# Stable distributions and pseudo-processes related to fractional Airy
functions

Manfred Marvin Marchione
Department of Statistical Sciences
Sapienza University of Rome
<EMAIL_ADDRESS>
&Enzo Orsingher
Department of Statistical Sciences
Sapienza University of Rome
<EMAIL_ADDRESS>
###### Abstract

In this paper we study pseudo-processes related to odd-order heat-type
equations composed with Lévy stable subordinators. The aim of the article is
twofold. We first show that the pseudo-density of the subordinated pseudo-
process can be represented as an expectation of damped oscillations with
generalized gamma distributed parameters. This stochastic representation also
arises as the solution to a fractional diffusion equation, involving a higher-
order Riesz-Feller operator, which generalizes the odd-order heat-type
equation. We then prove that, if the stable subordinator has a suitable
exponent, the time-changed pseudo-process becomes a genuine Lévy stable
process. This result permits us to obtain a power series representation for
the probability density function of an arbitrary asymmetric stable process of
exponent $\nu>1$ and skewness parameter $\beta$, with $0<\lvert\beta\lvert<1$.
The methods we use in order to carry out our analysis are based on the study
of a fractional Airy function which emerges in the investigation of the
higher-order Riesz-Feller operator.

_Keywords_ Pseudo-processes $\cdot$ Higher-order heat equation $\cdot$ Stable
processes $\cdot$ Fractional Airy function $\cdot$ Riesz-Feller derivative

##  1 Introduction

While the analysis of higher-order heat-type equations was first tackled in
some special cases by Bernstein [3] and Burwell [5], a probabilistic insight
on the topic was established some decades later by Krylov [11]. In his paper,
the author constructed a signed measure on the space $\Omega$ of measurable
functions $x=x(t),\;t>0,$ by defining the cylinder sets

$I_{n}=\\{x:\;a_{k}\leq x(t_{k})\leq b_{k},\;k=1,...,n\\},\qquad 0\leq
t_{0}<...<t_{n-1}.$

A finitely additive signed measure $\mathbb{P}$ on $\Omega$ is constructed by
the rule

$\mathbb{P}(I_{n})=\int_{a_{1}}^{b_{1}}\ldots\int_{a_{n}}^{b_{n}}\prod_{k=1}^{n}u(x_{k}-x_{k-1},t_{k}-t_{k-1})\;dx_{1}\ldots
dx_{n}$

where $x_{0}=0$, $t_{0}=0$, $u(x,t)$ is the fundamental solution to the
higher-order heat equation

$\frac{\partial u}{\partial t}=c_{m}\frac{\partial^{m}u}{\partial
x^{m}},\qquad m\geq 2$

and $c_{m}$ is a suitable coefficient. As pointed out by Miyamoto [17], the
signed measure $\mathbb{P}$ is well defined on the algebra $\mathcal{F}$
consisting of all cylinder sets and it is finitely additive on $\mathcal{F}$.
Since $\mathbb{P}$ is signed, Kolmogorov’s extension theorem cannot be applied
to extend $\mathbb{P}$ to the $\sigma$-algebra generated by $\mathcal{F}$.
Moreover, Krylov [11] proved that $\mathbb{P}$ has infinite total variation.
Therefore $\mathbb{P}$ cannot be extended to $\sigma\left(\mathcal{F}\right)$.
In Krylov’s paper only the even-order case $m=2n$ was considered. Hochberg
[10] also developed an Itô-type stochastic calculus for the even-order pseudo-
processes. Similar ideas for the construction of pseudo-processes were
proposed by Ladokhin [15], Daletsky and Fomin [6] and Miyamoto [17] who
investigated equations in which two space derivatives appear. Debbi [7, 8]
proposed a space-fractional extension of the even-order heat-type equation
involving the Riesz operator.
The pseudo-processes related to odd-order heat-type equations were first
considered by Orsingher [21] for the third order heat equation and were then
extended to all possible orders by Lachal [12]. In these papers the main
concern was to evaluate the distribution of some functionals of pseudo-
processes like the sojourn time or the maximum. Nakajima and Sato [18] have
obtained the joint distribution of the hitting time and hitting place for the
third order pseudo-process. The sample paths of pseudo-process $x(t)$ display
a sort of continuity or moderate discontinuity, as stated by Lachal [13] and
Nishioka [19, 20] by studying the distribution of $x(\tau_{a})$ with

$\tau_{a}=\inf\\{t\geq 0:\;x(t)>a\\}.$

Alternative approaches have been proposed in the literature for the
probabilistic construction of pseudo-processes. Bonaccorsi and Mazzucchi [4]
and Lachal [14] obtained the solutions to higher order heat-type equations in
terms of the scaling limit of suitable random walks. Orsingher and Toaldo [23]
constructed the pseudo-processes as the limit of compound Poisson processes
with steps represented by pseudo-random variables with signed measures.
The starting point of the present research is the paper by Orsingher and
D’Ovidio [22] in which the fundamental solution to the odd-order heat-type
equation

$\frac{\partial u}{\partial t}(x,t)=(-1)^{n}\frac{\partial^{2n+1}u}{\partial
x^{2n+1}}(x,t),\qquad x\in\mathbb{R},\;t>0,\;n\in\mathbb{N}$ (1)

is expressed as

$u_{2n+1}(x,t)=\frac{1}{\pi
x}\mathbb{E}\left[e^{-b_{n}x\;G_{2n+1}(1/t)}\sin\left(a_{n}x\;G_{2n+1}(1/t)\right)\right]$
(2)

where $G_{\gamma}(\tau)$ is a random variable with generalized gamma
distribution having probability density function

$g_{\gamma}(y;\tau)=\gamma\frac{y^{\gamma-1}}{\tau}\exp\left(-\frac{y^{\gamma}}{\tau}\right),\qquad
y,\gamma,\tau>0$

and

$a_{n}=\cos\frac{\pi}{2(2n+1)},\qquad b_{n}=\sin\frac{\pi}{2(2n+1)}.$

It is known (see Orsingher [21]) that, in the case $n=1$, the solution to
equation (1) admits the following representation in terms of the Airy function
of the first kind:

$u_{3}(x,t)=\frac{1}{\sqrt[3]{3t}}\text{Ai}\left(\frac{x}{\sqrt[3]{3t}}\right).$

For general values of $n$, we start our analysis by exploring the connection
between the probabilistic representation (2) and the higher-order Airy
functions

$\text{Ai}_{2n+1}(x)=\frac{1}{\pi}\int_{0}^{+\infty}\cos\left(sx+\frac{s^{2n+1}}{2n+1}\right)ds\qquad
n\in\mathbb{N}$ (3)

which were profoundly investigated by Askari and Ansari [2]. We then analyze
the pseudo-process $X_{2n+1}(t)$ time-changed with an independent stable
subordinator $S_{\nu}(t)$ of exponent $\nu,\;0<\nu<1$. We show that, as a
consequence of the subordination, the probabilistic representation (2) is
transformed into

