# Integral representations and zeros of the Lommel function and the
hypergeometric ${}_{1}F_{2}$ function.

Federico Zullo DICATAM, Università degli Studi di Brescia, Brescia, Italy &
INFN, & INFN, Sezione di Milano-Bicocca, Milano, Italy

###### Abstract

We give different integral representations of the Lommel function
$s_{\mu,\nu}(z)$ involving trigonometric and hypergeometric ${}_{2}F_{1}$
functions. By using classical results of Pólya, we give the distribution of
the zeros of $s_{\mu,\nu}(z)$ for certain regions in the plane $(\mu,\nu)$.
Further, thanks to a well known relation between the functions
$s_{\mu,\nu}(z)$ and the hypergeometric ${}_{1}F_{2}$ function, we describe
the distribution of the zeros of ${}_{1}F_{2}$ for specific values of its
parameters.

##  1 Introduction

The Lommel function $s_{\mu,\nu}(z)$ is a particular solution of the
inhomogeneous Bessel differential equation:

$z^{2}\frac{d^{2}y}{dz^{2}}+z\frac{dy}{dz}+(z^{2}-\nu^{2})y=z^{\mu+1}.$ (1)

More precisely, $s_{\mu,\nu}(z)$ is the solution of equation (1) satisfying
$s_{\mu,\nu}(z)\sim\frac{z^{\mu+1}}{(\mu+1)^{2}-\nu^{2}}(1+O(z^{2}))$ around
$z=0$ when $\mu\pm\nu$ is not an odd negative integer. The explicit Taylor
series of $s_{\mu,\nu}(z)$ around $z=0$ is given by [24]

$s_{\mu,\nu}(z)=\frac{z^{\mu+1}}{(\mu+1)^{2}-\nu^{2}}\left(1-\frac{z^{2}}{(\mu+3)^{2}-\nu^{2}}+\frac{z^{4}}{\left((\mu+3)^{2}-\nu^{2}\right)\left((\mu+5)^{2}-\nu^{2}\right)}+\cdots\right)$
(2)

The previous expression can be equivalently written in terms of hypergeometric
${}_{1}F_{2}$ function as [24]

$s_{\mu,\nu}(z)=\frac{z^{\mu+1}}{(\mu+1)^{2}-\nu^{2}}\,_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};-\frac{z^{2}}{4}\right)$
(3)

Notice that the function ${}_{1}F_{2}$ in (3), or $z^{-\mu-1}s_{\mu,\nu}(z)$,
is an entire function of $z$ with order of growth equal to 1. The Lommel
function has many applications in mathematical physics and applied sciences:
for example, it is fundamental in the description of piezoelectricity
phenomena [1], optics [2], radiative transfer processes in the atmosphere [3],
mechanics [17], elastic scattering theory [22].

In this paper we will analyze the distribution of the zeros of the function
$s_{\mu,\nu}(z)$. The identification of the values $(\mu,\nu)$ giving a non
negative function $s_{\mu,\nu}(z)$ on the real axis and the description of the
location of the zeros have been investigated by many authors. In [5] Cooke
showed that if $\nu\geq 0$ and $\nu\leq\mu-1$ then $s_{\mu,\nu}(z)>0$ for
$z>0$. Equally, if $\nu\geq 1/2$ and $\nu\leq\mu$ $s_{\mu,\nu}(z)>0$ for
$z>0$, unless $\mu=\nu=1/2$ when $s_{\mu,\nu}\geq 0$ for $z>0$. Successively,
Steinig [19] showed that $s_{\mu,\nu}(z)>0$ for $z>0$ if $|\nu|<\mu$ and
$\mu\geq 1/2$ except when $|\nu|=\mu=1/2$, where one has $s_{\mu,\nu}(z)\geq
0$ for $z>0$. Also, Steinig established the following properties of
$s_{\mu,\nu}(z)$: if $\mu<1/2$ or if $\mu=1/2$ and $|\nu|>1/2$ then
$s_{\mu,\nu}(z)$ has an infinite number of zeros for $z\in(0,\infty)$. An
extension of the positivity results of Cooke and Steinig has been given more
recently by Cho & Chung [4], since they showed that $s_{\mu,\nu}>0$ also in
the region $\mu>1/2$ and $\nu^{2}\leq(\mu+1)^{2}-2$, whereas $s_{\mu,\nu}\geq
0$ for $\mu=|\nu|=1/2$, giving also interlacing properties among the zeros of
$s_{\mu,\nu}$ and those of the Bessel function $J_{\nu}$ for certain values of
the parameters $(\mu,\nu)$. In [12] Koumandos and Lamprecht give estimates
about the location of the zeros of $s_{\mu,\nu}(z)$ for $\mu\in(-1/2,1/2)$ and
$\nu=1/2$, whereas in [11] Koumandos, extends these to $\mu\in(-3/2,1/2)$,
$\mu\neq-1/2$ and $\nu=1/2$. For more properties of the Lommel functios the
reader can look for example at [14], Ch. 11.9 or [24], Ch. 10.7.

One of the tools that we will use is a result due to Pólya [15] about the
zeros of entire functions possessing suitable integral representations. The
theorem is the following:

###### Theorem 1.1

[Pólya, 1918 [15]] Suppose that the function $f(t)$ is positive and not
decreasing in $(0,1)$. Then the functions of $z$ defined by

$V(z)=\int_{0}^{1}\sin(zt)f(t)dt,\quad U(z)=\int_{0}^{1}\cos(zt)f(t)dt$ (4)

possesses only real zeros. Further, if $f(t)$ grows steadily these zeros are
simple and the intervals $(k\pi,(k+1)\pi)$, $k=1,2,\ldots$ contain the
positive zeros of $V(z)$, whereas the intervals
$(\frac{(2k+1)\pi}{2},\frac{(2k+3)\pi}{2})$, $k=0,1,2,\ldots$ contain the
zeros of $U(z)$, each interval containing just one zero in both cases.

By growing steadily it is meant that the function is not piecewise constant
with a finite number of rational points of discontinuity in $(0,1)$.

