sum_{|\alpha|\leq
m}(-1)^{|\alpha|}\left(\frac{|\alpha|!}{\alpha!}{\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}T,D^{\alpha}p\right).$

Therefore the series
$\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{k}T$ converges for any copolynomial $T\in
K[x_{1},...,x_{n}]^{\prime}$, the operator
$I-\sum\limits_{j=1}^{n}\frac{\partial}{\partial x_{j}}{\mathcal{G}}_{j}$ is
bijective and its inverse operator has the form:

$\left(I-\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{-1}=\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{k}.$

Now (5.3) implies the bijectivity of the operator ${\mathcal{F}}$ and

${\mathcal{F}}^{-1}=a_{0}^{-1}\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{k}=a_{0}^{-1}\sum_{k=0}^{\infty}(I-a_{0}^{-1}{\mathcal{F}})^{k}.$
(5.4)

Therefore the representation (5.2) is true for the inverse operator
${\mathcal{F}}^{-1}$. Hence, the differential equation (5.1) has a unique
solution $u\in K[x_{1},...,x_{n}]^{\prime}$ and, moreover,
$u={\mathcal{F}}^{-1}T$. By the equality (5.4) and Corollary 2.15 we obtain
the following representation for this solution:

$u={\mathcal{F}}^{-1}T=a_{0}^{-1}\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{k}T$
$=a_{0}^{-1}\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{k}(\delta*T)=\left(a_{0}^{-1}\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j}\right)^{k}\delta\right)*T={\mathcal{F}}^{-1}(\delta)*T$

(see also Example 2.12).

Now the continuity of the operator ${\mathcal{F}}^{-1}$ follows from the
continuity of the convolution (see Theorem 2.14). The theorem is proved. ∎

######  Corollary 5.2.

Let $a_{0}$ be an invertible element of the ring $K$. Then the differential
equation ${\mathcal{F}}u=\delta$ has the unique solution

${\mathcal{E}}={\mathcal{F}}^{-1}\delta.$ (5.5)

######  Definition 5.3.

The copolynomial defined by the formula (5.5) is called the fundamental
solution of the linear differential operator ${\mathcal{F}}$.

Theorem 5.1 implies the following assertion.

######  Corollary 5.4.

Let $a_{0}$ be an invertible element of the ring $K$. Then the unique solution
of Equation (5.1) is the convolution of the fundamental solution
${\mathcal{E}}$ and the copolynomial $T$: $\ u={\mathcal{E}}*T$.

######  Remark 5.5.

In all previous results it was supposed that $a_{0}$ is an invertible element
of the ring $K$. We note that this condition is necessary for the existence of
the fundamental solution of the differential operator ${\mathcal{F}}$. Indeed,
if ${\mathcal{E}}$ is the fundamental solution of this operator, then applying
the left-hand and right-hand sides of Equation
${\mathcal{F}}{\mathcal{E}}=\delta$ to 1, we get that
$a_{0}({\mathcal{E}},1)=1$, i.e. $a_{0}$ is an invertible element of the ring
$K$.

######  Remark 5.6.

Let $a_{0}$ be an invertible element of $K$. Then the differential operator
${\mathcal{F}}:K[x_{1},...,x_{n}]\to K[x_{1},...,x_{n}]$ is bijective and for
any polynomial $p\in K[x_{1},...,x_{n}]$ the differential equation

${\mathcal{F}u}=p$

has a unique solution $u\in K[x_{1},...,x_{n}]$, moreover ${\rm deg}u\leq{\rm
deg}p$. Furthermore, this solution is a convolution of the fundamental
solution ${\mathcal{E}}$ of the differential operator ${\mathcal{F}}$ and the
polynomial $p$: $\ u={\mathcal{E}}*p$. The proof of this assertion is similar
to the proof of Theorem 5.1.

######  Example 5.7.

Let $n=1$ and $a\in K$. Consider the linear differential operator
${\mathcal{F}}=a\frac{d}{dx}+I$ on $K[x]^{\prime}$. By Corollary 5.2 the
operator ${\mathcal{F}}$ has the fundamental solution

${\mathcal{E}}_{a}={\mathcal{F}}^{-1}\delta=\left(I+a\frac{d}{dx}\right)^{-1}\delta=\sum\limits_{j=0}^{\infty}{(-1)}^{j}a^{j}\delta^{(j)},$

i.e. $a\frac{d{\mathcal{E}}_{a}}{dx}+{\mathcal{E}}_{a}=\delta$ (see also
Example 4.4). If $K=\mathbb{R}$ and $a>0$, then we obtain that the fundamental
solution ${\mathcal{E}}_{a}$ regarded as a regular copolynomial coincides with
the classical fundamental solution $\frac{1}{a}\ \theta(x)e^{-x/a}$ of the
differential operator ${\mathcal{F}}$, where $\theta(x)$ is the Heaviside
function (see also Examples 4 and 5 in [8]).

######  Example 5.8.

The linear differential operator of infinite order
${\mathcal{F}}=\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}\right)^{k}$ has the inverse operator
$I-\sum\limits_{j=1}^{n}\frac{\partial}{\partial x_{j}}$. Therefore
${\mathcal{E}}=\delta-\sum\limits_{j=1}^{n}\frac{\partial\delta}{\partial
x_{j}}$ is the fundamental solution of the operator ${\mathcal{F}}$. By
Theorem 5.1 for every copolynomial $T\in K[x_{1},...,x_{n}]^{\prime}$ the
differential equation of infinite order

$\sum_{k=0}^{\infty}\left(\sum\limits_{j=1}^{n}\frac{\partial}{\partial
x_{j}}\right)^{k}u=T$

has a unique solution
$u={\mathcal{F}}^{-1}T=T-\sum\limits_{j=1}^{n}\frac{\partial T}{\partial
x_{j}}$.

