# On the Dirichlet’s type of Eulerian polynomials

Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>,  Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>and  Deyao Gao
Yuren Lab., No. 8 Tongsheng Road, Changsha, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In the present paper, we introduce Eulerian polynomials attached to $\chi$ by
using $p$-adic $q$-integral on $\mathbb{Z}_{p}$. Also, we give new interesting
identities via the generating functions of Dirichlet’s type of Eulerian
polynomials. After, by applying Mellin transformation to this generating
function of Dirichlet’s type of Eulerian polynomials, we derive $L$-function
for Eulerian polynomials which interpolates of Dirichlet’s type of Eulerian
polynomials at negative integers.

2010 Mathematics Subject Classification. 11S80, 11B68.

Keywords and phrases. Eulerian polynomials, $p$-adic $q$-integral on
$\mathbb{Z}_{p}$, Mellin transformation, $L$-function.

##  1\. Introduction

Recently, Kim $et$ $al.$ have studied to Eulerian polynomials. They gave not
only Witt’s formula for Eulerian polynomials but also relations between
Genocchi, Tangent and Euler numbers (for more details, see [1]). In arithmetic
works of T. Kim have introduced many different generating functions of
families of Bernoulli, Euler, Genocchi numbers and polynomials by using
$p$-adic $q$-integral on $\mathbb{Z}_{p}$ (see [1-5]). After, many
mathematicians are motivated from his papers and introduced new generating
function for special functions (for more information about this subject, see
[16-28]). Y. Simsek also gave new $q$-twisted Euler numbers and polynomials
and ($h,q$)-Bernoulli numbers and polynomials by using Kim’s $p$-adic
$q$-integral on $\mathbb{Z}_{p}$. He also derived some interesting properties
in his works [26], [27], [28].

The $p$-adic $q$-integral on $\mathbb{Z}_{p}$ was originally defined by Kim.
He also investigated that $p$-adic $q$-integral on $\mathbb{Z}_{p}$ is related
to non-Archimedean combinatorial analysis in mathematical physics. That is,
the functional equation of the $q$-zeta function, the $q$-Stirling numbers and
$q$-Mahler theory and so on (for details, see[5], [6]).

We firstly list some properties of familiar Eulerian polynomials for sequel of
this paper as follows:

As it is well-known, the Eulerian polynomials, $\mathcal{A}_{n}\left(x\right)$
are given by means of the following generating function:

(1)
$e^{\mathcal{A}\left(x\right)t}=\sum_{n=0}^{\infty}\mathcal{A}_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{1-x}{e^{t\left(1-x\right)}-x}$

where $\mathcal{A}^{n}\left(x\right):=\mathcal{A}_{n}\left(x\right)$ as
symbolic. To find Eulerian polynomials, it has the following recurrence
relation:

(2)
$\left(\mathcal{A}\left(t\right)+\left(t-1\right)\right)^{n}-t\mathcal{A}_{n}\left(t\right)=\left\\{\begin{array}[]{cc}1-t&\text{if
}n=0\\\ 0&\text{if }n\neq 0,\end{array}\right.$

(for details, see [1]).

Suppose that $p$ be a fixed odd prime number. Throughout this paper, we use
the following notations. By $\mathbb{Z}_{p}$, we denote the ring of $p$-adic
rational integers, $\mathbb{Q}$ denotes the field of rational numbers,
$\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$.

The $p$-adic absolute value is defined by

$\left|p\right|_{p}=\frac{1}{p}\text{.}$

In this paper we assume $\left|q-1\right|_{p}<1$ as an indeterminate. Let
$UD\left(\mathbb{Z}_{p}\right)$ be the space of uniformly differentiable
functions on $\mathbb{Z}_{p}$. For a positive integer $d$ with
$\left(d,p\right)=1$, set

$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{m}=\lim_{\overleftarrow{m}}\mathbb{Z}/dp^{m}\mathbb{Z}\text{,}$
$\displaystyle X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathbb{Z}_{p}$

and

$a+dp^{m}\mathbb{Z}_{p}=\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}dp^{m}\right)\right\\}\text{,}$

where $a\in\mathbb{Z}$ satisfies the condition $0\leq a<dp^{m}$.