##  2 Some integral representations and comments

In this section we will report some known facts about integral representations
of the function $s_{\mu,\nu}(z)$. In particular, in 1936 Szymanski [21] got an
integral representation for the Lommel function in the case $\nu-\mu$ is an
even positive integer. Actually, Szymanski firstly considers equation (1) for
imaginary argument:

$z^{2}\frac{d^{2}T}{dz^{2}}+z\frac{dT}{dz}-(z^{2}+\nu^{2})T=z^{\mu+1}.$ (5)

and noticed that the solution of (5) satisfying
$T=t_{\mu,\nu}(z)\sim\frac{z^{\mu+1}}{(\mu+1)^{2}-\nu^{2}}(1+O(z))$ can be
expressed as

$t_{\mu,\nu}=z^{\mu+1}\int_{0}^{1}\cosh(zt)W_{\mu,\nu}(t)dt,$ (6)

where $W_{\mu,\nu}$ is the solution of the following differential equation

$(1-t^{2})\frac{d^{2}W_{\mu,\nu}}{dt^{2}}+(2\mu-1)t\frac{dW_{\mu,\nu}}{dt}+(\nu^{2}-\mu^{2})W_{\mu,\nu}=0,$
(7)

with the following boundary conditions

$W(1)=0,\quad W^{\prime}(0)=-1.$ (8)

The solution of equation (7) with the boundary conditions (8) is written by
Szymanski in terms of the Gegenbauer functions $C_{n}^{k}$:

$W_{\mu,\nu}(t)=-(1-t^{2})^{\mu+\frac{1}{2}}\frac{C_{\nu-\mu-1}^{\mu+1}(t)}{\left(C_{\nu-\mu-1}^{\mu+1}(t)\right)^{\prime}_{t=0}}.$
(9)

where, for integers values of $\nu-\mu-1$ (odd values, for even positive
integers the denominator vanishes), the functions $C_{n}^{k}$ are defined by
the generating function $(1-2xt+x^{2})^{-k}$, i.e. $C_{n}^{k}(t)$ is the
coefficients of $x^{n}$ in the expansion of $(1-2xt+x^{2})$ in powers of $x$:

$(1-2xt+x^{2})^{-k}=\sum_{n=0}C_{n}^{k}(t)x^{n}.$ (10)

For the solution $s_{\mu,\nu}(z)$ of (1) equally one finds, for $\nu-\mu$ even
integers:

$s_{\mu,\nu}=z^{\mu+1}\int_{0}^{1}\cos(zt)W_{\mu,\nu}(t)dt,$ (11)

where $W_{\mu,\nu}(t)$ is again given by the expression (9).

In [25] another integral representation of the function $s_{0,v}(z)$ is given.
In particular, by a direct check, it is possible to show that the function

$s_{0,\nu}(z)=\frac{1}{1+\cos(\pi\nu)}\int_{0}^{\pi}\sin(z\sin(t))\cos(\nu
t)dt.$ (12)

solves equation (1) when $\mu=0$ with
$s_{0,\nu}(z)\sim\frac{z}{1-\nu^{2}}(1+O(z))$. Equivalently, the function
$s_{0,\nu}(z)$ can be also represented as

$s_{0,\nu}(z)=\frac{1}{1+\cos(\pi\nu)}\int_{0}^{1}\sin(zt)\frac{\cos(\nu\arcsin(t))+\cos(\nu\pi-\nu\arcsin(t))}{\sqrt{1-t^{2}}}dt,$
(13)

where the range of $\arcsin$ is $(-\pi/2,\pi/2)$. It is possible to show that
the function multiplying $\sin(zt)$ in (13) is positive and increasing for
$|\nu|<1$ and $t\in(0,1)$. It is then possible to apply the Theorem (1.1) to
get the following

###### Corollary 2.1

The function $s_{0,\nu}(z)$, for $|\nu|<1$, possesses only real zeros. The
zeros are simple and the intervals $(k\pi,(k+1)\pi)$, $k=0,1,2,\ldots$ contain
the non-negative zeros of $s_{0,\nu}(z)$, each interval containing just one
zero.

In the next section we will extend the integral representation (12) and the
results described in Corollary (2.1) to other values of $\mu$. Actually, the
existence of a formula generalizing (12), given by formula (21) below, has
been pointed out to us after the writing of this work: it is a special case of
an equality given by Prudnikov et al. [16]. For the sake of completeness, we
re-derive such formula in a manner that is closer to what Szymanski did in
[21] and to what is reported in formula (13). Some comments on the formula
given in [16] will be also given. The application to this formula to get a
generalization of (2.1) seems instead to be new to us.

##  3 Other integral representations and the zeros

In this section we will assume that $\mu\pm\nu$ is not an odd negative integer
so that the series (2) is defined. By generalizing the case $\mu=0$ considered
in [25], we make the following ansatz for the function $s_{\mu,\nu}(z)$:

$s_{\mu,\nu}(z)=z^{\mu}\int_{0}^{1}\sin(zt)f_{\mu,\nu}(t)dt.$ (14)

We assume that the functions $f_{\mu,\nu}(t)$ are finite in $z=0$ whereas may
be unbounded for $t\to 1$. By inserting (14) in (1) we get that the functions
$f_{\mu,\nu}(t)$ must be the solutions of the following differential equations

$(1-t^{2})\frac{d^{2}f_{\mu,\nu}}{dt^{2}}+(2\mu-3)t\frac{df_{\mu,\nu}}{dt}+(\nu^{2}-(\mu-1)^{2})f_{\mu,\nu}=0,$
(15)

with the boundary conditions:

$f_{\mu,\nu}(0)=1,\quad\lim_{t\to 1^{-}}(1-t)f_{\mu,\nu}(t)=0,\quad\lim_{t\to
1^{-}}\left((1-t^{2})\frac{df_{\mu,\nu}}{dt}+(2\mu-1)tf_{\mu,\nu}\right)=0.$
(16)

The general solution of equation (15) can be represented in terms of
hypergeometric functions as:

$A(1-t)^{\mu-\frac{1}{2}}\,_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\frac{1-t}{2}\right)+B\,_{2}F_{1}\left(1-\nu-\mu,1+\nu-\mu;\frac{3}{2}-\mu;\frac{1-t}{2}\right)$
(17)

where $A$ and $B$ are two arbitrary constants. The condition $\lim_{t\to
1^{-}}(1-t)f_{\mu,\nu}(t)=0$ gives