$\mathbb{P}(X_{2n+1}(S_{\nu}(t))\in dx)/dx=\frac{1}{\pi
x}\mathbb{E}\left[e^{-b_{n}x\;G_{\nu(2n+1)}(1/t)}\sin\left(a_{n}x\;G_{\nu(2n+1)}(1/t)\right)\right]$

where $G_{\gamma}(\tau)$, $a_{n}$ and $b_{n}$ are defined as in formula (2).
In the final section of the paper, we examine a fractional extension of
equation (1)

$\begin{dcases}\frac{\partial u}{\partial
t}(x,t)={}_{x}D_{\alpha,\theta}u(x,t),\qquad
x\in\mathbb{R},\;t>0,\;\alpha>1\\\ u(x,0)=\delta(x).\end{dcases}$ (4)

where the space operator $D_{\alpha,\theta}$ is a higher-order Riesz-Feller
derivative of which we give an explicit representation. The analysis of
equation (4) is carried out by defining a fractional generalization of the
Airy function

${\text{Ai}}_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{+\infty}\cos\left(x\gamma+\frac{\gamma^{\alpha}}{\alpha}\right)d\gamma,\qquad\alpha>1$

for which we also give the following power series representation:

${\text{Ai}}_{\alpha}(x)=\frac{1}{\pi\alpha^{\frac{\alpha-1}{\alpha}}}\sum_{k=0}^{\infty}\frac{x^{k}\alpha^{\frac{k}{\alpha}}}{k!}\Gamma\left(\frac{k+1}{\alpha}\right)\sin\left(\pi\frac{(k+1)(\alpha+1)}{2\alpha}\right).$

For $\theta=1$ and $\alpha>1$, the solution to the Cauchy problem (4) is given
by

$u_{\alpha}(x,t)=\frac{1}{(\alpha
t)^{\frac{1}{\alpha}}}{\text{Ai}}_{\alpha}\left(\frac{x}{(\alpha
t)^{\frac{1}{\alpha}}}\right),\qquad x\in\mathbb{R},\;t>0.$ (5)

The extension to the case $0<\theta<1$ can be achieved by time-changing the
pseudo-process $X_{\alpha}(t)$, having pseudo-distribution (5), with a stable
subordinator $S_{\theta}(t)$ of exponent $0<\theta<1$.
We conclude our analysis by proving that the subordinated pseudo-process

$Y_{\alpha,\theta}(t)=X_{\alpha}(S_{\theta}(t))$

is, for $0<\alpha\theta<2$, a stable process with characteristic function

$\mathbb{E}\left[e^{i\gamma
Y_{\alpha,\theta}(t)}\right]=e^{-t\lvert\gamma\lvert^{\alpha\theta}\cos\frac{\pi\theta}{2}\left(1+i\tan\frac{\pi\theta}{2}\operatorname{sgn}\gamma\right)}$

and its pseudo-density becomes a genuine probability density function. This
implies that, by giving a power series representation for the distribution of
$Y_{\alpha,\theta}(t)$, we are able to obtain the exact distribution of an
arbitrary asymmetric stable process of exponent $1<\nu<2$ and skewness
parameter $0<\beta<1$. Our result is consistent with that reported in the book
by Zolotarev [26] and shows that stable processes can be represented as
pseudo-processes time-changed with stable subordinators.

##  2 Odd-order pseudo-processes and Airy functions

Explicit representations for the solution to the odd-order heat-type equation

$\begin{dcases}\frac{\partial u}{\partial
t}(x,t)=(-1)^{n}\frac{\partial^{2n+1}u}{\partial x^{2n+1}}(x,t),\qquad
x\in\mathbb{R},\;t>0,\;n\in\mathbb{N}\\\ u(x,0)=\delta(x)\end{dcases}$ (6)

have been proposed in different forms in the literature. The connection
between the solution to equation (6) and the higher-order Airy function

$\text{Ai}_{2n+1}(x)=\frac{1}{\pi}\int_{0}^{+\infty}\cos\left(sx+\frac{s^{2n+1}}{2n+1}\right)ds\qquad
n\in\mathbb{N}$ (7)

was highlighted by Askari and Ansari [2]. The authors studied the generalized
Airy function (7) as a solution to the ordinary differential equation

$y^{(2n)}(x)+(-1)^{n}xy(x)=0.$ (8)

Similarly to the classical Airy equation, equation (8) can be solved by
applying the generalized Laplace transform method. By splitting then the
complex plane into $2(2n+1)$ sectors of amplitude $\frac{\pi}{2(2n+1)}$, $2n$
linearly independent solutions are obtained among which the solution (7)
arises.
In the same paper the authors pointed out that, by solving equation (6) with a
Fourier transform approach, the solution reads

$\displaystyle u_{2n+1}(x,t)=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{-i\gamma
x-it\gamma^{2n+1}}d\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{+\infty}\cos\left(\gamma
x+t\gamma^{2n+1}\right)d\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{t^{\frac{1}{2n+1}}\;(2n+1)^{\frac{1}{2n+1}}}\;\text{Ai}_{2n+1}\left(\frac{x}{t^{\frac{1}{2n+1}}\;(2n+1)^{\frac{1}{2n+1}}}\right).$
(9)

The representation (9) is crucial for our purposes since most of our
probabilistic considerations emerge from the study of generalized Airy
functions.
For $n=1$, the classical Airy function admits the well-known power series
representation

$\text{Ai}_{3}(z)=\frac{2}{3^{\frac{7}{6}}}\sum_{k=0}^{+\infty}\left(\frac{z}{3^{\frac{2}{3}}}\right)^{k}\frac{\sin\left(\frac{2\pi}{3}(k+1)\right)}{\Gamma\left(\frac{k}{3}+1\right)\Gamma\left(\frac{k+2}{3}\right)}.$
(10)

The power series expansion (10) can be extended to the higher-order Airy
function (7) as shown by Ansari and Askari [1]. In the following theorem we
provide a different proof which generalizes the one proposed by Watson [25],
pag. 189, for the classical case $n=1$.

######  Theorem 1.

The higher-order Airy function (7) admits the following series representation
for $x\in\mathbb{R}$:

${\text{Ai}}_{2n+1}(x)=\frac{1}{\pi(2n+1)^{\frac{2n}{2n+1}}}\sum_{k=0}^{\infty}\frac{x^{k}(2n+1)^{\frac{k}{2n+1}}}{k!}\sin\left(\pi\frac{(k+1)(n+1)}{2n+1}\right)\Gamma\left(\frac{k+1}{2n+1}\right)$
(11)

###### Proof.

We start by observing that

$\text{Ai}_{2n+1}(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\exp\left(itx+i\frac{t^{2n+1}}{2n+1}\right)dt.$
(12)

Consider a contour enclosing two circular sectors of radius $R$, centered at
the origin, such that the arc of the first sector starts at the point $R$ and
ends at $R\exp\\{i\pi\frac{1}{4n+2}\\}$ while the second sector has an arc
starting at $R\exp\\{i\pi\frac{4n+1}{4n+2}\\}$ and ending at $-R$ (see figure
1). By the Cauchy integral theorem, the integrand of formula (12) has null
integral along the considered contour. Moreover, because of Jordan’s lemma,
the integral along the arcs of the circular sectors converge to 0 as
$R\to+\infty$.