Firstly, for introducing fermionic $p$-adic $q$-integral, we need some basic
information which we state here. A measure on $\mathbb{Z}_{p}$ with values in
a $p$-adic Banach space $B$ is a continuous linear map

$f\mapsto\int f(x)\mu=\int_{\mathbb{Z}_{p}}f(x)\mu(x)$

from $C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$, (continuous function on
$\mathbb{Z}_{p}$ ) to $B$. We know that the set of locally constant functions
from $\mathbb{Z}_{p}$ to $\mathbb{Q}_{p}$ is dense in
$C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$ so.

Explicitly, for all $f\in C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$, the locally
constant functions

$f_{n}=\sum_{i=0}^{p^{m}-1}f(i)1_{i+p^{m}\mathbb{Z}_{p}}\rightarrow f\text{ in
}C^{0}$

Now, set
$\mu(i+p^{m}\mathbb{Z}_{p})=\int_{\mathbb{Z}_{p}}1_{i+p^{m}\mathbb{Z}_{p}}\mu$.
Then $\int_{\mathbb{Z}_{p}}f\mu$, is given by the following Riemannian sum

$\int_{\mathbb{Z}_{p}}f\mu=\lim_{m\rightarrow\infty}\sum_{i=0}^{p^{m}-1}f(i)\mu{(i+p^{m}\mathbb{Z}_{p})}$

The following $q$-Haar measure is defined by Kim in [3] and [5]:

$\mu_{q}(a+p^{m}\mathbb{Z}_{p})=\frac{q^{a}}{[p^{m}]_{q}}$

So, for $f\in UD\left(\mathbb{Z}_{p}\right)$, the $p$-adic $q$-integral on
$\mathbb{Z}_{p}$ is defined by Kim as follows:

(3)
$I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\eta\right)d\mu_{q}\left(\eta\right)=\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{q}}\sum_{\eta=0}^{p^{n}-1}q^{\eta}f\left(\eta\right)\text{.}$

The bosonic integral is considered as the bosonic limit $q\rightarrow 1,$
$I_{1}\left(f\right)=\lim_{q\rightarrow 1}I_{q}\left(f\right)$. In [10], [11]
and [12], similarly, the $p$-adic fermionic integration on $\mathbb{Z}_{p}$
defined by Kim as follows:

(4)
$I_{-q}\left(f\right)=\lim_{q\rightarrow-q}I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q}\left(x\right)\text{.}$

By (4), we have the following well-known integral eguation:

(5)
$q^{n}I_{-q}\left(f_{n}\right)+\left(-1\right)^{n-1}I_{-q}\left(f\right)=\left[2\right]_{q}\sum_{l=0}^{n-1}\left(-1\right)^{n-1-l}q^{l}f\left(l\right)$

Here $f_{n}\left(x\right):=f\left(x+n\right)$. By (5), we have the following
equalities:

If $n$ odd, then

(6)
$q^{n}I_{-q}\left(f_{n}\right)+I_{-q}\left(f\right)=\left[2\right]_{q}\sum_{l=0}^{n-1}\left(-1\right)^{l}q^{l}f\left(l\right)\text{.}$

If $n$ even, then we have

(7)
$I_{-q}\left(f\right)-q^{n}I_{-q}\left(f_{n}\right)=\left[2\right]_{q}\sum_{l=0}^{n-1}\left(-1\right)^{l}q^{l}f\left(l\right)\text{.}$

Substituting $n=1$ into (6), we readily see the following

(8)
$qI_{-q}\left(f_{1}\right)+I_{-q}\left(f\right)=\left[2\right]_{q}f\left(0\right)\text{.}$

Replacing $q$ by $q^{-1}$ in (8), we easily derive the following

(9)
$I_{-q^{-1}}\left(f_{1}\right)+qI_{-q^{-1}}\left(f\right)=\left[2\right]_{q}f\left(0\right)\text{.}$