$\mu+\frac{1}{2}>0,$ (18)

whereas the series around $t=1$ of the function
$(1-t^{2})\frac{df_{\mu,\nu}}{dt}+(2\mu-1)tf_{\mu,\nu}$ is given by

$(1-t^{2})\frac{df_{\mu,\nu}}{dt}+(2\mu-1)tf_{\mu,\nu}\sim\left(A(1-t)^{\mu+\frac{1}{2}}\frac{\nu^{2}-\mu^{2}}{2\mu+1}+B(2\mu+1)\right)\left(1+O(1-t)\right),$
(19)

and results in $B=0$. Finally, from the boundary condition (16)
$f_{\mu,\nu}(0)=1$ we get

$A\,_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\frac{1}{2}\right)=1.$
(20)

The previous result is summarized in the following

###### Proposition 3.1

For $\mu>-\frac{1}{2}$ the Lommel function $s_{\mu,\nu}(z)$ possess the
following integral representation:

$s_{\mu,\nu}(z)=z^{\mu}\int_{0}^{1}\sin(zt)f_{\mu,\nu}(t)dt=z^{\mu}\int_{0}^{1}\frac{\sin(zt)}{(1-t)^{\frac{1}{2}-\mu}}\frac{{}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\frac{1-t}{2}\right)}{{}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\frac{1}{2}\right)}dt.$
(21)

Notice that the function
${}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\frac{1-t}{2}\right)$
is proportional to the Ferrers functions
$P_{\nu-\frac{1}{2}}^{-\mu+\frac{1}{2}}(t)$. Indeed, by definition, one has
[16]

$P_{\nu}^{\mu}(t)=\frac{1}{\Gamma(1-\mu)}\left(\frac{1+t}{1-t}\right)^{\frac{\mu}{2}}\,_{2}F_{1}\left(-\nu,\nu+1,1-\mu,\frac{1-t}{2}\right),$
(22)

giving

$s_{\mu,\nu}(z)=z^{\mu}\int_{0}^{1}\frac{\sin(zt)}{\left(1-t^{2}\right)^{\frac{1}{4}-\frac{\mu}{2}}}\frac{P_{\nu-\frac{1}{2}}^{-\mu+\frac{1}{2}}(t)}{P_{\nu-\frac{1}{2}}^{-\mu+\frac{1}{2}}(0)}dt.$
(23)

The previous equation can be derived also from a formula given in [16]. Indeed
one has (see [16] equation 2.17.9.1)

$\begin{split}&\int_{0}^{1}\frac{t^{\alpha-1}}{(1-t^{2})^{\frac{\mu}{2}}}\sin(zt)P_{\nu}^{\mu}(t)dt=z\frac{\sqrt{\pi}}{2^{\alpha-\mu+1}}\frac{\Gamma(\alpha+1)}{\Gamma(\frac{2+\alpha-\mu-\nu}{2})\Gamma(\frac{3+\alpha-\mu+\nu}{2})}\,\\\
&{}_{2}F_{3}\left(\frac{\alpha+1}{2},\frac{\alpha}{2}+1;\frac{3}{2},\frac{2+\alpha-\mu-\nu}{2},\frac{3+\alpha-\mu+\nu}{2};-\frac{z^{2}}{4}\right)\end{split}$
(24)

By setting $\alpha=1$ and changing $\mu\to-\mu+\frac{1}{2}$ and
$\nu\to\nu-\frac{1}{2}$, equation (24) gives (23) via (3). In this work,
however, since we are going to use an old result of Hurwitz [9] on the zeros
of hypergeometric functions, we prefer to keep the integral (21) as it is.

We are interested in the monotonicity of the integrand $f_{\mu,\nu}(t)$ for
$t\in(0,1)$ and $\mu>-1/2$. We notice that the derivative of $f_{\mu,\nu}$
with respect to $t$ is proportional again to a hypergeometric function:

$\frac{df_{\mu,\nu}}{dt}=\frac{1-2\mu}{2(1-t)^{\frac{3}{2}-\mu}}\frac{{}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu-\frac{1}{2};\frac{1-t}{2}\right)}{{}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\frac{1}{2}\right)}.$
(25)

The previous relation can be written in a more compact form as

$\frac{df_{\mu,\nu}}{dt}+a_{\mu,\nu}f_{\mu-1,\nu}=0,$ (26)

where we set

$a_{\mu,\nu}\doteq
2\frac{\Gamma\left(\frac{\mu+1+\nu}{2}\right)\Gamma\left(\frac{\mu+1-\nu}{2}\right)}{\Gamma\left(\frac{\mu+\nu}{2}\right.)\Gamma\left(\frac{\mu-\nu}{2}\right)}.$
(27)

Notice that $a_{\mu,\nu}$ is related to the integral of $f_{\mu,\nu}(t)$
between $0$ and $1$: indeed, from (26) we get for $\mu+1/2>0$

$\int_{0}^{1}f_{\mu,\nu}(t)dt=\frac{1}{a_{\mu+1,\nu}}=\frac{\Gamma\left(\frac{\mu+1+\nu}{2}\right)\Gamma\left(\frac{\mu+1-\nu}{2}\right)}{2\Gamma\left(\frac{\mu+\nu}{2}+1\right.)\Gamma\left(\frac{\mu-\nu}{2}+1\right)}.$
(28)

The previous quantity is finite under the given assumptions that $\mu\pm\nu$
is not an odd negative integer.

As regards the monotonicity, from (26) we see that the zeros of
$\frac{df_{\mu,\nu}}{dt}$ between $0$ and $1/2$ are the zeros of
${}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu-\frac{1}{2};\frac{1-t}{2}\right)$,
since the constant coefficient is zero only for $\mu+\frac{1}{2}=-n$,
$n=0,1,2,...$ as can be seen directly from the explicit expression

${}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu+\frac{1}{2};\frac{1}{2}\right)=2^{1/2-\mu}\sqrt{\pi}\frac{\Gamma(\mu+\frac{1}{2})}{\Gamma(\frac{\mu+\nu+1}{2})\Gamma(\frac{\mu-\nu+1}{2})}$
(29)

The number of zeros of the hypergeometric functions ${}_{2}F_{1}(a,b;c;x)$ for
$x\in(0,1)$ have been analyzed by Klein [10] and Hurwitz [9]. For
completeness, we report these results as given by Hurwitz. They are summarized
in Table (1) for $a>b$, $c\leq a+b$111These choices are not restrictive, since
$a$ and $b$ can be interchanged and, for $c>a+b$ one can use the equivalence
${}_{2}F_{1}(a,b;c;x)=(1-x)^{c-a-b}_{2}F_{1}(c-a,c-b;c;x)$.