Figure 1: the contour used for the proof of theorem 1 encloses two circular
sectors of radius $R$ centered at the origin. The arc of the first sector
starts at the point $R$ and ends at $R\exp\\{i\pi\frac{1}{4n+2}\\}$ while the
second sector has an arc starting at $R\exp\\{i\pi\frac{4n+1}{4n+2}\\}$ and
ending at $-R$. The integral along the arcs converges to 0 as $R\to+\infty$.

Thus, by summing the integrals along the two contours and taking the limit for
$R\to+\infty$, we can write

$\displaystyle\text{Ai}_{2n+1}$
$\displaystyle(x)=\frac{1}{2\pi}e^{i\pi\frac{1}{4n+2}}\int_{0}^{+\infty}\exp\left\\{rxe^{i\pi\frac{n+1}{2n+1}}-\frac{r^{2n+1}}{2n+1}\right\\}dr$
$\displaystyle\hskip
22.76219pt-\frac{1}{2\pi}e^{i\pi\frac{4n+1}{4n+2}}\int_{0}^{+\infty}\exp\left\\{rxe^{i\pi\frac{3n+1}{2n+1}}-\frac{r^{2n+1}}{2n+1}\right\\}dr$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}e^{i\pi\frac{1}{4n+2}}\int_{0}^{+\infty}\exp\left\\{rxe^{i\pi\frac{n+1}{2n+1}}-\frac{r^{2n+1}}{2n+1}\right\\}dr$
$\displaystyle+\frac{1}{2\pi}e^{-i\pi\frac{1}{4n+2}}\int_{0}^{+\infty}\exp\left\\{rxe^{-i\pi\frac{n+1}{2n+1}}-\frac{r^{2n+1}}{2n+1}\right\\}dr.$
(13)

The exponential functions $f(r)=\exp\left\\{rxe^{\pm
i\pi\frac{n+1}{2n+1}}\right\\}$ in formula (13) can be substituted by their
Taylor expansions and the order of integration and summation can then be
interchanged in force of the dominated convergence theorem. In order to apply
the dominated convergence theorem, observe that
$f(r)=\lim_{n\to+\infty}f_{n}(r)$ where $f_{n}(r)$ represents the sum of the
first $n$ terms of the Taylor expansion of $f(r)$. Since $\lvert
f_{n}(r)\lvert\leq e^{\lvert rx\lvert}$ for all $n$ and $e^{\lvert rx\lvert}$
is integrable on $(0,+\infty)$ with respect to the measure
$\mu(dr)=e^{-\frac{r^{2n+1}}{2n+1}}dr$, the dominated convergence theorem can
be applied. Thus, we can write

$\displaystyle\text{Ai}_{2n+1}(x)=$
$\displaystyle\frac{1}{2\pi}e^{i\pi\frac{1}{4n+2}}\sum_{k=0}^{\infty}\frac{x^{k}e^{ik\pi\frac{n+1}{2n+1}}}{k!}\int_{0}^{+\infty}r^{k}\exp\left\\{-\frac{r^{2n+1}}{2n+1}\right\\}dr$
$\displaystyle+\frac{1}{2\pi}e^{-i\pi\frac{1}{4n+2}}\sum_{k=0}^{\infty}\frac{x^{k}e^{-ik\pi\frac{n+1}{2n+1}}}{k!}\int_{0}^{+\infty}r^{k}\exp\left\\{-\frac{r^{2n+1}}{2n+1}\right\\}dr$
$\displaystyle=$
$\displaystyle\frac{1}{\pi(2n+1)^{\frac{2n}{2n+1}}}\sum_{k=0}^{\infty}\frac{e^{i\pi\frac{2k(n+1)+1}{4n+2}}+e^{-ik\pi\frac{2k(n+1)+1}{4n+2}}}{2}\;x^{k}(2n+1)^{\frac{k}{2n+1}}\frac{\Gamma\left(\frac{k+1}{2n+1}\right)}{k!}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi(2n+1)^{\frac{2n}{2n+1}}}\sum_{k=0}^{\infty}\frac{x^{k}(2n+1)^{\frac{k}{2n+1}}}{k!}\cos\left(\pi\frac{2k(n+1)+1}{4n+2}\right)\Gamma\left(\frac{k+1}{2n+1}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\pi(2n+1)^{\frac{2n}{2n+1}}}\sum_{k=0}^{\infty}\frac{x^{k}(2n+1)^{\frac{k}{2n+1}}}{k!}\sin\left(\pi\frac{(k+1)(n+1)}{2n+1}\right)\Gamma\left(\frac{k+1}{2n+1}\right)$

where in the last step we have used the formula
$\cos\theta=\sin\left(\theta+\frac{\pi}{2}\right)$. ∎

We observe that formula (11) is consistent with the result obtained by
Orsingher and D’Ovidio [22] who proved the following representation for the
pseudo-density:

$u_{2n+1}(x,t)=-\frac{1}{\pi
x}\sum_{k=1}^{\infty}\frac{1}{k!}\sin\left(\frac{n\pi
k}{2n+1}\right)\Gamma\left(1+\frac{k}{2n+1}\right)\left(-\frac{x}{t^{\frac{1}{2n+1}}}\right)^{k}.$
(14)

The expression (14) is immediately obtained by combining formulas (9) and
(11). Moreover, by using the triplication formula for the gamma function

$\Gamma(3z)=\frac{1}{2\pi}3^{3z-\frac{1}{2}}\Gamma\left(z\right)\Gamma\left(z+\frac{1}{3}\right)\Gamma\left(z+\frac{2}{3}\right)$

with $z=\frac{k+1}{3}$, formula (11) reduces to (10) for $n=1$.
Orsingher and D’Ovidio [22] discussed the behaviour of the function
$u_{2n+1}(x,t)$ and they observed that, while for $n=1$ the pseudo-density is
non-negative for $x>0$, the non-negativity on the positive semi-axis is lost
for $n>1$ due to the oscillating behaviour of the function. They also pointed
out that the asymmetry of $u_{2n+1}(x,t)$ seems to reduce as $n$ increases.
This assertion can be supported by observing that

$\displaystyle\lim_{n\to+\infty}u_{2n+1}(x,t)=$ $\displaystyle-\frac{1}{\pi
x}\sum_{k=1}^{+\infty}\frac{(-x)^{k}}{k!}\sin\left(k\frac{\pi}{2}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi
ix}\sum_{k=0}^{+\infty}\frac{(-x)^{k}}{k!}\left(e^{ik\frac{\pi}{2}}-e^{-ik\frac{\pi}{2}}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi
ix}\left(e^{-ix}-e^{ix}\right)=\frac{\sin x}{\pi x}.$

##  3 Pseudo-processes time-changed with stable subordinators

In this section we study the pseudo-process $X_{2n+1}(t)$, governed by the
higher-order equation (6), time-changed with stable subordinators. In
particular, we consider stable subordinators $S_{\theta}(t)$ having
characteristic function