######  Example 5.9.

Let $c$ be an invertible element of the ring $K$. We consider in the module
$K[x_{1},x_{2},x_{3}]^{\prime}$ the Helmholtz equation

$\triangle{\mathcal{E}}+c{\mathcal{E}}=\delta,$ (5.6)

where $\triangle=\frac{\partial^{2}}{\partial
x_{1}^{2}}+\frac{\partial^{2}}{\partial
x_{2}^{2}}+\frac{\partial^{2}}{\partial x_{3}^{2}}$ is the Laplace operator.
By Theorem 5.1, Equation (5.6) has the unique solution (see Formula (5.2)):

${\mathcal{E}}=(cI+\triangle)^{-1}\delta=\sum_{k=0}^{\infty}(-1)^{k}c^{-k-1}\triangle^{k}\delta.$
(5.7)

For any $\beta=(\beta_{1},\beta_{2},\beta_{3})\in\mathbb{N}_{0}^{3}$ and
$k\in\mathbb{N}_{0}$ we have:

$\triangle^{k}x^{\beta}=\sum_{|\alpha|=k}\frac{k!}{\alpha!}D^{2\alpha}x^{\beta},\quad
x^{\beta}=x_{1}^{\beta_{1}}x_{2}^{\beta_{2}}x_{3}^{\beta_{3}}.$

Therefore

$(\triangle^{|\alpha|}\delta,x^{\beta})=(\delta,\triangle^{|\alpha|}x^{\beta})=\left\\{\begin{array}[]{ccc}\frac{|\alpha|!(2\alpha)!}{\alpha!},\quad\beta=2\alpha,\\\
0,\quad\beta\not=2\alpha.\end{array}\right.$

Substituting this expression into the formula (5.7), we obtain

$({\mathcal{E}},x^{\beta})=\left\\{\begin{array}[]{ccc}(-1)^{|\alpha|}\frac{|\alpha|!(2\alpha)!}{\alpha!c^{|\alpha|+1}},\quad\beta=2\alpha,\\\
0,\quad\beta\not=2\alpha.\end{array}\right.$

This formula gives the fundamental solution of the Helmholtz operator
$\triangle+cI$. In the case $K=\mathbb{R}$ and $c>0$ this solution is
connected with classical fundamental solutions $-\frac{1}{4\pi}\frac{e^{\pm
i\sqrt{c}|x|}}{|x|},\ (|x|=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}})$ of the
Helmholtz operator by equalities

$({\mathcal{E}},x^{\beta})=\lim\limits_{b\to+0}\int\limits_{\mathbb{R}^{3}}e^{-b|x|}\left(-\frac{\cos(\sqrt{c}|x|)}{4\pi|x|}\right)x^{\beta}dx,\quad\beta\in\mathbb{N}_{0}^{3},$
(5.8)

where the integral in the right-hand side (5.8) is calculated with the help of
the converting to the spherical coordinates.

######  Example 5.10.

Let $a,c\in K$ and let $c$ be an invertible element of the ring $K$. We find
the fundamental solution of the linear differential operator
${\mathcal{F}}=\frac{\partial}{\partial t}-a\frac{\partial^{2}}{\partial
x^{2}}+cI$. We have

${\mathcal{F}}=c\left(I-\left(ac^{-1}\frac{\partial^{2}}{\partial
x^{2}}-c^{-1}\frac{\partial}{\partial t}\right)\right).$

Therefore taking into account (5.2) and (5.5), we obtain the following
expression for the fundamental solution of the operator ${\mathcal{F}}$:

${\mathcal{E}}={\mathcal{F}}^{-1}\delta=\sum_{k=0}^{\infty}c^{-k-1}\left(a\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial}{\partial t}\right)^{k}\delta$
$=\sum_{k=0}^{\infty}c^{-k-1}\sum_{j=0}^{k}\left(\begin{array}[]{c}k\\\
j\end{array}\right)(-1)^{j}a^{k-j}\frac{\partial^{2k-j}\delta}{\partial
t^{j}\partial x^{2k-2j}}.$

This implies that for every $s,l\in\mathbb{N}_{0}$

$({\mathcal{E}},x^{m}t^{l})=\left\\{\begin{array}[]{ccc}\frac{(2s)!(l+s)!}{s!}a^{s}c^{-l-s-1},\quad
m=2s,\\\ 0,\quad m=2s+1.\end{array}\right.$

This result also can be obtained with the help of the Laplace transform.
Assume that the ring $K$ contains the field of rational numbers $\mathbb{Q}$.
We apply the Laplace transform to both sides of Equation

$\frac{\partial{\mathcal{E}}}{\partial
t}-a\frac{\partial^{2}{\mathcal{E}}}{\partial
x^{2}}+c{\mathcal{E}}=\delta(t,x).$

By Proposition 4.5 we obtain

$(c-az_{2}^{2}-z_{1})\widetilde{{\mathcal{E}}}(z_{1},z_{2})=1.$

Since $c$ is an invertible element of the ring $K$, the polynomial
$c-az_{2}^{2}-z_{1}$ is an invertible element of the ring $K[[z_{1},z_{2}]]$.
Then

$\widetilde{{\mathcal{E}}}(z_{1},z_{2})=\frac{1}{c-az_{2}^{2}-z_{1}}=\sum_{k=0}^{\infty}c^{-k-1}\sum_{j=0}^{k}\left(\begin{array}[]{c}k\\\
j\end{array}\right)a^{k-j}z_{1}^{j}z_{2}^{2k-2j}.$

Let $p(t,x)=x^{m}t^{l}$. Then
$\widetilde{p}(z_{1},z_{2})=\frac{m!l!}{z_{1}^{l+1}z_{2}^{m+1}}$ and

$\widetilde{{\mathcal{E}}}(z_{1},z_{2})\widetilde{p}(z_{1},z_{2})=\sum_{k=0}^{\infty}c^{-k-1}\sum_{j=0}^{k}\left(\begin{array}[]{c}k\\\
j\end{array}\right)a^{k-j}m!l!z_{1}^{j-l-1}z_{2}^{2k-2j-m-1}.$

Now by Proposition 4.2, we obtain

$({\mathcal{E}}(t,x),p(t,x))=Res(\widetilde{{\mathcal{E}}}(z_{1},z_{2})\widetilde{p}(z_{1},z_{2}))=\left\\{\begin{array}[]{ccc}\frac{(2s)!(l+s)!}{s!}a^{s}c^{-l-s-1},\quad
m=2s,\\\ 0,\quad m=2s+1.\end{array}\right.$

Now let $K=\mathbb{R}$, $a>0$ and $c>0$. Note that

$\int\limits_{0}^{\infty}dt\int\limits_{-\infty}^{\infty}t^{l}e^{-ct}x^{m}\frac{e^{-\frac{x^{2}}{4at}}}{\sqrt{4\pi
at}}dx=\left\\{\begin{array}[]{ccc}\frac{(2s)!(l+s)!}{s!}a^{s}c^{-l-s-1},\quad
m=2s,\\\ 0,\quad m=2s+1.\end{array}\right.$

Therefore in the space $\mathbb{R}[t,x]^{\prime}$ the fundamental solution of
the differential operator ${\mathcal{F}}$, regarded as a regular copolynomial,
coincide with the function $\frac{\theta(t)e^{-ct}}{\sqrt{4\pi
at}}e^{-\frac{x^{2}}{4at}}$.