$a$ | $b$ | $c$ | $N$
---|---|---|---
+ | + | + | 0
+ | + | - | $\frac{1+(-1)^{\lfloor-c\rfloor}}{2}$
+ | - | + | $1+\lfloor-b\rfloor$
+ | - | - |  $\lfloor-b\rfloor-\lfloor-c\rfloor$, for $\lfloor-b\rfloor>\lfloor-c\rfloor$
+ | - | - |  $\frac{1-(-1)^{\lfloor-b\rfloor+\lfloor-c\rfloor}}{2}$, for $\lfloor-b\rfloor\leq\lfloor-c\rfloor$
- | - | - | $\frac{1+(-1)^{\lfloor-a\rfloor+\lfloor-b\rfloor+\lfloor-c\rfloor}}{2}$
Table 1: The number of zeros $N$ of ${}_{2}F_{1}(a,b;c;x)$ for $x\in(0,1)$,
$a>b$, $c\leq a+b$.

We notice however that from table (1) we can provide only partial results
about the region of the parameters $(\nu,\mu)$ where the function
${}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu-\frac{1}{2};\frac{1-t}{2}\right)$
is free from zeros for $t\in(0,1)$. Indeed, the table (1) gives this
information for $t\in(-1,1)$. We need to extend the results of Hurwitz and
Klein and look at the zeros of ${}_{2}F_{1}(a,b;c;x)$ in the region
$x\in(0,1/2)$, i.e. the zeros of
${}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu-\frac{1}{2};\frac{1-t}{2}\right)$
for $t\in(0,1)$. To this aim, the quadratic (in the independent variable)
functional identities for the hypergeometric functions are very useful. The
identity we need is the following (see [7], formula 41 at page 120):

$\frac{{}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu-\frac{1}{2};x\right)}{(1-2x)\left(1-x\right)^{\mu-3/2}}=\,_{2}F_{1}\left(\frac{\mu+\nu}{2},\frac{\mu-\nu}{2};\mu-\frac{1}{2};4x(1-x)\right)$
(30)

where $x\in(0,1/2)$. Equation (30) provides an explicit relation between the
zeros of
${}_{2}F_{1}\left(\frac{\mu+\nu}{2},\frac{\mu-\nu}{2};\mu-\frac{1}{2};x\right)$
and the zeros of
${}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu-\frac{1}{2};x\right)$.
In particular it follows that if
${}_{2}F_{1}\left(\frac{\mu+\nu}{2},\frac{\mu-\nu}{2};\mu-\frac{1}{2};x\right)$
is free from zeros for $x\in(0,1)$,
${}_{2}F_{1}\left(\frac{1}{2}+\nu,\frac{1}{2}-\nu;\mu-\frac{1}{2};x\right)$ is
free from zeros for $x\in(0,1/2)$. By looking at table (1) for
${}_{2}F_{1}\left(\frac{\mu+\nu}{2},\frac{\mu-\nu}{2};\mu-\frac{1}{2};x\right)$
we get the following

###### Proposition 3.2

For $\mu>-1/2$ the function $f_{\mu,\nu}(t)$ is monotonic for $t\in(0,1)$ iff
the following conditions on the parameters are satisfied:

  * •

$\mu>1/2$ and $|\nu|\in(0,\mu)$.

  * •

$\mu<1/2$ and $|\nu|\in(|\mu|,\mu+2)$

The region in the plane $(\mu,\nu)$ corresponding to monotonic function
$f_{\mu,\nu}(t)$ for $t\in(0,1)$ is illustrated in figure (1).

Figure 1: The region of monotonicity of $f_{\mu,\nu}(t)$ for $t\in(0,1)$.

The function $f_{\mu,\nu}(t)$ can be decreasing or increasing for $t\in(0,1)$.
Since $f_{\mu,\nu}(0)=1$, by looking at what happens in a neighborhood of
$t=1$ it is simple to get the following

###### Proposition 3.3

For $\mu>-1/2$ the function $f_{\mu,\nu}(t)$ has the following monotonicity
properties going from $t=0$ to $t=1$:

  * •

For $\mu>1/2$ and $|\nu|\in(0,\mu)$ the function is decreasing from 1 to 0.

  * •

For $\mu<1/2$ and $|\nu|\in(|\mu|,\mu+1)$ the function is increasing from 1 to
$+\infty$.

  * •

For $\mu<1/2$ and $|\nu|\in(\mu+1,\mu+2)$ the function is decreasing from 1 to
$-\infty$.

From Proposition (3.3) and Theorem (1.1) we immediately get the following

###### Corollary 3.4

Apart the branch point at $z=0$, the function $s_{\mu,\nu}(z)$ for
$\mu\in(-1/2,1/2)$ and $|\nu|\in(|\mu|,\mu+1)$ possesses only real zeros. The
zeros are simple and the intervals $(k\pi,(k+1)\pi)$, $k=1,2,\ldots$ contain
the non-negative zeros, each interval containing just one zero.

The previous generalizes the Corollary (2.1) given in [25]. Further, since for
$\mu>1/2$ and $|\nu|\in(0,\mu)$ the function $f_{\mu,\nu}(t)$ is positive and
decreasing, we get the Corollary:

###### Corollary 3.5

The function $s_{\mu,\nu}(z)$ for $\mu>1/2$ and $|\nu|\in(0,\mu)$ is positive
on the positive real axis and possesses only complex zeros.