In [1], Kim $et$ $al.$ is considered $f(x)=e^{-x\left(1+q\right)t}$ in (9),
then they gave Witt’s formula of Eulerian polynomials as follows:

For $n\in\mathbb{N}^{\ast}$,

(10)
$I_{-q^{-1}}\left(x^{n}\right)=\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n}\left(-q\right)\text{.}$

Now also, we consider $I_{-q^{-1}}\left(\chi\left(x\right)x^{n}\right)$ in the
next section. We shall call as Dirichlet’s type of Eulerian polynomials. After
we shall give arithmetic properties for Dirichlet’s type of Eulerian
polynomials.

##  2\. On the Dirichlet’s type of Eulerian polynomials

Firstly, we consider the following equality by using (6): For $d$ odd natural
numbers,

(11)
$\displaystyle\int_{\mathbb{Z}_{p}}f\left(x+d\right)d\mu_{-q^{-1}}\left(x\right)+q^{d}\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q^{-1}}\left(x\right)$
$\displaystyle=\left[2\right]_{q}\sum_{0\leq l\leq
d-1}\left(-1\right)^{l}q^{d-l+1}f\left(l\right)\text{.}$

Let $\chi$ be a Dirichlet’s character of conductor $d,$ which is any multiple
of $p$ (=$odd$). Then, substituting
$f(x)=\chi\left(x\right)e^{-x\left(1+q\right)t}$ in (11), then we compute as
follows:

$\displaystyle\int_{\mathbb{Z}_{p}}\chi\left(x+d\right)e^{-\left(x+d\right)\left(1+q\right)t}d\mu_{-q^{-1}}\left(x\right)+q^{d}\int_{\mathbb{Z}_{p}}\chi\left(x\right)e^{-x\left(1+q\right)t}d\mu_{-q^{-1}}\left(x\right)$
$\displaystyle=\left[2\right]_{q}\sum_{0\leq l\leq
d-1}\left(-1\right)^{l}q^{d-l+1}\chi\left(l\right)e^{-l\left(1+q\right)t}$

After some applications, we discover the following

(12)
$\int_{\mathbb{Z}_{p}}\chi\left(x\right)e^{-x\left(1+q\right)t}d\mu_{-q^{-1}}\left(x\right)=\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{d-l+1}\chi\left(l\right)\frac{e^{-l\left(1+q\right)t}}{e^{-d\left(1+q\right)t}+q^{d}}\text{.}$

Let
$\mathcal{F}_{q}\left(t\mid\chi\right)=\sum_{n=0}^{\infty}\mathcal{A}_{n,\chi}\left(-q\right)\frac{t^{n}}{n!}$.
Then, we introduce the following definition of generating function of
Dirichlet’s type of Eulerian polynomials.

######  Definition 1.

For $n\in\mathbb{N}^{\ast}$, then we define the following:

(13)
$\sum_{n=0}^{\infty}\mathcal{A}_{n,\chi}\left(-q\right)\frac{t^{n}}{n!}=\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{d-l+1}\chi\left(l\right)\frac{e^{-l\left(1+q\right)t}}{e^{-d\left(1+q\right)t}+q^{d}}\text{.}$

By (12) and (13), we state the following theorem which is the Witt’s formula
for Dirichlet’s type of Eulerian polynomials.

######  Theorem 2.1.

The following identity holds true:

(14)
$I_{-q^{-1}}\left(\chi\left(x\right)x^{n}\right)=\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n,\chi}\left(-q\right)\text{.}$

By using (13), we discover the following applications:

$\displaystyle\sum_{n=0}^{\infty}\mathcal{A}_{n,\chi}\left(-q\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{d-l+1}\chi\left(l\right)\frac{e^{-l\left(1+q\right)t}}{e^{-d\left(1+q\right)t}+q^{d}}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{-l+1}\chi\left(l\right)e^{-l\left(1+q\right)t}\sum_{m=0}^{\infty}\left(-1\right)^{m}q^{-md}e^{-md\left(1+q\right)t}$
$\displaystyle=$ $\displaystyle
q\left[2\right]_{q}\sum_{m=0}^{\infty}\sum_{l=0}^{d-1}\left(-1\right)^{l+md}\chi\left(l+md\right)q^{-\left(l+md\right)}e^{-\left(l+md\right)\left(1+q\right)t}$
$\displaystyle=$ $\displaystyle
q\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}\chi\left(m\right)q^{-m}e^{-m\left(1+q\right)t}\text{.}$

Thus, we get the following theorem.