######  Example 5.11.

We find the fundamental solution of the differential operator
${\mathcal{F}}=\frac{\partial^{2}}{\partial x\partial
t}+\frac{\partial}{\partial x}-\frac{\partial}{\partial t}-I$. By Definition
5.3 it is a solution of the differential equation

$\frac{\partial^{2}{\mathcal{E}}}{\partial x\partial
t}+\frac{\partial{\mathcal{E}}}{\partial
x}-\frac{\partial{\mathcal{E}}}{\partial t}-{\mathcal{E}}=\delta.$

Then the sequence $C_{sl}=({\mathcal{E}},x^{s}t^{l})\ (l,s\in\mathbb{N}_{0})$
is a solution of the following problem for the difference equation

$C_{sl}=slC_{s-1,l-1}-sC_{s-1,l}+lC_{s,l-1},\quad s,l\in\mathbb{N};$ $\quad
C_{s0}=(-1)^{s+1}s!,\quad C_{0l}=-l!,\quad s,l\in\mathbb{N}_{0}.$

This problem has a unique solution

$C_{sl}=(-1)^{s+1}l!s!,\quad s,l=0,1,2,....$

We note that

$-\int_{0}^{\infty}dt\int_{-\infty}^{0}e^{-t+x}t^{l}x^{s}dx=(-1)^{s+1}l!s!.$

Therefore, in the space $\mathbb{R}[t,x]^{\prime}$ the fundamental solution of
the differential operator ${\mathcal{F}}$, regarded as a regular copolynomial,
coincide with the function $-\theta(t)\theta(-x)e^{x-t}$.

######  Example 5.12.

We find the fundamental solution of the transport operator
${\mathcal{F}}=\frac{\partial}{\partial
t}+\sum\limits_{i=1}^{n}s_{i}\frac{\partial}{\partial x_{i}}+I$, where
$s_{i}\in K$. Therefore, taking into account formulas (5.2) and (5.5) we
obtain the expression for the fundamental solution of the differential
operator ${\mathcal{F}}$:

${\mathcal{E}}(t,x)=({\mathcal{F}}^{-1}\delta)(t,x)=\sum_{k=0}^{\infty}(-1)^{k}\left(\frac{\partial}{\partial
t}+\sum_{i=1}^{n}s_{i}\frac{\partial}{\partial x_{i}}\right)^{k}\delta(t,x)$
$=\sum_{k=0}^{\infty}(-1)^{k}\sum_{j=0}^{k}\left(\begin{array}[]{c}k\\\
j\end{array}\right)\frac{\partial^{j}}{\partial
t^{j}}\left(\sum_{i=1}^{n}s_{i}\frac{\partial}{\partial
x_{i}}\right)^{k-j}\delta(t,x)=$
$=\sum_{k=0}^{\infty}(-1)^{k}\sum_{j=0}^{k}\left(\begin{array}[]{c}k\\\
j\end{array}\right)\frac{\partial^{j}}{\partial
t^{j}}\sum_{|\alpha|=k-j}\frac{|\alpha|!}{\alpha!}s^{\alpha}D^{\alpha}\delta(t,x),\quad
s=(s_{1},...,s_{n}).$

Then, for every $l\in\mathbb{N}_{0}$ and $\beta\in\mathbb{N}_{0}^{n}$ we have:

$({\mathcal{E}},t^{l}x^{\beta})=\sum_{k=0}^{\infty}(-1)^{k}\sum_{j=0}^{k}\left(\begin{array}[]{c}k\\\
j\end{array}\right)\sum_{|\alpha|=k-j}\frac{|\alpha|!}{\alpha!}s^{\alpha}(-1)^{|\alpha|+j}\frac{\partial^{j}}{\partial
t^{j}}D^{\alpha}(t^{l}x^{\beta})|_{t=0,x=0}$ $=s^{\beta}(|\beta|+l)!.$

Now let $K=\mathbb{R}$. Note that

$\int_{0}^{\infty}e^{-t}t^{l}(\delta(x-ts),x^{\beta})dt=s^{\beta}(|\beta|+l)!,\quad
l\in\mathbb{N}_{0},\ \beta\in\mathbb{N}_{0}^{n},\ s\in\mathbb{R}^{n}.$

Thus a connection between the fundamental solution of the transport operator
and the classical fundamental solution $\theta(t)e^{-t}\delta(x-ts)$ of this
operator is established:

$({\mathcal{E}},t^{l}x^{\beta})=\int_{0}^{\infty}e^{-t}t^{l}(\delta(x-ts),x^{\beta})dt,\quad
l\in\mathbb{N}_{0},\beta\in\mathbb{N}_{0}^{n}.$

######  Example 5.13.

Now we consider the $m$-th order ordinary linear differential equation in the
module $K[x]^{\prime}$ of copolynomials of one variable

$\sum_{j=0}^{m}a_{j}\frac{d^{j}u}{dx^{j}}=T,$ (5.9)

where $a_{j}\in K,j=0,...,m$, $a_{0}\not=0,\ a_{m}\not=0$ and $T,u\in
K[x]^{\prime}$ are known and unknown copolynomials of one variable
respectively. This equation is a particular case of Equation (5.1) with the
differential operator
${\mathcal{F}}=\sum\limits_{j=0}^{m}a_{j}\frac{d^{j}}{dx^{j}}$. Assume that
$a_{0}$ is an invertible element of the ring $K$. Then by Theorem 5.1,
Equation (5.9) has a unique solution. This solution has the form