Corollary (3.5) is not new, it has been given by [19] and then appears in [6].
The proof given here is however simpler. The zeros of $s_{\mu,\nu}(z)$ can be
better characterized by looking at the asymptotic expansion of the
hypergeometric ${}_{1}F_{2}$ function, related to $s_{\mu,\nu}(z)$ by the
formula (3). Indeed, by using the results given in [13] and with the aid of
(3) we get, for $z$ real and positive, the asymptotic relation

$\frac{s_{\mu,\nu}(z)}{z^{\mu+1}}\sim\frac{1}{z^{2}}\left(1+O(\frac{1}{z})\right)+\frac{\Gamma\left(\frac{\mu+\nu+1}{2}\right)\Gamma\left(\frac{\mu-\nu+1}{2}\right)}{4\sqrt{\pi}}\left(\frac{2}{z}\right)^{\mu+\frac{3}{2}}\cos\left(z-\frac{\pi}{2}(\mu+\frac{3}{2})\right)(1+O(\frac{1}{z}))$
(31)

and if $\mu<1/2$ we see that the zeros are asymptotically given by
$z_{k}\sim\pi\left(k+\frac{2\mu+5}{4}\right)$, for $k$ suitable large
integers.

The behavior of $f_{\mu,\nu}(t)$ for $t\in(0,1)$, $\mu\in(-1/2,1/2)$ and
$|\nu|\in(\mu+1,\mu+2)$ gives the following result: yhe function
$c-f_{\mu,\nu}(t)$ is positive and increasing for $t\in(0,1)$ for any $c\geq
1$. Indeed, if we introduce the function

$V_{\mu,\nu}(z)\doteq
z^{\mu}\int_{0}^{1}(c-f_{\mu,\nu}(t))\sin(zt)dt=z^{\mu-1}c(1-\cos(z))-s_{\mu,\nu}(z),$
(32)

we can apply the Pólya Theorem (1.1) and get the following

###### Corollary 3.6

For any $c\geq 1$, the function
$V_{\mu,\nu}(z)=cz^{\mu-1}(1-\cos(z))-s_{\mu,\nu}(z)$ for $\mu\in(-1/2,1/2)$
and $|\nu|\in(\mu+1,\mu+2)$ possesses only real zeros. The zeros are simple
and the intervals $(k\pi,(k+1)\pi)$, $k=1,2,\ldots$ contain the non-negative
zeros, each interval containing just one zero.

Actually, (21) is not the only integral representation in terms of a
trigonometric kernel. Indeed, it is also possible to give a cosine integral
representation. Let us assume $\mu>1/2$. Then, by integrating by parts
equation (21) we get

$s_{\mu,\nu}=z^{\mu}\left(\frac{1}{z}-\frac{a_{\mu,\nu}}{z}\int_{0}^{1}\cos(zt)f_{\mu-1,\nu}(t)\right),$
(33)

where we used equation (26). Let us set

$c_{\mu,\nu}\doteq z^{\mu+1}a_{\mu+1,\nu}\int_{0}^{1}\cos(zt)f_{\mu,\nu}(t).$
(34)

Equation (34) is well defined for $\mu>-1/2$. Again, by integrating by parts
(34) and by using the equation $a_{\mu+1,\nu}a_{\mu,\nu}=(\mu^{2}-\nu^{2})$ we
get, for $\mu>+1/2$

$c_{\mu,\nu}=z^{\mu}\left(\mu^{2}-\nu^{2}\right)\int_{0}^{1}\sin(zt)f_{\mu-1,\nu}(t)=(\mu^{2}-\nu^{2})zs_{\mu-1,\nu}.$
(35)

From (35) it follows that, for $\mu>-1/2$

$s_{\mu,\nu}=\frac{1}{z((\mu+1)^{2}-\nu^{2})}c_{\mu+1,\nu}=\frac{a_{\mu+2,\nu}}{((\mu+1)^{2}-\nu^{2})}z^{\mu+1}\int_{0}^{1}\cos(zt)f_{\mu+1,\nu}(t).$
(36)

Notice that the previous integral representation is still convergent for
$\mu>-3/2$. It is also possible to check directly, by using the series for the
hypergeometric function around $t=1$ that indeed the integral in (36) gives
$s_{\mu,\nu}$ also for $\mu\in(-3/2,-1/2)$, so we get the following

###### Proposition 3.7

For $\mu>-\frac{3}{2}$ the Lommel function $s_{\mu,\nu}(z)$ possesses the
following integral representation:

$s_{\mu,\nu}(z)=\frac{a_{\mu+2,\nu}}{((\mu+1)^{2}-\nu^{2})}z^{\mu+1}\int_{0}^{1}\cos(zt)f_{\mu+1,\nu}(t)=\frac{z^{\mu+1}}{a_{\mu+1,\nu}}\int_{0}^{1}\cos(zt)f_{\mu+1,\nu}(t).$
(37)

where $a_{\mu,\nu}$ is given in equation (27) and $f_{\mu,\nu}(t)$ by equation
(21).

It is possible to get the integral representation (37) also by directly making
the ansatz $s_{\mu,\nu}=z^{\mu+1}\int_{0}^{1}\cos(zt)f(t)$ for some $f(t)$ and
then by looking at the differential equation and boundary conditions that
$f(t)$ must obey so that $s_{\mu,\nu}$ satisfies equation (1), like we did
with the integral representation (21). Since in the integral (37) it appears
the function $f_{\mu+1,\nu}(t)$, i.e. the same function appearing in the
integral (21) but with $\mu\to\mu+1$, from Proposition (3.7) we get directly
the following

###### Proposition 3.8

For $\mu>-3/2$ the function $f_{\mu+1,\nu}(t)$ is monotonic for $t\in(0,1)$
iff the following conditions on the parameters are satisfied:

  * •

$\mu>-1/2$ and $|\nu|\in(0,\mu+1)$.

  * •

$\mu<-1/2$ and $|\nu|\in(|\mu+1|,\mu+3)$

To understand the behavior of $\frac{f_{\mu+1,\nu}(t)}{a_{\mu+1,\nu}}$, we
look at the end-points $t=0$ and $t=1$. For $t=0$ we get

$\frac{f_{\mu+1,\nu}(0)}{a_{\mu+1,\nu}}=\frac{1}{a_{\mu+1,\nu}}=\frac{\Gamma\left(\frac{\mu+1+\nu}{2}\right)\Gamma\left(\frac{\mu+1-\nu}{2}\right)}{2\Gamma\left(1+\frac{\mu+\nu}{2}\right.)\Gamma\left(1+\frac{\mu-\nu}{2}\right)}$
(38)