$\mathbb{E}\left[e^{i\gamma
S_{\theta}(t)}\right]=e^{-t\lvert\gamma\lvert^{\theta}e^{-i\frac{\theta\pi}{2}\operatorname{sgn}{\gamma}}}.$

In the proof of the following theorem we need, as a preliminary result, the
Mellin transform of the Wright function (see Prudnikov et al. [24], pag. 355)

$\int_{0}^{+\infty}\text{W}_{a,b}(-x)\;x^{\eta-1}dx=\frac{\Gamma(\eta)}{\Gamma(b-a\eta)},\qquad\eta>0$

where

$\text{W}_{a,b}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{k!\;\Gamma(ak+b)},\qquad
z\in\mathbb{C},\;a>-1,\;b\in\mathbb{C}.$

######  Theorem 2.

Let $X_{2n+1}(t)$ be the pseudo-process governed by equation (6) and
$S_{\theta}(t)$ a stable subordinator of exponent $\theta$, $0<\theta<1$,
independent of $X_{2n+1}(t)$. For $\theta(2n+1)>1$ the following formula
holds:

$\mathbb{P}(X_{2n+1}(S_{\theta}(t))\in dx)/dx=\frac{1}{\pi
x}\mathbb{E}\left[e^{-b_{n}x\;G_{\theta(2n+1)}(1/t)}\sin\left(a_{n}x\;G_{\theta(2n+1)}(1/t)\right)\right]$
(15)

where $G_{\gamma}(\tau)$ is a random variable with generalized gamma
distribution having probability density function

$g_{\gamma}(y;\tau)=\gamma\frac{y^{\gamma-1}}{\tau}\exp\left(-\frac{y^{\gamma}}{\tau}\right),\qquad
y,\gamma,\tau>0$

and

$a_{n}=\cos\frac{\pi}{2(2n+1)},\qquad b_{n}=\sin\frac{\pi}{2(2n+1)}.$

###### Proof.

We use the Wright function representation of the probability density function
of the stable subordinator

$\displaystyle h_{\theta}(x,t)=$ $\displaystyle\frac{\theta
t}{x^{\theta+1}}W_{-\theta,1-\theta}\left(-\frac{t}{x^{\theta}}\right)$
$\displaystyle=$
$\displaystyle\frac{\theta}{x}\sum_{k=0}^{\infty}\frac{(-1)^{k}t^{k+1}}{x^{\theta(k+1)}k!\;\Gamma(-\theta(k+1)+1)}.$
(16)

Thus we have

$\displaystyle\mathbb{P}$ $\displaystyle(X_{2n+1}(S_{\theta}(t))\in
dx)/dx=\int_{0}^{+\infty}u_{2n+1}(x,s)\;h_{\theta}(s,t)\;ds$ $\displaystyle=$
$\displaystyle-\frac{\theta t}{\pi
x}\sum_{k=1}^{\infty}\frac{\left(-x\right)^{k}}{k!}\sin\left(\frac{n\pi
k}{2n+1}\right)\Gamma\left(1+\frac{k}{2n+1}\right)\int_{0}^{+\infty}s^{-\frac{k}{2n+1}-\theta-1}\text{W}_{-\theta,1-\theta}\left(-\frac{t}{s^{\theta}}\right)ds$

Observe that, in the last step, we have used the power series representation
(14) for $u_{2n+1}(x,s)$ and we have interchanged the order of summation and
integration by the dominated convergence theorem. We note that
$u_{2n+1}(x,s)=\lim_{m\to+\infty}f_{m}(x,s)$ where $f_{m}(x,s)$ represents the
sum of the first $m$ terms of the power series (14). Of course $\lvert
f_{m}(x,s)\lvert\leq g(x,s)$ for all $m$, where $g(x,s)=\frac{1}{\pi
x}\sum_{k=1}^{\infty}\frac{1}{k!}\Gamma\left(1+\frac{k}{2n+1}\right)\left(\frac{\lvert
x\lvert}{s^{\frac{1}{2n+1}}}\right)^{k}$. By applying the monotone convergence
theorem and integrating termwise, it can be shown that
$\int_{0}^{+\infty}g(x,s)\;h_{\theta}(s,t)\;ds<+\infty$ for $\theta(2n+1)>1$.
Thus, the dominated convergence theorem can be applied and we can write

$\displaystyle\mathbb{P}$ $\displaystyle(X_{2n+1}(S_{\theta}(t))\in dx)/dx$
$\displaystyle=$ $\displaystyle-\frac{1}{\pi
x}\sum_{k=1}^{\infty}\frac{1}{k!}\sin\left(\frac{n\pi
k}{2n+1}\right)\Gamma\left(1+\frac{k}{\theta(2n+1)}\right)\left(-\frac{x}{t^{\frac{1}{\theta(2n+1)}}}\right)^{k}$
$\displaystyle=$ $\displaystyle-\frac{(2n+1)t}{\pi
x}\int_{0}^{+\infty}s^{2n}e^{-ts^{2n+1}}\sum_{k=1}^{\infty}\frac{1}{k!}\sin\left(\frac{n\pi
k}{2n+1}\right)\left(-xs^{\frac{1}{\theta}}\right)^{k}\;ds.$

By using the relationship

$e^{x\cos\phi}\sin(x\sin\phi)=\sum_{k=0}^{\infty}\sin(\phi k)\frac{x^{k}}{k!}$
(17)

we finally obtain

$\displaystyle\mathbb{P}(X_{2n+1}($ $\displaystyle S_{\theta}(t))\in dx)/dx$
$\displaystyle=$ $\displaystyle\frac{(2n+1)t}{\pi
x}\int_{0}^{+\infty}e^{-xs^{\frac{1}{\theta}}\cos\frac{n\pi}{2n+1}}\sin\left(xs^{\frac{1}{\theta}}\sin\frac{n\pi}{2n+1}\right)s^{2n}e^{-ts^{2n+1}}ds$
$\displaystyle=$ $\displaystyle\frac{(2n+1)t}{\pi
x}\int_{0}^{+\infty}e^{-xs^{\frac{1}{\nu}}\sin\frac{\pi}{2(2n+1)}}\sin\left(xs^{\frac{1}{\theta}}\cos\frac{\pi}{2(2n+1)}\right)s^{2n}e^{-ts^{2n+1}}ds.$

The change of variables $y=s^{\frac{1}{\theta}}$ completes the proof. ∎

Formula (15) provides a probabilistic representation for the pseudo-density of
the subordinated pseudo-process $X_{2n+1}(S_{\theta}(t))$ in terms of an
expected value of damped oscillations with generalized gamma distributed
parameters. Theorem 2 is an extension of the probabilistic representation (2)
for the odd-order Airy function obtained by Orsingher and D’Ovidio [22]. As a
corollary of our result we can write

$\mathbb{P}(X_{2n+1}(S_{\theta}(t))\in dx)/dx=-\frac{1}{\pi
x}\sum_{k=1}^{\infty}\frac{1}{k!}\sin\left(\frac{n\pi
k}{2n+1}\right)\Gamma\left(1+\frac{k}{\theta(2n+1)}\right)\left(-\frac{x}{t^{\frac{1}{\theta(2n+1)}}}\right)^{k}.$
(18)

The condition $\theta(2n+1)>1$ imposed in theorem 2 ensures that the power
series (18) has infinite radius of convergence. This can be proved by using
the Stirling approximation formula for the gamma function.
In the next section we extend formula (18) to non-integer values of $n$ by
introducing a suitable fractional generalization of the pseudo-process
$X_{2n+1}(t)$. Moreover, we show that the pseudo-density (18) and its
fractional extension are genuine non-negative probability density functions if
the parameters are chosen in a suitable way.