$u=a_{0}^{-1}\sum_{k=0}^{\infty}(I-a_{0}^{-1}{\mathcal{F}})^{k}T.$ (5.10)

Now for every $k\in\mathbb{N}_{0}$ we have

$(I-a_{0}^{-1}{\mathcal{F}})^{k}=(-1)^{k}\frac{d^{k}}{dx^{k}}\left(\sum_{j=1}^{m}a_{0}^{-1}a_{j}\frac{d^{j-1}}{dx^{j-1}}\right)^{k}$
$=(-1)^{k}\sum_{|\gamma|=k}\frac{k!}{\gamma!}a_{0}^{-k}a_{1}^{\gamma_{1}}\cdot\ldots\cdot
a_{m}^{\gamma_{m}}\frac{d^{k+\sum\limits_{j=1}^{m}(j-1)\gamma_{j}}}{dx^{k+\sum\limits_{j=1}^{m}(j-1)\gamma_{j}}}.$

Substituting this expression into (5.10), we obtain the following
representation for the unique solution of Equation (5.9):

$u(x)=\sum_{k=0}^{\infty}\sum_{|\gamma|=k}(-1)^{k}\frac{k!}{\gamma!}a_{0}^{-k-1}a_{1}^{\gamma_{1}}\cdot\ldots\cdot
a_{m}^{\gamma_{m}}T^{\left(k+\sum\limits_{j=1}^{m}(j-1)\gamma_{j}\right)}(x).$

(see [13, Section 4], where a similar formula has been obtained in the other
situation). In particular, the first order equation
$a_{1}u^{\prime}(x)+a_{0}u(x)=T(x)$ has the unique solution

$u(x)=\sum_{k=0}^{\infty}(-1)^{k}a_{0}^{-k-1}a_{1}^{k}T^{(k)}(x),$

and the second order equation
$a_{2}u^{\prime\prime}(x)+a_{1}u^{\prime}(x)+a_{0}u(x)=T(x)$ has the unique
solution

$u(x)=\sum_{k=0}^{\infty}\sum_{j=0}^{k}(-1)^{k}\left(\begin{array}[]{c}k\\\
j\end{array}\right)a_{0}^{-k-1}a_{1}^{k-j}a_{2}^{j}T^{(k+j)}(x)$
$=\sum_{j=0}^{\infty}\sum_{k=j}^{\infty}(-1)^{k}\left(\begin{array}[]{c}k\\\
j\end{array}\right)a_{0}^{-k-1}a_{1}^{k-j}a_{2}^{j}T^{(k+j)}(x)$
$=\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}(-1)^{k+j}\left(\begin{array}[]{c}k+j\\\
j\end{array}\right)a_{0}^{-j-k-1}a_{1}^{k}a_{2}^{j}T^{(k+2j)}(x)$
$=\sum_{s=0}^{\infty}\left(\sum_{j=0}^{\left[\frac{s}{2}\right]}(-1)^{s-j}\left(\begin{array}[]{c}s-j\\\
j\end{array}\right)a_{0}^{j-s-1}a_{1}^{s-2j}a_{2}^{j}\right)T^{(s)}(x)$

(see Formula (4.10) in [7]).

##  6 Fundamental solution of the Cauchy problem for a linear differential
equation in the module of copolynomials

###  6.1 Formal power series over the module of copolynomials.

The module of formal power series of the form
$u(t,x)=\sum\limits_{k=0}^{\infty}u_{k}(x)t^{k}$ with coefficients
$u_{k}(x)\in K[x_{1},...,x_{n}]^{\prime}$ will be denoted by
$K[x_{1},...,x_{n}]^{\prime}[[t]]$.

The partial derivative with respect to $t$ of the series $u(t,x)\in
K[x_{1},....,x_{n}]^{\prime}[[t]]$ is defined by the formula

$\frac{\partial u}{\partial t}=\sum_{k=1}^{\infty}ku_{k}(x)t^{k-1}.$

The partial derivatives $D^{\alpha}$ with respect to variables
$x_{1},...,x_{n}$ of the series $u(t,x)\in K[x_{1},...,x_{n}]^{\prime}[[t]]$
is defined as follows:

$D^{\alpha}u(t,x)=\sum_{k=0}^{\infty}(D^{\alpha}u_{k})(x)t^{k}.$

The action of the $K$-linear operator
${\mathcal{A}}:K[x_{1},....,x_{n}]^{\prime}\to K[x_{1},....,x_{n}]^{\prime}$
on a formal power series $u(t,x)=\sum\limits_{k=0}^{\infty}u_{k}(x)t^{k}\in
K[x_{1},...,x_{n}]^{\prime}[[t]]$ is defined coefficient-wisely:

$({\mathcal{A}}u)(t,x)=\sum_{k=0}^{\infty}({\mathcal{A}}u_{k})(x)t^{k}.$

Obviously, that if ${\mathcal{A}}$ is an invertible $K$-linear operator on the
module $K[x_{1},....,x_{n}]^{\prime}$, then its extension on the module
$K[x_{1},...,x_{n}]^{\prime}[[t]]$ is also invertible.

We denote by $(u(t,x),p(x))$ the action of $u(t,x)\in
K[x_{1},...,x_{n}]^{\prime}[[t]]$ on $p(x)\in K[x_{1},...,x_{n}]$, which is
defined coefficient-wisely:

$(u(t,x),p(x))=\sum\limits_{k=0}^{\infty}(u_{k}(x),p(x))t^{k}.$

Thus, $(u(t,x),p(x))\in K[[t]]$.

######  Definition 6.1.

Let $u(t,x)=\sum\limits_{k=0}^{\infty}u_{k}(x)t^{k}\in
K[x_{1},...,x_{n}]^{\prime}[[t]]$. The convolution  of a copolynomial $T\in
K[x_{1},...,x_{n}]^{\prime}$ and a formal power series $u(t,x)$ is also
defined coefficient-wisely:

$(T*u)(t,x)=\sum_{k=0}^{\infty}(T*u_{k}(x))t^{k},$

Thus, $(T*u)(t,x)\in K[x_{1},...,x_{n}]^{\prime}[[t]]$.