For $\mu>-1/2$ $f_{\mu+1,\nu}(1)=0$, giving a decreasing function for
$|\nu|\in(0,\mu+1)$ since the values of the function (38) is positive in this
interval. For $\mu<-1/2$ $f_{\mu+1,\nu}(t)$ diverges at $t=1$, the sign being
negative. Also, for $|\nu|\in(|\mu+1|,\mu+2)$ the values of the function (38)
are negative, whereas for $|\nu|\in(\mu+2,\mu+3)$ are positive. It follows
that the function

$-\frac{1}{a_{\mu+1,\nu}}f_{\mu+1,\nu}(t)$ (39)

is positive and increasing for $\mu\in(-3/2,-1/2)$ and
$|\nu|\in(|\mu+1|,\mu+2)$. The previous results let to expand the region in
the $(\mu,\nu)$ plane for the Corollary (3.4). Indeed we immediately get the
following

###### Corollary 3.9

Apart the branch point at $z=0$, the function $s_{\mu,\nu}(z)$ for
$\mu\in(-3/2,-1/2)$ and $|\nu|\in(|\mu+1|,\mu+2)$ possesses only real zeros.
The zeros are simple and the intervals
$(\frac{(2k+1)\pi}{2},\frac{(2k+3)\pi}{2})$, $k=0,1,2,\ldots$ contain the non-
negative zeros, each interval containing just one zero.

With the same considerations given after Corollary (3.5) it is possible to
show that the zeros in Corollary (3.9) are asymptotically given by
$z_{k}\sim\pi\left(k+\frac{2\mu+5}{4}\right)$ for large $k$.

Figure 2: The regions described in Corollary (3.4) and Corollary (3.9).

As we did with Corollary (3.6), we can introduce the function

$U_{\mu,\nu}(z)=z^{\mu}(c+\frac{1}{a_{\mu+1,\nu}})\sin(z)-s_{\mu,\nu}(z),$
(40)

and, from the Pólya Theorem (1.1) we get the following

###### Corollary 3.10

For any $c\geq 0$, the function $U_{\mu,\nu}(z)$ (40) for $\mu\in(-3/2,-1/2)$
and $|\nu|\in(\mu+2,\mu+3)$ or $\mu>-1/2$ and $|\nu|\in(0,\mu+1)$ possesses
only real zeros. The zeros are simple and the intervals
$(\frac{(2k+1)\pi}{2},\frac{(2k+3)\pi}{2})$, $k=0,1,2,\ldots$ contain the non-
negative zeros, each interval containing just one zero.

The Corollaries (3.6) and (3.10) are trivial when the value of $c$ is large
and for large real values of $z$, due to the asymptotic (31). They are less
obvious for smaller values of $z$.

By looking closer at the paper [15], one sees that Theorem (1.1) can be
actually extended to linear combinations of the functions $U(z)$ and $V(z)$ as
also underline by Pólya himself (see also [20]). Under the same assumptions
about $f(t)$ of Theorem (1.1) indeed, one has [20] that the entire function

$\cos(\theta)U(z)+\sin(\theta)V(z)=\int_{0}^{1}f(t)\cos(zt-\theta)dt,\quad\theta\in(0,\pi)$
(41)

possesses only real simple zeros, each zero belonging to the intervals

$\left(\left(k-\frac{1}{2}\right)\pi+\theta,\left(k+\frac{1}{2}\right)\pi+\theta\right),\quad
k=0,\pm 1,\pm 2,\ldots$ (42)

Due to the results given in Proposition (3.3), the Corollaries (3.4) and (3.9)
can be summarized in the following

###### Corollary 3.11

The function
$z^{-\mu}\left(a_{\mu,\nu}\cos(\theta)s_{\mu-1,\nu}(z)+\sin(\theta)s_{\mu,\nu}(z)\right)$
for $\mu\in(-1/2,1/2)$ and $|\nu|\in(|\mu|,\mu+1)$ possesses only real zeros
for any $\theta\in[0,\pi]$. The zeros are simple and each of the intervals
(42) contain just one zero.

Let us finally notice that for particular values of the parameters $\mu$ and
$\nu$ it is possible to get algebraic functions for the kernel of the integral
(21). Also, when $\mu$ is a positive integer, it is possible to give explicit
formulae for the hypergeometric kernel in terms of trigonometric functions.
These observations will be further investigated in a separate paper.

##  4 Some properties of the hypergeometric ${}_{1}F_{2}$ function

Pólya [15] and Hille [8] have been two of the few authors to investigate about
the distribution of the zeros of the hypergeometric function
${}_{1}F_{2}(1;b,c;z)$. For uniformity of notation, we use the same set of
parameters for ${}_{1}F_{2}$ as in (3). The results of Pólya and Hille can be
summarized as follows:

###### Proposition 4.1

The function
${}_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};z\right)$
possesses only real zeros for $\nu=1/2$ and $\mu\in(-5/2,-1/2)$ or
$\mu\in(-7/2,-5/2)$. For $\nu=1/2$ and $\mu>1/2$ it has only complex zeros.
For $\nu=1/2$ and $\mu<1/2$ there are infinitely many real zeros.

Very recently Sokal [18] gives certain conditions ensuring that
${}_{p}F_{q}(a_{1},...a_{p};b_{1},...b_{q};x)$, $p\leq q$ possesses only real,
non positive roots. In particular he showed that
${}_{p}F_{q}(a_{1},...a_{p};b_{1},...b_{q};x)$, $p\leq q$ are entire functions
belonging to the Laguerre–Pólya class $LP^{+}$ (i.e. they can be obtained as a
limit, uniformly on compact subsets of $\mathbb{C}$, of a sequence of
polynomials with roots on $(-\infty,0]$) for arbitrarily large
$b_{p+1},...,b_{q}$ if and only if, after a possible reordering, the
differences $a_{i}-b_{i}$, $i=1,...,p$, are nonnegative integers. When the
differences $a_{i}-b_{i}$, $i=1,...,p$, are not integers it is still possible
for the functions ${}_{p}F_{q}$ to belong to the Laguerre-Pólya class $LP^{+}$
for some finite values of $b_{p+1},...b_{q}$, as it is shown also by the
results of Pólya and Hille stated above.