##  4 Fractional Airy functions and stable processes

In this section we generalize the results obtained so far by studying a family
of fractional-order pseudo-processes. In particular, we are interested in the
solution to the fractional partial differential equation

$\begin{dcases}\frac{\partial u}{\partial
t}(x,t)={}_{x}D^{\alpha}_{\theta}u(x,t),\qquad\alpha>1,\;\theta\in(0,1]\\\
u(x,0)=\delta(x)\end{dcases}$ (19)

where ${}_{x}D^{\alpha}_{\theta}$ represents the Riesz-Feller fractional
derivative (originally defined by Feller [9]) for which we modify the usual
restrictions $0<\alpha\leq 2$ and
$\lvert\theta\lvert\leq\min\\{\alpha,2-\alpha\\}$. As pointed out by Mainardi
[16], these restrictions are usually imposed in order to ensure the
probabilistic interpretability of the Riesz-Feller operator. For the
restricted parameters, the solution to equation (19) is indeed the probability
density function of an asymmetric stable process. However, by eliminating the
upper bound on $\alpha$, we are able to obtain an explicit series
representation for the density function of asymmetric stable processes of
exponent $\nu>1$ and skewness parameter $\beta$, with $0<\lvert\beta\lvert<1$.
By using the notation

$\mathcal{F}\\{f(x)\\}(\gamma)=\int_{-\infty}^{+\infty}e^{i\gamma x}f(x)dx$

the Riesz-Feller fractional operator can be defined implicitely by means of
its Fourier transform

$\mathcal{F}\\{{}_{x}D^{\alpha}_{\theta}f(x)\\}(\gamma)=-\lvert\gamma\lvert^{\alpha}e^{\frac{i\pi\theta}{2}\operatorname{sgn}(\gamma)}\mathcal{F}\\{f(x)\\}(\gamma).$
(20)

For functions $f:\mathbb{R}\to\mathbb{R}$, $f\in\mathcal{C}^{m}(\mathbb{R})$,
with derivatives decaying for $\lvert x\lvert\to+\infty$, the Riesz-Feller
derivative of order $\alpha$ admits, for all values of $\theta$, the explicit
integral representation

$\displaystyle{}_{x}D^{\alpha}_{\theta}f(x)=\frac{\Gamma(\alpha-m+1)}{\pi}\frac{d^{m}}{dx^{m}}\left[\sin\frac{\pi(\alpha+\theta)}{2}\int_{0}^{+\infty}\frac{f(x+z)}{z^{\alpha-m+1}}dz\right.$
$\displaystyle\left.+(-1)^{m}\sin\frac{\pi(\alpha-\theta)}{2}\int_{0}^{+\infty}\frac{f(x-z)}{z^{\alpha-m+1}}dz\right]$
(21)

with $m-1<\alpha<m$, $m\in\mathbb{N}$.
The integral representation (21) can be proved by checking that its Fourier
transform coincides with the expression (20) (see Orsingher and Toaldo [23]).

We start our analysis by studying equation (19) for $\theta=1$:

$\begin{dcases}\frac{\partial u}{\partial
t}(x,t)={}_{x}D^{\alpha}_{1}u(x,t),\qquad\alpha>1\\\
u(x,0)=\delta(x).\end{dcases}$ (22)

Denoting by $u_{\alpha}(x,t)$ the solution to equation (22), its Fourier
transform with respect to $x$ is

$\mathcal{F}\\{u_{\alpha}(x,t)\\}(\gamma,t)=e^{-it\operatorname{sgn}(\gamma)\lvert\gamma\lvert^{\alpha}}$
(23)

from which we obtain

$\displaystyle u_{\alpha}(x,t)=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{-i\gamma
x-it\operatorname{sgn}(\gamma)\lvert\gamma\lvert^{\alpha}}d\gamma$
$\displaystyle=$ $\displaystyle\frac{1}{\pi}\int_{0}^{+\infty}\cos\left(\gamma
x+t\lvert\gamma\lvert^{\alpha}\right)d\gamma.$ (24)

In analogy with formula (9), by defining the generalized Airy function

${\text{Ai}}_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{+\infty}\cos\left(xs+\frac{s^{\alpha}}{\alpha}\right)ds,\qquad\alpha>1$
(25)

we express the pseudo-density (24) in the form

$u_{\alpha}(x,t)=\frac{1}{(\alpha
t)^{\frac{1}{\alpha}}}{\text{Ai}}_{\alpha}\left(\frac{x}{(\alpha
t)^{\frac{1}{\alpha}}}\right).$ (26)

The first problem we tackle is the convergence of the integral defining the
generalized Airy function (25).

######  Theorem 3.

For $\alpha>1$, the improper integral defining the generalized Airy function
(25) is convergent $\forall x\in\mathbb{R}$.

###### Proof.

For any $x\in\mathbb{R}$ there exists $s_{0}$ such that the function

$u(s)=xs+\frac{s^{\alpha}}{\alpha}$

is increasing for $s\geq s_{0}$. Denote by $s(u)$ the inverse of $u(s)$ for
$s>s_{0}$. For $c>s_{0}$ we have that

$\displaystyle\int_{s_{0}}^{c}\cos(u(s))ds=\int_{u(s_{0})}^{u(c)}\frac{\cos(u)}{x+s(u)^{\alpha-1}}du.$
(27)

Since $\lim_{u\to+\infty}s(u)=+\infty$ and $\alpha>1$, for any
$x\in\mathbb{R}$ the function $u\mapsto\frac{1}{x+s(u)^{\alpha-1}}$ converges
to 0 for $u\to+\infty$. Thus, by taking the limit for $c\to+\infty$ of the
integrals in formula (27), the Dirichlet test implies the convergence of the
integral

$\int_{s_{0}}^{+\infty}\cos(u(s))ds.$

The convergence of the integral in formula (25) follows immediately.∎

In the following theorem we obtain a power series representation for the
generalized Airy function.