######  Theorem 2.2.

The following

(15)
$\mathcal{F}_{q}\left(t\mid\chi\right)=\sum_{n=0}^{\infty}\mathcal{A}_{n,\chi}\left(-q\right)\frac{t^{n}}{n!}=\left[2\right]_{q}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)q^{-m}e^{-m\left(1+q\right)t}}{q^{m-1}}$

is true.

By considering Taylor expansion of $e^{-m\left(1+q\right)t}$ in (15), we
procure the following theorem.

######  Theorem 2.3.

For $n\in\mathbb{N}$, then we have

(16)
$\frac{\left(-1\right)^{n}}{q\left(1+q\right)^{n+1}}\mathcal{A}_{n,\chi}\left(-q\right)=\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)m^{n}}{q^{m}}\text{.}$

From (14) and (16), we easily derive the following corollary:

######  Corollary 2.4.

For $n\in\mathbb{N}$, then we procure the following

$\lim_{m\rightarrow\infty}\sum_{x=1}^{p^{m}-1}\frac{\left(-1\right)^{x}\chi\left(x\right)x^{n}}{q^{x}}=2\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)m^{n}}{q^{m-2}}\text{.}$

Now, we give distribution formula for Dirichlet’s type of Eulerian polynomials
by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$, as follows:

$\displaystyle\int_{\mathbb{Z}_{p}}\chi\left(x\right)x^{n}d\mu_{-q^{-1}}\left(x\right)$
$\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{\left[dp^{m}\right]_{-q^{-1}}}\sum_{x=0}^{dp^{m}-1}\left(-1\right)^{x}\chi\left(x\right)x^{n}q^{-x}$
$\displaystyle=$
$\displaystyle\frac{d^{n}}{\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\left(\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{-q^{-d}}}\sum_{x=0}^{p^{m}-1}\left(-1\right)^{x}\left(\frac{a}{d}+x\right)^{n}q^{-dx}\right)$
$\displaystyle=$
$\displaystyle\frac{d^{n}}{\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\int_{\mathbb{Z}_{p}}\left(\frac{a}{d}+x\right)^{n}d\mu_{-q^{-d}}\left(x\right).$

Thus, we state the following theorem.

######  Theorem 2.5.

The following identity holds true:

(17)
$\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n,\chi}\left(-q\right)=\frac{d^{n}}{\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\int_{\mathbb{Z}_{p}}\left(\frac{a}{d}+x\right)^{n}d\mu_{-q^{-d}}\left(x\right)\text{.}$

From this, we notice that the above equation is related to $q$-Genocchi
polynomials with weight zero, $\widetilde{G}_{n,q}\left(x\right)$, and
$q$-Euler polynomials with weight zero, $\widetilde{E}_{n,q}\left(x\right)$,
which is defined by Araci $et$ $al.$ and Kim and Choi in [25] and [15] as
follows:

(18)
$\frac{\widetilde{G}_{n+1,q}\left(x\right)}{n+1}=\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{-q}}\sum_{y=0}^{p^{m}-1}\left(-1\right)^{y}\left(x+y\right)^{n}q^{y}$

and

(19)
$\widetilde{E}_{n,q}\left(x\right)=\int_{\mathbb{Z}_{p}}\left(x+y\right)^{n}d\mu_{-q}\left(y\right).$

By expressions of (17), (18) and (19), we easily discover the following
corollary.