###  6.2 The Cauchy problem for a linear partial differential equation in the
module of copolynomials.

Let ${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ be a
linear differential equation of infinite order on
$K[x_{1},...,x_{n}]^{\prime}$ with coefficients $a_{\alpha}\in K$. In the
module $K[x_{1},...,x_{n}]^{\prime}[[t]]$, we consider the Cauchy problem

$\frac{\partial u(t,x)}{\partial t}=({\mathcal{F}}u)(t,x),$ (6.1)
$u(0,x)=Q(x)\in K[x_{1},...,x_{n}]^{\prime}.$ (6.2)

The following example shows that if $a_{0}$ is invertible, then this Cauchy
problem may has no solutions.

######  Example 6.2.

Let $K=\mathbb{Z}$, ${\mathcal{F}}=I$ and $Q(x)=\delta(x)$. Then the Cauchy
problem (6.1), (6.2) is written in the form:

$\frac{\partial u(t,x)}{\partial t}=u(t,x),$ (6.3) $u(0,x)=\delta(x).$ (6.4)

Any solution of this problem can be represented in the form of a formal power
series $u(t,x)=\sum\limits_{k=0}^{\infty}u_{k}(x)t^{k}$ with coefficients
$u_{k}(x)\in\mathbb{Z}[x_{1},...,x_{n}]^{\prime}$. Substituting this
representation into (6.3), (6.4), we get

$u_{0}(x)=\delta(x),\quad(k+1)u_{k+1}(x)=u_{k}(x),\quad k=0,1,2,....$

This implies $2(u_{2},1)=1$, which contradicts the condition
$(u_{2},1)\in\mathbb{Z}$.

The following theorem shows that in the case $a_{0}=0$ the Cauchy problem
(6.1), (6.2) has a unique solution.

######  Theorem 6.3.

Let $a_{0}=0$ and let the ring $K$ be of characteristic 0. Then for any
copolynomial $Q\in K[x_{1},...,x_{n}]^{\prime}$ the formal power series

$u(t,x)=\sum_{k=0}^{\infty}\frac{({\mathcal{F}}^{k}Q)(x)}{k!}t^{k}$ (6.5)

is well defined and is a unique solution of the Cauchy problem (6.1), (6.2).
Furthermore, for every $t\in K$ the series (6.5) converges in the topology of
the module $K[x_{1},...,x_{n}]^{\prime}$.

###### Proof.

We shows first that $K[x_{1},...,x_{n}]^{\prime}$ is a torsion-free
$\mathbb{Z}$-module [1, Section VII, §2]. Suppose that an element $T\in
K[x_{1},...,x_{n}]^{\prime}$ satisfies the equality $kT=0$ for some natural
$k$. Then $(kT,p)=k(T,p)=0$ for every polynomial $p\in K[x_{1},...,x_{n}]$.
Since the integral domain $K$ is of characteristic 0, we have $(T,p)=0$, i.e.
$T=0$.

Now we prove that the formal power series (6.5) is well defined and is a
unique solution of the Cauchy problem (6.1), (6.2). Since $a_{0}=0$, we obtain
the following representation for the operator ${\mathcal{F}}$:

${\mathcal{F}}=\sum_{j=1}^{n}\frac{\partial}{\partial
x_{j}}{\mathcal{G}}_{j},$

where ${\mathcal{G}}_{j}\ (j=1,...,n)$ are some differential
operators.Therefore

${\mathcal{F}}^{k}=k!\sum_{|\alpha|=k}\frac{D^{\alpha}{\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}}{\alpha!},\quad
k\in\mathbb{N}.$

Since copolynomials
$\frac{D^{\alpha}{\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}Q}{\alpha!}$
are well-defined (see (2.2)), the element ${\mathcal{F}}^{k}Q$ in the module
$K[x_{1},...,x_{n}]^{\prime}$ is divided by $k!$. Since the module
$K[x_{1},...,x_{n}]^{\prime}$ is a torsion-free $\mathbb{Z}$-module, we get

$\frac{{\mathcal{F}}^{k}Q}{k!}=\sum_{|\alpha|=k}\frac{D^{\alpha}{\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}Q}{\alpha!},\quad
k\in\mathbb{N}.$ (6.6)

By Theorem 2.3 [6] the series (6.5) is well-defined, the Cauchy problem (6.1),
(6.2) has a unique solution and this solution has the form (6.5). Now we show
that for any $t\in K$ the series (6.5) converges in
$K[x_{1},...,x_{n}]^{\prime}$. We consider the partial sums
$u_{N}(t,x)=\sum\limits_{k=0}^{N}\frac{({\mathcal{F}}^{k}Q)(x)}{k!}t^{k}$ of
this series and show that they are stabilized on every polynomial. By
equalities (2.2) and (6.6) for any $p\in K[x_{1},...,x_{n}]$ we have

$(u_{N}(t,x),p(x))=\sum_{k=0}^{N}\sum_{|\alpha|=k}\left(\frac{D^{\alpha}{\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}Q}{\alpha!},p\right)t^{k}$
$=\sum_{k=0}^{N}\sum_{|\alpha|=k}(-1)^{|\alpha|}\left({\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}Q,\frac{D^{\alpha}p}{\alpha!}\right)t^{k}$
$=\sum_{k=0}^{m}\sum_{|\alpha|=k}(-1)^{|\alpha|}\left({\mathcal{G}}_{1}^{\alpha_{1}}\cdots{\mathcal{G}}_{n}^{\alpha_{n}}Q,\frac{D^{\alpha}p}{\alpha!}\right)t^{k}=\sum_{k=0}^{m}\left(\frac{({\mathcal{F}}^{k}Q)(x)}{k!},p(x)\right)t^{k},\quad
N\geq m,$

where $m={\rm deg}p$. The theorem is proved. ∎

######  Remark 6.4.

The condition that $K$ has the characteristic 0 is essentially for a
uniqueness of the solution of Cauchy problem (6.1), (6.2) even for $a_{0}=0$.
Indeed, let $K=\mathbb{Z}/2\mathbb{Z}$. This is a field of characteristic 2.
Then the Cauchy problem (6.1), (6.2) for $Q(x)=0$ has a solution
$u(t,x)=\sum\limits_{k=0}^{\infty}u_{k}(x)t^{k}$, where $u_{0}(x)=u_{1}(x)=0$,
$u_{2k}(x)$ is an arbitrary element of $K[x_{1},...,x_{n}]^{\prime}$ and
$u_{2k+1}(x)=({\mathcal{F}}u_{2k})(x)$ for any $k\in\mathbb{N}$. Therefore the
considered Cauchy problem has nontrivial solutions.