######  Theorem 4.

The generalized Airy function

${\text{Ai}}_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{+\infty}\cos\left(xs+\frac{s^{\alpha}}{\alpha}\right)ds,\qquad\alpha>1$

admits the power series representation

${\text{Ai}}_{\alpha}(x)=\frac{1}{\pi\alpha^{\frac{\alpha-1}{\alpha}}}\sum_{k=0}^{\infty}\frac{x^{k}\alpha^{\frac{k}{\alpha}}}{k!}\Gamma\left(\frac{k+1}{\alpha}\right)\sin\left(\pi\frac{(k+1)(\alpha+1)}{2\alpha}\right).$
(28)

###### Proof.

We start by observing that

${\text{Ai}}_{\alpha}(x)=\frac{1}{2\pi}\left(\int_{0}^{+\infty}\exp\left\\{itx+i\frac{t^{\alpha}}{\alpha}\right\\}dt+\int_{0}^{+\infty}\exp\left\\{-itx-i\frac{t^{\alpha}}{\alpha}\right\\}dt\right).$
(29)

By setting

$f_{1}(t)=\exp\left\\{itx+i\frac{t^{\alpha}}{\alpha}\right\\},\qquad
f_{2}(t)=\exp\left\\{-itx-i\frac{t^{\alpha}}{\alpha}\right\\}$

the expression (29) can be reformulated by integrating $f_{1}$ and $f_{2}$
along two suitable contours encircling two different sectors of a circular
annulus with inner radius $\varepsilon$ and outer radius $R$. The contours are
described in detail in figure 2.

Figure 2: the functions $f_{1}$ and $f_{2}$ are integrated respectively along
the contours $(a)$ and $(b)$. Both contours enclose sectors of a circular
annulus with inner radius $\varepsilon$ and outer radius $R$. The contour
$(a)$ has arcs with angle ranging from $0$ to $\frac{\pi}{2\alpha}$, while the
arcs of the contour $(b)$ have angle ranging from $-\frac{\pi}{2\alpha}$ to
$0$. The integrals along the arcs converge to 0 as $\varepsilon\to 0$ and
$R\to+\infty$.

By applying the Cauchy integral theorem and by taking the limits for
$\varepsilon\to 0$ and $R\to+\infty$, we obtain

$\displaystyle{\text{Ai}}_{\alpha}(x)$
$\displaystyle=\frac{1}{2\pi}e^{i\frac{\pi}{2\alpha}}\int_{0}^{+\infty}\exp\left\\{ixre^{i\frac{\pi}{2\alpha}}-\frac{r^{\alpha}}{\alpha}\right\\}dr$
$\displaystyle\qquad+\frac{1}{2\pi}e^{-i\frac{\pi}{2\alpha}}\int_{0}^{+\infty}\exp\left\\{-ixre^{-i\frac{\pi}{2\alpha}}-\frac{r^{\alpha}}{\alpha}\right\\}dr$
$\displaystyle=\frac{1}{2\pi}e^{i\frac{\pi}{2\alpha}}\sum_{k=0}^{\infty}\frac{x^{k}e^{ik\pi\frac{\alpha+1}{2\alpha}}}{k!}\int_{0}^{+\infty}r^{k}e^{-\frac{r^{\alpha}}{\alpha}}dr$
$\displaystyle\qquad+\frac{1}{2\pi}e^{-i\frac{\pi}{2\alpha}}\sum_{k=0}^{\infty}\frac{x^{k}e^{-ik\pi\frac{\alpha+1}{2\alpha}}}{k!}\int_{0}^{+\infty}r^{k}e^{-\frac{r^{\alpha}}{\alpha}}dr$
$\displaystyle=\frac{1}{\pi\alpha^{\frac{\alpha-1}{\alpha}}}\sum_{k=0}^{\infty}\frac{e^{i\pi\frac{k(\alpha+1)+1}{2\alpha}}+e^{-i\pi\frac{k(\alpha+1)+1}{2\alpha}}}{2}\frac{x^{k}\alpha^{\frac{k}{\alpha}}}{k!}\Gamma\left(\frac{k+1}{\alpha}\right)$
$\displaystyle=\frac{1}{\pi\alpha^{\frac{\alpha-1}{\alpha}}}\sum_{k=0}^{\infty}\frac{x^{k}\alpha^{\frac{k}{\alpha}}}{k!}\Gamma\left(\frac{k+1}{\alpha}\right)\cos\left(\pi\frac{k(\alpha+1)+1}{2\alpha}\right)$
$\displaystyle=\frac{1}{\pi\alpha^{\frac{\alpha-1}{\alpha}}}\sum_{k=0}^{\infty}\frac{x^{k}\alpha^{\frac{k}{\alpha}}}{k!}\Gamma\left(\frac{k+1}{\alpha}\right)\sin\left(\pi\frac{(k+1)(\alpha+1)}{2\alpha}\right)$

where in the last step we have used the formula
$\cos\theta=\sin\left(\theta+\frac{\pi}{2}\right)$. The inversion of the order
of summation and integration performed in the proof can be justified again as
in theorem 1. ∎

By using formula (26) and by applying theorem 4 we can now express the pseudo-
density of the fractional order pseudo-process in the form

$u_{\alpha}(x,t)=-\frac{1}{\pi
x}\sum_{k=1}^{\infty}\left(-\frac{x}{t^{\frac{1}{\alpha}}}\right)^{k}\frac{\Gamma\left(1+\frac{k}{\alpha}\right)}{k!}\sin\left(k\pi\frac{\alpha-1}{2\alpha}\right).$

Our last step is to time-change the fractional order pseudo-process by means
of stable subordinators. Similarly to the the odd-order case, the following
result holds.

######  Theorem 5.

Let $X_{\alpha}(t)$ be the pseudo-process governed by the fractional equation
(22) and $S_{\theta}(t)$ be a stable subordinator of exponent $\theta$,
$0<\theta<1$, independent of $X_{\alpha}(t)$. For $\alpha\theta>1$ the
following formula holds:

$\mathbb{P}(X_{\alpha}(S_{\theta}(t))\in dx)/dx=\frac{1}{\pi
x}\mathbb{E}\left[e^{-b_{\alpha}x\;G_{\alpha\theta}(1/t)}\sin\left(a_{\alpha}x\;G_{\alpha\theta}(1/t)\right)\right]$
(30)

where $G_{\gamma}(\tau)$ is a random variable with generalized gamma
distribution having probability density function

$g_{\gamma}(y;\tau)=\gamma\frac{y^{\gamma-1}}{\tau}\exp\left(-\frac{y^{\gamma}}{\tau}\right),\qquad
y,\gamma,\tau>0$

and

$a_{\alpha}=\sin\frac{\pi}{2\alpha},\qquad
b_{\alpha}=\cos\frac{\pi}{2\alpha}.$

###### Proof.

As in theorem 2, we have that

$\displaystyle\mathbb{P}$ $\displaystyle(X_{\alpha}(S_{\theta}(t))\in
dx)/dx=\int_{0}^{+\infty}u_{\alpha}(x,s)\;h_{\theta}(s,t)\;ds$
$\displaystyle=$ $\displaystyle-\frac{\theta t}{\pi
x}\sum_{k=1}^{\infty}\frac{\left(-x\right)^{k}}{k!}\sin\left(k\pi\frac{\alpha-1}{2\alpha}\right)\Gamma\left(1+\frac{k}{\alpha}\right)\int_{0}^{+\infty}s^{-\frac{k}{\alpha}-\theta-1}\text{W}_{-\theta,1-\theta}\left(-\frac{t}{s^{\theta}}\right)ds$
$\displaystyle=$ $\displaystyle-\frac{1}{\pi
x}\sum_{k=1}^{\infty}\frac{1}{k!}\sin\left(k\pi\frac{\alpha-1}{2\alpha}\right)\Gamma\left(1+\frac{k}{\theta\alpha}\right)\left(-\frac{x}{t^{\frac{1}{\theta\alpha}}}\right)^{k}.$