As far as we know, if one considers the function
${}_{1}F_{2}(a_{1};b_{1},b_{2};z)$, there are no results in the literature
about two or three dimensional sets of $(a_{1},b_{1},b_{2})\in\mathbb{R}^{3}$
for which ${}_{1}F_{2}(a_{1};b_{1},b_{2};z)$ belong to $LP^{+}$ (see also at
the end of [18]). However, due to the equivalence (3), any result about the
zeros of $s_{\mu,\nu}(z)$ can be directly transferred to
${}_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};z\right)$.
Actually, it would be preferable to investigate the zeros of the function
${}_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};z\right)$ rather
then those of $s_{\mu,\nu}(z)$ since
${}_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};z\right)$ is an
entire function of $z$ for any choice of the parameters $(\mu,\nu)$. In this
section, for completeness, we report the Corollaries for
${}_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};z\right)$
corresponding to the Corollaries (3.4), (3.5), (3.6), (3.9) and (3.10) for
$s_{\mu,\nu}(z)$. Also, for completeness, we will give the corresponding
regions in the plane $(b_{1},b_{2})$ for which
${}_{1}F_{2}(a_{1};b_{1},b_{2};z)$ belong to $LP^{+}$.

###### Corollary 4.2

The function
${}_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};z\right)$ for
$\mu\in(-1/2,1/2)$ and $|\nu|\in(|\mu|,\mu+1)$ possesses only real negative
zeros. The zeros are simple and are contained in the intervals
$(-\frac{k^{2}\pi^{2}}{4},-\frac{(k+1)^{2}\pi^{2}}{4})$, $k=1,2,\ldots$, each
interval containing just one zero.

###### Corollary 4.3

The function
${}_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};z\right)$ for
$\mu\in(-3/2,-1/2)$ and $|\nu|\in(|\mu+1|,\mu+2)$ possesses only real negative
zeros. The zeros are simple and are contained in the intervals
$(-\frac{(2k+1)^{2}\pi^{2}}{16},-\frac{(2k+3)^{2}\pi^{2}}{16})$,
$k=0,1,2,\ldots$, each interval containing just one zero.

From the previous two Corollaries, by setting $a_{1}=1$, $b_{1}=\frac{\mu-
nu+3}{2}$, $b_{2}=\frac{\mu+\nu+3}{2}$ and by noticing that
${}_{1}F_{2}(a_{1};b_{1},b_{2};z)$ is an entire function of fractional order
of growth and hence possesses an infinite number of zeros [23], we get the
following

###### Proposition 4.4

In the regions enclosed by the lines $b_{1}=\frac{1}{2}+\delta$,
$b_{1}=1+\delta$, $b_{2}=1+\delta$ and $b_{1}+b_{2}=\frac{5}{2}+\delta$, where
$\delta=0,\frac{1}{2}$, and in the regions enclosed by the lines obtained by
the exchange $b_{1}\leftrightarrow b_{2}$, the function
${}_{1}F_{2}(a_{1};b_{1},b_{2};z)$ belong to the Laguerre-Pólya class
$LP^{+}$.

The regions described in Proposition (4.4) are reported in the following
figure. Clearly it is a simple rotation and translation of the regions in
figure (2).

Figure 3: The regions (in grey) described in Proposition (4.4).

Finally, we give the following results as consequences of the Corollaries
(3.5), (3.6) and (3.10).

###### Corollary 4.5

The function
${}_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};z\right)$ for
$\mu>1/2$ and $|\nu|\in(0,\mu)$ is positive on the positive real axis and
possesses only complex zeros.

###### Corollary 4.6

For any $c\geq 1$, the function

$c\left((\mu+1)^{2}-\nu^{2}\right)\frac{(1-\cos(z))}{z^{2}}-\,_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};-\frac{z^{2}}{4}\right)$
(43)

for $\mu\in(-1/2,1/2)$ and $|\nu|\in(\mu+1,\mu+2)$ possesses only real zeros.
The zeros are simple and the intervals $(k\pi,(k+1)\pi)$, $k=1,2,\ldots$
contain the non-negative zeros, each interval containing just one zero.

###### Corollary 4.7

For any $c\geq 0$, the function

$a_{\mu+2,\nu}(ca_{\mu+1,\nu}+1)\frac{\sin(z)}{z}-\,_{1}F_{2}\left(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};-\frac{z^{2}}{4}\right)$
(44)

for $\mu\in(-3/2,-1/2)$ and $\nu\in(\mu+2,\mu+3)$ or $\mu>-1/2$ and
$\nu\in(0,\mu+1)$ possesses only real zeros. The zeros are simple and the
intervals $(\frac{(2k+1)\pi}{2},\frac{(2k+3)\pi}{2})$, $k=0,1,2,\ldots$
contain the non-negative zeros, each interval containing just one zero.

Acknowledgments

I wish to acknowledge the support of Università degli Studi di Brescia, GNFM-
INdAM and INFN, Gr. IV - Mathematical Methods in NonLinear Physics.

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# An Arecibo 327 MHz Search for Radio Pulsars and Bursts in the Dwarf
Irregular Galaxies Leo A and T

Fronefield Crawford Department of Physics and Astronomy, Franklin and Marshall
College, P.O. Box 3003, Lancaster, PA 17604, USA  Kohei Hayashi Astronomical
Institute, Tohoku University, Sendai, Miyagi, 980-8578, Japan  National
Institute of Technology, Ichinoseki College, Takanashi, Hagisho, Ichinoseki,
Iwate, 021-8511, Japan

###### Abstract

We have conducted an Arecibo 327 MHz search of two dwarf irregular galaxies in
the Local Group, Leo A and T, for radio pulsars and single pulses from fast
radio bursts and other giant pulse emitters. We detected no astrophysical
signals in this search, and we estimate flux density limits on both periodic
and burst emission. Our derived luminosity limits indicate that only the most
luminous radio pulsars known in our Galaxy and in the Magellanic Clouds (MCs)
would have been detectable in our search if they were at the distances of Leo
A and T. Given the much smaller stellar mass content and star formation rates
of Leo A and T compared to the Milky Way and the MCs, there are likely to be
few (if any) extremely luminous pulsars in these galaxies. It is therefore not
surprising that we detected no pulsars in our search.