Example 6.2 shows that the Cauchy problem (6.1), (6.2) may has no solutions
when $a_{0}\not=0$. The following theorem shows that under an additional
restriction on the ring $K$ there exists a solution of this problem even when
$a_{0}\not=0$.

######  Theorem 6.5.

For any linear differential operator
${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ of
infinite order with coefficients $a_{\alpha}\in K$ and for any copolynomial
$Q\in K[x_{1},...,x_{n}]^{\prime}$ there exists a solution of the Cauchy
problem (6.1), (6.2) has a solution if and only if the ring $K$ contains the
field of rational numbers. Furthermore, a solution of this Cauchy problem is
unique and has the form (6.5).

###### Proof.

Sufficiency. Since the ring $K$ contains the field of rational numbers, this
ring is of characteristic 0 and $\frac{1}{k!}\in K$ for any $k\in\mathbb{N}$.
Arguing as in the proof of Theorem 6.3, we obtain that the module
$K[x_{1},...,x_{n}]^{\prime}$ is also a torsion-free $\mathbb{Z}$-module and
the element ${\mathcal{F}}^{k}Q$ is divided by $k!$ in the module
$K[x_{1},...,x_{n}]^{\prime}$ for any $k\in\mathbb{N}$. By Theorem 2.3 [6],
the series (6.5) is well defined, the Cauchy problem (6.1), (6.2) has a unique
solution and this solution has the form (6.5).

Necessity. Suppose that for any copolynomial $Q\in
K[x_{1},...,x_{n}]^{\prime}$ and for any differential operator
${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ of
infinite order with coefficients $a_{\alpha}\in K$ there exists a solution
$u(t,x)=\sum\limits_{k=0}^{\infty}u_{k}(x)t^{k}$ of the Cauchy problem (6.1),
(6.2). We put $a_{0}=1$ and $Q(x)=\delta(x)$. Then coefficients $u_{k}(x)$ of
the corresponding solution $u(t,x)$ satisfy equalities

$u_{0}(x)=\delta(x),\quad(k+1)u_{k+1}(x)=({\mathcal{F}}u_{k})(x),\quad
k=0,1,2,....$

Therefore $(k+1)(u_{k+1},1)=(u_{k},1)$ and $(k+1)!(u_{k+1},1)=k!(u_{k},1)=1$
for any $k\in\mathbb{N}_{0}$. This implies that elements $k!\in\mathbb{N}$,
regarded as elements of $K$, are invertible. Therefore $\frac{1}{k!}\in K,\
k\in\mathbb{N}$. Then $K$ contains the field of rational numbers. ∎

Theorems 6.3 and 6.5 leads to the following assertion.

######  Corollary 6.6.

Assume that one of the following two conditions is satisfied:

  1. 1.

The ring $K$ is of characteristic 0 and $a_{0}=0$.

  2. 2.

The ring $K$ contains the field of rational numbers.

Then the Cauchy problem

$\frac{\partial u(t,x)}{\partial t}=({\mathcal{F}}u)(t,x),\quad
u(0,x)=\delta(x)$

has a unique solution in the module $K[x_{1},...,x_{n}]^{\prime}[[t]]$. This
solution has the form

${\mathcal{E}}_{C}(t,x)=\sum_{k=0}^{\infty}\frac{({\mathcal{F}}^{k}\delta)(x)}{k!}t^{k}.$
(6.7)

######  Definition 6.7.

The formal power series ${\mathcal{E}}_{C}(t,x)\in
K[x_{1},...,x_{n}]^{\prime}[[t]]$ defined in (6.7) is called the fundamental
solution of the Cauchy problem (6.1), (6.2).

Arguing as in the proof of Theorem 6.5, we obtain the following criterion of
existence of a fundamental solution of the Cauchy problem (6.1), (6.2) for any
differential operator
${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ of
infinite order.

######  Theorem 6.8.

There exists a fundamental solution of the Cauchy problem (6.1), (6.2) for any
linear differential operator
${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ of
infinite order with coefficients $a_{\alpha}\in K$ if and only if the ring $K$
contains the field of rational numbers. Moreover, a fundamental solution of
this Cauchy problem is unique and has the form (6.7).

The following assertion shows that, under the assumptions of Corollary 6.6 the
unique solution of the Cauchy problem (6.1), (6.2) is represented as the
convolution of the fundamental solution ${\mathcal{E}}_{C}(t,x)$ and the
copolynomial $Q$.

######  Theorem 6.9.

Let the assumptions of Corollary 6.6 hold. Then a unique solution of the
Cauchy problem (6.1), (6.2) can be represented in the form

$u(t,x)={\mathcal{E}}_{C}(t,x)*Q.$

###### Proof.

Indeed, a unique solution of the Cauchy problem (6.1), (6.2) is defined by
(6.5). On the other hand, in view of Definition 6.1 and Theorem 3.2 we have

${\mathcal{E}}_{C}(t,x)*Q=\sum_{k=0}^{\infty}\frac{({\mathcal{F}}^{k}\delta)*Q}{k!}t^{k}=\sum_{k=0}^{\infty}\frac{{\mathcal{F}}^{k}(\delta*Q)}{k!}t^{k}=\sum_{k=0}^{\infty}\frac{{\mathcal{F}}^{k}Q}{k!}t^{k}=u(t,x)$

(see also Example 2.12). ∎

######  Corollary 6.10.

Let the assumptions of Theorem 6.3 hold. Then for every fixed $t\in K$ the sum
of the series (6.5) is continuously depends on $Q$ in the topology of the
module $K[x_{1},...,x_{n}]^{\prime}$.