The obtained power series can be shown to be convergent for $\alpha\theta>1$
by using the Stirling approximation formula for the gamma function. By using
now the integral representation of the gamma function and formula (17) as in
theorem 2, we finally obtain

$\displaystyle\mathbb{P}(X_{\alpha}($ $\displaystyle S_{\theta}(t))\in dx)/dx$
$\displaystyle=$ $\displaystyle-\frac{\alpha t}{\pi
x}\int_{0}^{+\infty}s^{\alpha-1}e^{-ts^{\alpha}}\sum_{k=1}^{\infty}\frac{1}{k!}\sin\left(k\pi\frac{\alpha-1}{2\alpha}\right)\left(-xs^{\frac{1}{\theta}}\right)^{k}\;ds$
$\displaystyle=$ $\displaystyle\frac{\alpha t}{\pi
x}\int_{0}^{+\infty}e^{-xs^{\frac{1}{\theta}}\cos\frac{\pi(\alpha-1)}{2\alpha}}\sin\left(xs^{\frac{1}{\theta}}\sin\frac{\pi(\alpha-1)}{2\alpha}\right)s^{\alpha-1}e^{-ts^{\alpha}}ds$
$\displaystyle=$ $\displaystyle\frac{\alpha t}{\pi
x}\int_{0}^{+\infty}e^{-xs^{\frac{1}{\theta}}\sin\frac{\pi}{2\alpha}}\sin\left(xs^{\frac{1}{\theta}}\cos\frac{\pi}{2\alpha}\right)s^{\alpha-1}e^{-ts^{\alpha}}ds.$

The change of variables $y=s^{\frac{1}{\theta}}$ completes the proof. ∎

We conclude our analysis by showing that theorem 5 permits us to obtain a
series representation for the exact distribution of asymmetric stable
processes with exponent $1<\nu<2$, $\nu$ being related to $\alpha$ and
$\theta$, and skewness parameter $\beta$, $0<\lvert\beta\lvert<1$.
Consider the pseudo-process $X_{\alpha}(t)$ and an independent stable
subordinator $S_{\theta}(t)$ having characteristic function

$\mathbb{E}\left[e^{i\gamma
S_{\theta}(t)}\right]=e^{-t\lvert\gamma\lvert^{\theta}e^{-i\frac{\theta\pi}{2}\operatorname{sgn}{\gamma}}}.$
(31)

The characteristic function of the subordinated pseudo-process

$Y_{\alpha,\theta}(t)=X_{\alpha}(S_{\theta}(t))$

is given by

$\displaystyle\mathbb{E}\left[e^{i\gamma Y_{\alpha,\theta}(t)}\right]=$
$\displaystyle\mathbb{E}\left[\mathbb{E}\left[e^{i\gamma
X_{\alpha}(S_{\theta}(t))}\big{\lvert}S_{\theta}(t)\right]\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[e^{-iS_{\theta}(t)\lvert\gamma\lvert^{\alpha}\operatorname{sgn}\gamma}\right]$
$\displaystyle=$ $\displaystyle
e^{-t\lvert\gamma\lvert^{\alpha\theta}e^{i\frac{\theta\pi}{2}\operatorname{sgn}\gamma}}$
$\displaystyle=$ $\displaystyle
e^{-t\lvert\gamma\lvert^{\alpha\theta}\cos\frac{\pi\theta}{2}\left(1+i\tan\frac{\pi\theta}{2}\operatorname{sgn}\gamma\right)}$
(32)

where we have used formulas (23) and (31).
The characteristic function of a stable process $S_{\nu}(\sigma,\beta,\mu;t)$
of exponent $\nu\neq 1$ reads

$\mathbb{E}\left[e^{i\gamma
S_{\nu}(\sigma,\beta,\mu;t)}\right]=e^{-\sigma^{\nu}\lvert\gamma\lvert^{\nu}\left(1-i\beta\operatorname{sgn}\gamma\tan\frac{\pi\nu}{2}\right)+i\mu\gamma}$
(33)

where $0<\nu<2$ is the exponent of the stable process, $\sigma>0$ is the
dispersion parameter, $\beta\in[-1,1]$ is the skewness parameter and
$\mu\in\mathbb{R}$ is the location parameter. By comparing formulas (4) and
(33) we obtain that

$Y_{\alpha,\theta}(t)\overset{i.d.}{=}S_{\nu}(\sigma,\beta,\mu;t)$ (34)

with

$\nu=\alpha\theta,\qquad\beta=-\frac{\tan\frac{\pi\theta}{2}}{\tan\frac{\pi\alpha\theta}{2}},\qquad\sigma=\left(\cos\frac{\pi\theta}{2}\right)^{\frac{1}{\nu}},\qquad\mu=0.$
(35)

In order for the relationship (34) to hold, we must have that $\nu\in(0,2]$
and $\beta\in[-1,1]$. This poses no problem since, given $\nu\in(0,2)$ and
$\beta\in(0,1)$, it is always possible to choose $\theta\in(0,1)$ and
$\alpha>1$ such that the formulas in (35) are satisfied. As we will see, the
extension to negative values of $\beta$ is straightforward. This permits us to
represent stable processes as pseudo-processes time-changed with stable
subordinators. For stable processes with exponent $\nu$, $1<\nu<2$ and
skewness parameter $\beta\in(0,1)$, we obtain the following formula from the
proof of theorem 5:

$\mathbb{P}(Y_{\alpha,\theta}(t)\in dx)/dx=-\frac{1}{\pi
x}\sum_{k=1}^{\infty}\frac{1}{k!}\sin\left(k\pi\frac{\alpha-1}{2\alpha}\right)\Gamma\left(1+\frac{k}{\theta\alpha}\right)\left(-\frac{x}{t^{\frac{1}{\theta\alpha}}}\right)^{k}.$
(36)

Our representation (36) of the probability density function of an asymmetric
stable process is consistent with that reported by Zolotarev [26] (see theorem
2.4.2).
For $\nu=\alpha\theta>1$, the pseudo-distribution (36) represents the
fundamental solution to the fractional partial differential equation

$\begin{dcases}\frac{\partial u}{\partial
t}(x,t)={}_{x}D^{\alpha\theta}_{\theta}u(x,t),\qquad\alpha\theta>1,\;\theta\in(0,1)\\\
u(x,0)=\delta(x).\end{dcases}$ (37)