######  Corollary 2.6.

For $n\in\mathbb{N}^{\ast}$, then we have

$\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n,\chi}\left(-q\right)=\frac{d^{n}}{\left(n+1\right)\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\widetilde{G}_{n+1,q^{-d}}\left(\frac{a}{d}\right)$

Moreover,

$\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n,\chi}\left(-q\right)=\frac{d^{n}}{\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\widetilde{E}_{n,q^{-d}}\left(\frac{a}{d}\right)\text{.}$

##  3\. On the Eulerian-$L$ function

Our goal in this section is to introduce Eulerian-$L$ function by applying
Mellin transformation to the generating function of Dirichlet’s type of
Eulerian polynomials. By (15), for $s\in\mathbb{C}$, we define the following

$L_{E}\left(s\mid\chi\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}\mathcal{F}_{q}\left(t\mid\chi\right)dt$

where $\Gamma\left(s\right)$ is the Euler Gamma function. It becomes as
follows:

$\displaystyle L_{E}\left(s\mid\chi\right)$ $\displaystyle=$ $\displaystyle
q\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}\chi\left(m\right)q^{-m}\left\\{\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}e^{-m\left(1+q\right)t}dt\right\\}$
$\displaystyle=$
$\displaystyle\frac{q}{\left(1+q\right)^{s-1}}\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)}{q^{m}m^{s}}$

So, we give definition of Eulerian $L$-function as follows:

######  Definition 2.

For $s\in\mathbb{C}$, then we have

(20)
$L_{E}\left(s\mid\chi\right)=\frac{q}{\left(1+q\right)^{s-1}}\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)}{q^{m}m^{s}}\text{.}$

Substituting $s=-n$ into (16), then, relation between Eulerian $L$-function
and Dirichlet’s type of Eulerian polynomials are given by the following
theorem.

######  Theorem 3.1.

The following equality holds true:

$L_{E}\left(-n\mid\chi\right)=\left\\{\begin{array}[]{cc}-\mathcal{A}_{n,\chi}\left(-q\right)&\text{if
}n\text{ odd,}\\\ \mathcal{A}_{n,\chi}\left(-q\right)&\text{if }n\text{
even.}\end{array}\right.$

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# Mobility edge phenomenon in a Hubbard chain: A mean field study

Santanu K. Maiti<EMAIL_ADDRESS>Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700
108, India  Abraham Nitzan  School of Chemistry, Tel Aviv University, Ramat-
Aviv, Tel Aviv-69978, Israel

###### Abstract

We show that a tight-binding one-dimensional chain composed of interacting and
non-interacting atomic sites can exhibit multiple mobility edges at different
values of carrier energy in presence of external electric field. Within a mean
field Hartree-Fock approximation we numerically calculate two-terminal
transport by using Green’s function formalism. Several cases are analyzed
depending on the arrangements of interacting and non-interacting atoms in the
chain. The analysis may be helpful in designing mesoscale switching devices.

###### pacs:

73.63.Nm, 72.20.Ee, 73.21.-b

##  I Introduction

Electronic localization phenomena in one-dimensional ($1$D) quantum systems
have long been a central problem in condensed matter physics. It is well
established that in infinite $1$D systems with random site potentials,
irrespective of the strength of randomness, all the energy eigenstates are
exponentially localized anderson . Apart from this Anderson type localization
another kind of localization known as Wannier-Stark localization is also
observed in $1$D materials, even in absence of any disorder, when the system
is subjected to an external electric field wan . For both cases i.e., infinite
$1$D systems with random site potentials and $1$D chains in presence of
external electric field, one never encounters mobility edges separating the
localized energy eigenstates from the extended ones, since all eigenstates are
localized. However, there are some classes of $1$D systems such as, correlated
disordered models, quasi-periodic Aubry-Andre model where several classic
features of mobility edges at some specific values of energy are obtained dun
; sanch ; fa ; fm ; dom ; aubry ; san6 ; eco ; das ; rolf . Although the
existence of such mobility edges in one- or two-dimensional systems has been
described by several groups eco ; das ; rolf ; sch ; san1 ; san2 , a
comprehensive study of this phenomenon is still lacking, particularly in the
presence of electron-electron interaction. Still open fundamental questions
are whether some special features exist in disordered $1$D systems, or in the
response of $1$D systems to an externally applied electric field even in the
presence of electron-electron interaction.