###### Proof.

Indeed, for every $t\in K$ the fundamental solution ${\mathcal{E}}_{C}(t,x)$
is a copolynomial and by Theorem 6.9 the sum of the series (6.5) can be
represented as a convolution of copolynomials ${\mathcal{E}}_{C}(t,x)$ and
$Q(x)$. Now the assertion of the corollary follows from Theorem 2.14. ∎

######  Example 6.11.

Let the ring $K$ be of characteristic 0 and let $a\in K$. In the module
$K[x_{1},...,x_{n}]^{\prime}[[t]]$ we consider the heat equation

$\frac{\partial u(t,x)}{\partial t}=a\triangle
u(t,x),\quad\triangle=\sum_{j=1}^{n}\frac{\partial^{2}}{\partial x_{j}^{2}}$
(6.8)

which is a particular case of Equation (6.1) with the differential operator
${\mathcal{F}}=a\triangle$. The assumptions of Theorems 6.3, 6.9 and Corollary
6.6 are satisfied. By Theorem 6.3, for any $Q\in K[x_{1},...,x_{n}]^{\prime}$
the Cauchy problem (6.8), (6.2) has a unique solution and this solution has
the form (see Formula (6.5))

$u(t,x)=\sum_{k=0}^{\infty}a^{k}\frac{\triangle^{k}Q}{k!}t^{k}.$ (6.9)

By Corollary 6.6, the fundamental solution of this Cauchy problem exists and
has the form

${\mathcal{E}}_{C}(t,x)=\sum_{k=0}^{\infty}a^{k}\frac{\triangle^{k}\delta}{k!}t^{k}.$
(6.10)

As in Example 5.9, for any multi-index $\beta\in\mathbb{N}_{0}^{n}$ and
$k\in\mathbb{N}$ we obtain:

$\frac{\triangle^{k}x^{\beta}}{k!}=\sum_{|\alpha|=k}\frac{D^{2\alpha}x^{\beta}}{\alpha!}.$

Therefore

$\left(\frac{\triangle^{|\alpha|}\delta}{|\alpha|!},x^{\beta}\right)=\left(\delta,\frac{\triangle^{|\alpha|}x^{\beta}}{|\alpha|!}\right)=\left\\{\begin{array}[]{ccc}\frac{(2\alpha)!}{\alpha!},\quad\beta=2\alpha,\\\
0,\quad\beta\not=2\alpha.\end{array}\right.$

Substituting this expression into the formula (6.10), we obtain the following
representation for the fundamental solution of the Cauchy problem (6.8),
(6.2):

$({\mathcal{E}}_{C}(t,x),x^{\beta})=\left\\{\begin{array}[]{ccc}\frac{(2\alpha)!}{\alpha!}(at)^{|\alpha|},\quad\beta=2\alpha,\\\
0,\quad\beta\not=2\alpha.\end{array}\right.$ (6.11)

Now let $K=\mathbb{R}$ and $a>0$. We show that, in the space
$\mathbb{R}[x_{1},...,x_{n}]^{\prime}$ for every $t>0$ the sum of the series
(6.10), regarded as a regular copolynomial, has the form

$\sum_{k=0}^{\infty}a^{k}\frac{\triangle^{k}\delta}{k!}t^{k}=\frac{1}{(\sqrt{4\pi
at})^{n}}e^{-\frac{|x|^{2}}{4at}},\quad|x|^{2}=\sum_{j=1}^{n}x_{j}^{2}.$

For this purpose we first note that

$\frac{1}{\sqrt{4\pi
at}}\int\limits_{-\infty}^{\infty}y^{k}e^{-\frac{y^{2}}{4at}}dy=\left\\{\begin{array}[]{ccc}\frac{(2l)!}{l!}a^{l}t^{l},\quad
k=2l,\\\ 0,\quad k=2l+1.\end{array}\right.$

Now, taking into account (6.11), we obtain

$({\mathcal{E}}_{C}(t,x),x^{\beta})=\left\\{\begin{array}[]{ccc}\frac{(2\alpha)!}{\alpha!}(at)^{|\alpha|},\quad\beta=2\alpha,\\\
0,\quad\beta\not=2\alpha.\end{array}\right.=\frac{1}{(\sqrt{4\pi
at})^{n}}\int\limits_{\mathbb{R}^{n}}x^{\beta}e^{-\frac{|x|^{2}}{4at}}dx.$

######  Example 6.12.

Assume that the ring $K$ contains the field of rational numbers, $a\in K$ and
$Q(x)\in K[x_{1},...,x_{n}]^{\prime}$. In the module
$K[x_{1},...,x_{n}]^{\prime}[[t]]$ we consider the Cauchy problem for the
inhomogeneous heat equation

$\frac{\partial v(t,x)}{\partial t}=a\triangle v(t,x)+Q(x),$ (6.12)
$v(0,x)=0.$ (6.13)

It is easy to see that if $v(t,x)\in K[x_{1},...,x_{n}]^{\prime}[[t]]$ is a
solution of the Cauchy problem (6.12), (6.13), then the formal power series

$u(t,x)=\frac{\partial v(t,x)}{\partial t}$ (6.14)

is a solution of the Cauchy problem (6.8), (6.2). The unique solution of the
Cauchy problem (6.8), (6.2) has the form (6.9) (see Example 6.11). From (6.9),
(6.14) and (6.13) we uniquely restore the formal power series $v(t,x)$:

$v(t,x)=\sum_{k=0}^{\infty}a^{k}\frac{\triangle^{k}Q}{(k+1)!}t^{k+1}.$ (6.15)

Since

$\frac{\triangle^{k}Q}{(k+1)!}=\sum_{|\alpha|=k}\frac{D^{2\alpha}Q}{\alpha!(k+1)}\in
K[x_{1},...,x_{n}]^{\prime},\quad k\in\mathbb{N},$

this series is well defined. Substituting (6.15) into (6.12) and (6.13), we
see that the series $v(t,x)$ is a solution of the Cauchy problem (6.12),
(6.13). The uniqueness of a solution of the Cauchy problem (6.1), (6.13)
follows from Theorem 6.5.