This can be easily proved by showing that the Fourier transform of the
solution to equation (37) coincides with the characteristic function (4).
However, as formulas (35) show, the subordinated pseudo-process
$Y_{\alpha,\theta}(t)$ is identical in distribution to a genuine stochastic
process only for suitable choices of $\alpha$ and $\theta$. The Cauchy process
can be obtained as a particular case by setting $\theta=\frac{1}{\alpha}$,
which yields

$\displaystyle\mathbb{E}\left[e^{i\gamma Y_{\alpha,\theta}(t)}\right]=$
$\displaystyle
e^{-t\lvert\gamma\lvert\cos\frac{\pi}{2\alpha}\left(1+i\tan\frac{\pi}{2\alpha}\operatorname{sgn}\gamma\right)}$
$\displaystyle=$ $\displaystyle e^{-i\gamma
t\sin\frac{\pi}{2\alpha}-t\lvert\gamma\lvert\cos\frac{\pi}{2\alpha}}.$ (38)

Formula (4) represents the characteristic function of a Cauchy process with
probability density function

$\mathbb{P}(Y_{\alpha,\theta}(t)\in
dx)/dx=\frac{1}{\pi}\;\frac{t\cos\frac{\pi}{2\alpha}}{t^{2}+2xt\sin\frac{\pi}{2\alpha}+x^{2}}.$
(39)

The maximum of the Cauchy density (39) is located on the negative half-axis as
shown in figure 3.

Figure 3: the Cauchy distribution obtained as a subordinated pseudo-process
has its modal value on the negative half-axis.

We observe that the skewness parameter in (35)

$\beta=-\frac{\tan\frac{\pi\theta}{2}}{\tan\frac{\pi\alpha\theta}{2}}$

is positive for $0<\theta<1$ and $1<\alpha\theta<2$. Formula (36) thus
describes the probability density function of a stable process with positive
skewness parameter. By using the well-known property of stable processes

$-S_{\nu}(\sigma,\beta,\mu;t)\overset{i.d.}{=}S_{\nu}(\sigma,-\beta,-\mu;t),\qquad\nu\neq
1$

the series representation (36) can be immediately extended to stable processes
with negative skewness parameter.

## References

  * [1] Alireza Ansari and Hassan Askari, _On fractional calculus of $A_{2n+1}(x)$ function_, Applied Mathematics and Computation 232 (2014), 487–497.
  * [2] Hassan Askari and Alireza Ansari, _On Mellin transforms of solutions of differential equation $\chi$(n)(x ) +$\gamma$nx $\chi$ (x ) =0_, Analysis and Mathematical Physics 10 (2020), no. 4, 57.
  * [3] F. Bernstein, _Über das Fourierintegral $\int_{0}^{\infty}e^{-x^{4}}\cos txdx$_, Mathematische Annalen 79 (1919), 265–268.
  * [4] Stefano Bonaccorsi and Sonia Mazzucchi, _High order heat-type equations and random walks on the complex plane_ , Stochastic Processes and their Applications 125 (2014), 797–818.
  * [5] W. R. Burwell, _Asymptotic Expansions of Generalized Hyper-Geometric Functions_ , Proceedings of the London Mathematical Society 22 (1923), 57–72.
  * [6] Yu. L. Daletsky and S. V. Fomin, _Generalized measures in function spaces_ , Theory Prob. Appl. 10 (1965), no. 2, 304–316.
  * [7] L. Debbi, _Explicit solutions of some fractional partial differential equations via stable subordinators_ , Journal of Applied Mathematics and Stochastic Analysis 5 (2006).
  * [8] L. Debbi, _On some properties of a higher order fractional differential operator which is not in general selfadjoint_ , Applied Mathematical Sciences 1 (2007), no. 27, 1325–1339.
  * [9] William Feller, _On a generalization of Marcel Riesz’ Potentials and the Semi-Groups generated by them_ , Meddelanden Lunds Universitetes Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome Suppl. dédié a M. Riesz (1952), 73 – 81.
  * [10] Kenneth J. Hochberg, _A Signed Measure on Path Space Related to Wiener Measure_ , The Annals of Probability 6 (1978), no. 3, 433 – 458.
  * [11] V. Yu. Krylov, _Some properties of the distribution corresponding to equation $\frac{\partial u}{\partial t}=(-1)^{q+1}\frac{\partial^{2q}u}{\partial x^{2q}}$_, Soviet Math. Dokl. 1 (1960), 760–763.
  * [12] Aimé Lachal, _Distributions of Sojourn Time, Maximum and Minimum for Pseudo-Processes Governed by Higher-Order Heat-Type Equations_ , Electronic Journal of Probability 8 (2003), 1 – 53.
  * [13] Aimé Lachal, _First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^{N}u/\partial x^{N}$_, Electronic Journal of Probability 12 (2007), 300 – 353.
  * [14] Aimé Lachal, _From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval_ , International Journal of Stochastic Analysis 2014 (2014).
  * [15] V. I. Ladokhin, _On non-positive distributions_ , Kazan. Gos. Univ. Uĉen. Zap. 122 (1962), no. 4, 53–64.
  * [16] Francesco Mainardi, Yuri Luchko, and Gianni Pagnini, _The fundamental solution of the space-time fractional diffusion equation_ , Fractional Calculus and Applied Analysis 4 (2007).
  * [17] Munemi Miyamoto, _An extension of certain quasi-measure_ , Proceedings of the Japan Academy 42 (1966), no. 2, 70–74.
  * [18] Tadashi Nakajima and Sadao Sato, _An approach to the pseudoprocess driven by the equation $\frac{\partial}{\partial t}=-A\frac{\partial^{3}}{\partial x^{3}}$ by a random walk_, Kyoto Journal of Mathematics 54 (2014), no. 3, 507 – 528.
  * [19] Kunio Nishioka, _The first hitting time and place of a half-line by a biharmonic pseudo process_ , Japanese journal of mathematics. New series 23 (1997), no. 2, 235–280.
  * [20] Kunio Nishioka, _Boundary Conditions for One-Dimensional Biharmonic Pseudo Process_ , Electronic Journal of Probability 6 (2001), 1 – 27.
  * [21] Enzo Orsingher, _Processes governed by signed measures connected with third-order “heat-type” equations_ , Lithuanian Mathematical Journal 31 (1991), 220–231.
  * [22] Enzo Orsingher and Mirko D’Ovidio, _Probabilistic representation of fundamental solutions to $\frac{\partial u}{\partial t}=\kappa_{m}\frac{\partial^{m}u}{\partial x^{m}}$_, Electronic Communications in Probability 17 (2012), 1 – 12.
  * [23] Enzo Orsingher and Bruno Toaldo, _Pseudoprocesses Related to Space-Fractional Higher-Order Heat-Type Equations_ , Stochastic Analysis and Applications 32 (2014), no. 4, 619–641.
  * [24] A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, _Integrals and Series, Vol. 3: More Special Functions_ , Gordon and Breach, New York, 1989.
  * [25] G. N. Watson, _A treatise on the theory of bessel functions_ , Second ed., Cambridge University Press, Cambridge, 1951.
  * [26] V. M Zolotarev, _One-dimensional stable distributions_ , American Mathematical Society, Providence, 1986.