In the present article we investigate two-terminal electron transport through
a $1$D mesoscopic chain composed of interacting and non-interacting atomic
sites in presence of external electric field. Although some works have been
done in such superlattice structures pai1 ; pai2 ; pai3 ; pai4 ; wang , the
analogous representation of metallic multilayered structures which exhibit
several novel features multi1 ; multi2 ; multi3 , no rigorous effort has been
made so far, to the best of our knowledge, to unravel the effect of the
interplay of electron-electron interaction and an imposed external electric
field on electron transport in such systems. Here we show that a traditional
$1$D lattice with electron-electron interaction, evaluated at the Hartree-Fock
(HF) mean field (MF) level, is characterized by a mobility edge behavior at
finite bias voltage. Furthermore, a superlattice structure comprising sites on
which electron-electron interactions are expressed differently (some sites are
interacting and some sites are non-interacting) is characterized by multiple
occurrence of mobility edges at several values of the carrier energy. The
applicability of mean field approximation in such superlattice geometries has
already been reported in a recent work meanfield .

##  II Model and Calculation

We adopt a tight-binding (TB) framework to describe the model quantum system
and numerically calculate two-terminal transport within a mean field

Figure 1: (Color online). A $1$D mesoscopic chain, composed of interacting
(filled black circle) and non-interacting (filled green circle) atomic sites,
is attached to two semi-infinite $1$D metallic electrodes, representing source
and drain.

Hartree-Fock approximation using a Green’s function formalism. Several cases
characterized by different arrangements of interacting and non-interacting
atomic sites in the chain, are analyzed. For these models we calculate the
average density of states (ADOS) and the two-terminal transmission
probability, and find that sharp crossovers from completely opaque to fully or
partly transmitting zones take place at one or more specific electron
energies. This observation suggests the possibility of controlling the
transmission characteristics by gating the transmission zone, and using such
superlattice structures as switching devices.

Let us refer to Fig. 1 where a $1$D mesoscopic chain, composed of non-
interacting and interacting atomic sites, is attached to two semi-infinite
$1$D non-interacting source and drain electrodes. In the arrangement of the
two different atomic sites shown in Fig. 1, $M$ ($M\geq 1$) non-interacting
sites are placed between two interacting sites. Here and in what follows we
make a restriction that interacting atoms are not placed successively. In a
Wannier basis, the TB Hamiltonian for a $N$-site chain reads,

$\displaystyle H_{C}$ $\displaystyle=$
$\displaystyle\sum_{i,\sigma}\epsilon_{i\sigma}c_{i\sigma}^{\dagger}c_{i\sigma}+\sum_{\langle
ij\rangle,\sigma}t\left[c_{i\sigma}^{\dagger}c_{j\sigma}+c_{j\sigma}^{\dagger}c_{i\sigma}\right]$
(1) $\displaystyle+$
$\displaystyle\sum_{i}U_{i}c_{i\uparrow}^{\dagger}c_{i\uparrow}c_{i\downarrow}^{\dagger}c_{i\downarrow}$

where, $c_{i\sigma}^{\dagger}$ ($c_{i\sigma}$) is the creation (annihilation)
operator of an electron at the site $i$ with spin $\sigma$
($=\uparrow,\downarrow$), $t$ is the nearest-neighbor hopping element,
$\epsilon_{i\sigma}$ is the on-site energy of an electron at the site $i$ of
spin $\sigma$ and $U_{i}$ is the strength of on-site Coulomb interaction where
$U_{i}=0$ for the non-interacting sites. In presence of bias voltage $V$
between the two electrodes an electric field is developed and the site
energies become voltage dependent,
$\epsilon_{i\sigma}=\epsilon_{i}^{0}+\epsilon_{i}(V)$, where
$\epsilon_{i}^{0}$ is a voltage independent term. For the ordered chain
$\epsilon_{i}^{0}$ is a

Figure 2: (Color online). Variation of voltage dependent site energies in a
$1$D chain with $300$ lattice sites for three different electrostatic
potential profiles when the bias voltage $V$ is fixed at $0.2$.

constant independent of $i$ that can be chosen zero without loss of
generality, while for the disordered case we select it randomly from a uniform
“Box” distribution function in the range $-W/2$ to $W/2$.