######  Example 6.13.

Assume that $K$ is of characteristic 0 and $s_{1},...,s_{n}\in K$. We find the
fundamental solution of the Cauchy problem for the transport equation

$\frac{\partial u}{\partial t}=\sum_{j=1}^{n}s_{j}\frac{\partial u}{\partial
x_{j}}.$ (6.16)

Equation (6.16) is a particular case of Equation (6.1) with the differential
operator ${\mathcal{F}}=\sum\limits_{j=1}^{n}s_{j}\frac{\partial}{\partial
x_{j}}$. By Corollary 6.6 the fundamental solution of the Cauchy problem
(6.16),(6.2) has the form

${\mathcal{E}}_{C}(t,x)=\sum_{k=0}^{\infty}\frac{\left(\sum\limits_{j=1}^{n}s_{j}\frac{\partial}{\partial
x_{j}}\right)^{k}\delta}{k!}t^{k}=\sum_{k=0}^{\infty}t^{k}\sum_{|\alpha|=k}\frac{s^{\alpha}D^{\alpha}\delta}{\alpha!},$

where $s=(s_{1},...,s_{n})$. Since for any $\beta\in\mathbb{N}_{0}^{n}$

$\left(\frac{D^{\alpha}\delta}{\alpha!},x^{\beta}\right)=\left\\{\begin{array}[]{ccc}(-1)^{|\beta|},\quad\beta=\alpha,\\\
0,\quad\beta\not=\alpha.\end{array}\right.,$

we obtain by virtue the definition of the shift of a copolynomial

$({\mathcal{E}}_{C}(t,x),x^{\beta})=(-1)^{|\beta|}t^{|\beta|}s^{\beta}=(\delta(x+ts),x^{\beta}).$

Hence the fundamental solution of the Cauchy problem for Equation (6.16)
coincides with the copolynomial $\delta(x+ts)$.

###  6.3 Connections between fundamental solutions.

Let ${\mathcal{F}}=\sum\limits_{|\alpha|=0}^{\infty}a_{\alpha}D^{\alpha}$ be a
linear differential operator of infinite order on
$K[x_{1},...,x_{n}]^{\prime}$ with coefficients $a_{\alpha}\in K$ and let
$a_{0}$ be an invertible element of the ring $K$. We assume that the ring $K$
contains the field of rational numbers. By Corollaries 5.2 and 6.6, the
differential operator ${\mathcal{F}}$ and the Cauchy problem (6.1), (6.2) have
the fundamental solutions ${\mathcal{E}}(x)$ and ${\mathcal{E}}_{C}(t,x)$
respectively. Furthermore, by Theorem 5.1 the operator ${\mathcal{F}}$ is
invertible. Theorem 5.1 and Corollary 5.2 imply that the differential operator
$\frac{\partial}{\partial t}-{\mathcal{F}}:K[t,x_{1},...,x_{n}]^{\prime}\to
K[t,x_{1},...,x_{n}]^{\prime}$ is also invertible and this operator also has a
fundamental solution which will be denoted by $\tilde{{\mathcal{E}}}(t,x)$.

At first, we give the connections between fundamental solutions
${\mathcal{E}}(x)$ and ${\mathcal{E}}_{C}(t,x)$. By definitions of the
fundamental solutions of an operator and a Cauchy problem (see equalities
(5.5)) and (6.7)), we obtain the formula

${\mathcal{E}}(x)=({\mathcal{F}}^{-1}{\mathcal{E}}_{C})(0,x),$

which expresses the fundamental solution of the operator ${\mathcal{F}}$
through the fundamental solution of the Cauchy problem (6.1), (6.2). With the
help of (5.5) and (6.7) we obtain the formula

${\mathcal{E}}_{C}(t,x)=\sum_{k=0}^{\infty}\frac{({\mathcal{F}}^{k+1}{\mathcal{E}})(x)}{k!}t^{k},$

which expresses the fundamental solution of the Cauchy problem (6.1), (6.2)
through the fundamental solution of the operator ${\mathcal{F}}$.

Now we establishes connections between fundamental solutions
$\tilde{{\mathcal{E}}}(t,x)$ and ${\mathcal{E}}(x)$. We have

$\frac{\partial}{\partial
t}-{\mathcal{F}}={\mathcal{F}}\left({\mathcal{F}}^{-1}\frac{\partial}{\partial
t}-I\right).$

Then the operator $\left({\mathcal{F}}^{-1}\frac{\partial}{\partial
t}-I\right):K[t,x_{1},...,x_{n}]^{\prime}\to K[t,x_{1},...,x_{n}]^{\prime}$ is
invertible and the operator equality

$\left(\frac{\partial}{\partial
t}-{\mathcal{F}}\right)^{-1}=\left({\mathcal{F}}^{-1}\frac{\partial}{\partial
t}-I\right)^{-1}{\mathcal{F}}^{-1}$ (6.17)

holds. Applying the equality (6.17) to the copolynomial
$\delta(t,x)=\delta(t)\otimes\delta(x)\in K[t,x_{1},...,x_{n}]^{\prime}$, we
obtain the formulas

$\tilde{{\mathcal{E}}}(t,x)=\left({\mathcal{F}}^{-1}\frac{\partial}{\partial
t}-I\right)^{-1}{\mathcal{F}}^{-1}(\delta(t)\otimes\delta(x))=\left({\mathcal{F}}^{-1}\frac{\partial}{\partial
t}-I\right)^{-1}(\delta(t)\otimes({\mathcal{F}}^{-1}\delta)(x))$
$=\left({\mathcal{F}}^{-1}\frac{\partial}{\partial
t}-I\right)^{-1}(\delta(t)\otimes{\mathcal{E}}(x))=\left({\mathcal{F}}^{-1}\frac{\partial}{\partial
t}-I\right)^{-1}(\delta(t)\otimes({\mathcal{F}}^{-1}{\mathcal{E}}_{C})(0,x)),$

which express the fundamental solution of the operator
$\frac{\partial}{\partial t}-{\mathcal{F}}$ through either the fundamental
solution of the operator ${\mathcal{F}}$ or the fundamental solution of the
Cauchy problem (6.1), (6.2). Moreover, we obtain the formula

$\delta(t)\otimes{\mathcal{E}}(x)=\left({\mathcal{F}}^{-1}\frac{\partial}{\partial
t}-I\right)\tilde{{\mathcal{E}}}(t,x),$

which implicitly expresses the fundamental solution of the operator
${\mathcal{F}}$ through the fundamental solution of the operator
$\frac{\partial}{\partial t}-{\mathcal{F}}$.

### Acknowledgments.

This work was supported by the Akhiezer Foundation. The authors grateful to
Eugene Karolinsky and Sergey Favorov for useful discussions of the paper
results.

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