pulsars; radio transient sources; dwarf irregular galaxies

##  1 Introduction

The detection and study of extragalactic pulsars offers the rare opportunity
to probe the high end of the pulsar luminosity function in another galaxy.
Pulsar searches are also sensitive to bursting sources, such as fast radio
bursts (FRBs; Lorimer et al. 2007), and pulsed radio signals originating from
a nearby galaxy can be used to probe the ionized intergalactic medium and the
interstellar medium of the host galaxy. A number of searches for pulsars in
nearby galaxies have been attempted (e.g., McLaughlin & Cordes 2003; Rubio-
Herrera et al. 2013; Mikhailov & van Leeuwen 2016; van Leeuwen et al. 2020).
However, to date, the only confirmed extragalactic radio pulsar discoveries
have been in the Magellanic Clouds (MCs), the closest galaxies to our own
(Manchester et al., 2006; Ridley et al., 2013). Leo A and T are in the Local
Group and are classified as dwarf irregular galaxies (dIrrs), which are gas-
rich dwarfs with active star formation. Since many dIrrs are isolated (i.e.,
outside of the realm of the Milky Way and M31), their star formation
activities are thought not to be significantly interrupted by external effects
such as tidal disruption. The star formation rates (SFRs) of dIrrs are much
higher than those of dwarf spheroidal galaxies, which they resemble but which
have little gas and no current star forming activity. However, the SFRs of Leo
A and T (Ryan-Weber et al., 2008; Karachentsev & Kaisina, 2013) are still
several orders of magnitude smaller than the SFRs of the Milky Way and MCs
(Harris & Zaritsky, 2009) owing to their much smaller masses. We selected Leo
A and T to search from the list of dwarf galaxies in or around the Local Group
that were cataloged by McConnachie (2012). These two galaxies were selected
owing to their sky positions (visible from Arecibo), their relatively small
distances from Earth, and their small angular sizes (each galaxy fits within a
single 327 MHz Arecibo beam). Leo A is the larger of the two galaxies and so
should harbor a greater number of pulsars. However, Leo T is at half the
distance to Leo A (Table 1) and hence weaker pulsars would be more easily
detectable there. Neither Leo A or T appears to have been previously searched
for pulsars.

##  2 Observations and Analysis

We used the Arecibo 305-m telescope to search both Leo A and T for periodic
emission from pulsars and for single pulses from FRBs and other giant pulse
emitters. We used a center frequency of 327 MHz and the PUPPI backend, with an
effective bandwidth of 68.75 MHz split into 2816 channels. Each channel was
8-bit sampled every 81.92 $\mu$s, with total integration times of 40 min (for
Leo A) and 31 min (for Leo T). The Arecibo beam size at 327 MHz is 15 arcmin,
much larger than the angular sizes of the two galaxies (see Table 1). Thus, in
each case the entire galaxy could be observed with a single telescope
pointing. The data were processed using the PRESTO package (Ransom,
2001)111http://www.cv.nrao.edu/~sransom/presto/ for the periodicity search.
The data were inspected for radio frequency interference and then dedispersed
at trial dispersion measures (DMs) between 0 and 3000 pc cm-3. Each
dedispersed time series was Fourier transformed, and the resulting power
spectrum was filtered to remove red noise before it was harmonically summed
and searched for significant periodic signals. No acceleration search was
performed, which limited our sensitivity to any highly accelerated binary
pulsars. For the single pulse search, we dedispersed the data at trial DMs
between 0 and 1250 pc cm-3 and applied the HEIMDALL (Barsdell,
2012)222https://sourceforge.net/projects/heimdall-astro/ package to identify
significant pulses in each dedispersed time series. The detected pulses were
then passed to the single-pulse classifier FETCH (Agarwal et al.,
2020)333https://github.com/devanshkv/fetch/, and each detected pulse was
assigned a probability by FETCH of being a real, astrophysical signal.

##  3 Results and Discussion

No periodic or single-pulse signals of astrophysical origin were detected
above a signal-to-noise ratio of 7. Our 327 MHz flux density limit to single
pulses was $\sim 0.30$ Jy for a 1 ms pulse. Our 327 MHz flux density limits
for periodic emission from radio pulsars were $\sim 0.047$ and $\sim 0.053$
mJy for Leo A and T, respectively (Table 1), assuming a 5% pulsed duty cycle.
This was obtained from the radiometer equation and was checked empirically
using several test pulsar observations. Translating these periodicity flux
density limits to radio luminosity limits using the distance to each galaxy
(see Table 1) gives 327 MHz luminosity limits of 30000 and 9200 mJy kpc2 for
Leo A and T, respectively. These luminosity limits are higher than the
estimated 327 MHz luminosities for the vast majority of pulsars listed in the
ATNF catalog (Manchester et al.,
2005)444https://www.atnf.csiro.au/research/pulsar/psrcat/. The 327 MHz
luminosities for catalog pulsars were estimated using their measured 1400 MHz
flux densities, their cataloged distances, and an assumed spectral index of
$-1.4$ in each case (the expected average for the underlying pulsar
population; see Bates et al. 2013). A total of 6 cataloged Galactic pulsars
have estimated 327 MHz luminosities greater than our Leo T limit of 9200 mJy
kpc2, and one of these lies above our Leo A limit of 30000 mJy kpc2. For
cataloged pulsars in the MCs, where the distances are better known, we find
that one pulsar in the SMC has an estimated 327 MHz luminosity greater than
9200 mJy kpc2, and no pulsars have a luminosity greater than 30000 mJy kpc2.
Despite the uncertainties in these estimates, it is clear that the luminosity
limits for our periodicity search are near the limit of detectability for even
the most luminous pulsars currently known. The much smaller stellar mass
content and SFRs of both Leo A and T compared to the Milky Way and the MCs
suggest that few (if any) extremely luminous pulsars are likely to be present
in Leo A and T. It is therefore not surprising that no such pulsars were
detected in our search.

Table 1: Observed Galaxies and and Periodicity Search Limits

Name | Leo A | Leo T
---|---|---
Right Ascension (J2000) (hh:mm:ss) | 09:59:26 | 09:34:53
Declination (J2000) (dd:mm:ss) | +30:44:47 | +17:03:05
Distance (kpc) | 798 | 417
Angular Size (arcmin) | 5.1 x 3.1 | 1.4 x 1.4
Integration Time (s) | 2395 | 1885
327 MHz Flux Density Limit (mJy) | 0.047 | 0.053
327 MHz Luminosity Limit (mJy kpc2) | 30000 | 9200

Note. — From McConnachie (2012).

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