The voltage dependence of $\epsilon_{i}(V)$ reflects the bare electric field
in the bias junction as well as screening due to longer range e-e interaction
not explicitly accounted for in Eq. 1. In the absence of such screening the
electric field is uniform along the chain and $\epsilon_{i}(V)=V/2-iV/(N+1)$.
Below we consider this as well as screened electric field profiles, examples
of which are shown in Fig. 2. We will see that the appearance of multiple
mobility edges in superlattice geometries strongly depends on the existence of
finite bias and on the profile of the bias drop along the chain.

The Hamiltonian for the non-interacting ($U_{i}=0$) electrodes can be
expressed as,

$H_{\mbox{lead}}=\sum_{p}\epsilon_{0}c_{p}^{\dagger}c_{p}+\sum_{<pq>}t_{0}\left(c_{p}^{\dagger}c_{q}+c_{q}^{\dagger}c_{p}\right)$
(2)

with site energy and nearest-neighbor intersite coupling $\epsilon_{0}$ and
$t_{0}$, respectively. These electrodes are directly coupled to the $1$D chain
through the lattice sites $1$ and $N$. The hopping integrals between the
source and chain and between the chain and drain are denoted by $\tau_{S}$ and
$\tau_{D}$, respectively.

In the generalized HF approach san3 ; san4 ; san5 ; kato ; kam , the full
Hamiltonian is decoupled into its up-spin and down-spin components by
replacing the interaction terms by their mean field (MF) counterparts. This
redefines the on-site energies as
$\epsilon_{i\uparrow}^{\prime}=\epsilon_{i\uparrow}+U\langle
n_{i\downarrow}\rangle$ and
$\epsilon_{i\downarrow}^{\prime}=\epsilon_{i\downarrow}+U\langle
n_{i\uparrow}\rangle$ where, $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is
the number operator. With these site energies, the full Hamiltonian (Eq. 1)
can be written in the MF approximation in the decoupled form

$\displaystyle H_{MF}$ $\displaystyle=$
$\displaystyle\sum_{i}\epsilon_{i\uparrow}^{\prime}n_{i\uparrow}+\sum_{\langle
ij\rangle}t\left[c_{i\uparrow}^{\dagger}c_{j\uparrow}+c_{j\uparrow}^{\dagger}c_{i\uparrow}\right]$
(3) $\displaystyle+$
$\displaystyle\sum_{i}\epsilon_{i\downarrow}^{\prime}n_{i\downarrow}+\sum_{\langle
ij\rangle}t\left[c_{i\downarrow}^{\dagger}c_{j\downarrow}+c_{j\downarrow}^{\dagger}c_{i\downarrow}\right]$
$\displaystyle-$ $\displaystyle\sum_{i}U_{i}\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$ $\displaystyle=$
$\displaystyle H_{C,\uparrow}+H_{C,\downarrow}-\sum_{i}U_{i}\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$

where, $H_{C,\uparrow}$ and $H_{C,\downarrow}$ correspond to the effective

Figure 3: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for a $1$D non-interacting
($U_{i}=0$ $\forall$ $i$) ordered ($W=0$) chain with $N=300$ sites. The
electrostatic potential profile varies linearly (red curve in Fig. 2), with
the total potential drop across the chain to be (a) $V=0$ and (b) $V=0.2$.

TB Hamiltonians for the up and down spin electrons, respectively. The last
term provides a shift in the total energy that depends on the mean populations
of the up and down spin states.

Figure 4: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for an ordered ($W=0$) $1$D chain.
The left column corresponds to the case where all sites are interacting
($U_{i}=2$), while the right column represents the results for a $1$D
superlattice geometry where four non-interacting ($U_{i}=0$) atoms are placed
between two interacting ($U_{i}=2$) atoms. The 1st, 2nd and 3rd rows
correspond to $V=0$, $0.1$ and $0.2$, respectively. All these results are
shown for a linear bias drop along the chain.

With these decoupled Hamiltonians ($H_{C,\uparrow}$ and $H_{C,\downarrow}$) of
up and down spin electrons, we start our self consistent procedure considering
initial guess values of $\langle n_{i\uparrow}\rangle$ and $\langle
n_{i\downarrow}\rangle$. For these initial set of values of $\langle
n_{i\uparrow}\rangle$ and $\langle n_{i\downarrow}\rangle$, we numerically
diagonalize the up and down spin Hamiltonians. Then we calculate a new set of
values of $\langle n_{i\uparrow}\rangle$ and $\langle n_{i\downarrow}\rangle$.
These steps are repeated until a self consistent solution is achieved.

The converged mean field Hamiltonian is a sum of single electron up and down
spin Hamiltonians. The transmission function is therefore a sum
$T(E)=\sum_{\sigma}T_{\sigma}(E)$ where datta
$T_{\sigma}={\mbox{Tr}}\left[\Gamma_{S}\,G_{C,\sigma}^{r}\,\Gamma_{D}\,G_{C,\sigma}^{a}\right]$.
Here, $G_{C,\sigma}^{r}$ and $G_{C,\sigma}^{a}$ are the retarded and advanced
Green’s functions, respectively, of the chain including the effects of the
electrodes.
$G_{C,\sigma}=\left(E-H_{C,\sigma}-\Sigma_{S}-\Sigma_{D}\right)^{-1}$, where
$\Sigma_{S}$ and $\Sigma_{D}$ are the self-energies due to coupling of the
chain to the source and drain, respectively, while $\Gamma_{S}$ and
$\Gamma_{D}$ are their imaginary parts.

##  III Results and Discussion

In what follows we limit ourselves to absolute zero temperature and use the
units where $c=h=e=1$. For the numerical calculations we choose $t=1$,
$\epsilon_{0}=0$, $t_{0}=3$ and $\tau_{S}=\tau_{D}=1$. The energy scale is
measured in unit of $t$.

Before addressing the central problem i.e., the possibility of getting
multiple mobility edges in $1$D superlattice geometries, first we explore

Figure 5: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for a $1$D chain ($N=300$) in
absence of disorder ($W=0$) with on-site interaction $U_{i}=2$ and bias
voltage $V=0.2$ that varies linearly along the chain. Here we set $M=5$ i.e,
five non-interacting atoms are placed between two interacting atoms.

the effect of finite bias on electron transport in two simple systems, one for
a standard non-interacting chain and the other for a conventional Hubbard
chain where all sites are interacting.

In Fig. 3 we show the variation of total transmission probability (T) together
with the average density of states as a function of energy $E$ for an ordered
($\epsilon_{i}^{0}=0$ for all atomic sites $i$ in the chain) non-interacting
chain for two different magnitudes of the voltage bias, assuming a linear bias
drop (uniform electric field) across the chain. In the absence of electric
field electron transmission takes place throughout the energy band as clearly
seen from the spectrum Fig. 3(a), since in this case all the energy
eigenstates are extended. On the other hand, when a finite bias drop takes

Figure 6: (Color online). Same as Fig. 5, with $M=6$.

place along the chain, several energy eigenstates appear in the energy regions
around the band edges for which the transmission probability is exactly zero
(Fig. 3(b)). Therefore, the chain appears insulating when Fermi energy is
within the zone of zero transmission, while finite transmission, $T\neq 0$, is
seen more towards the band centre. The sharp transition between these regimes
illustrates the existence of a mobility edge phenomenon under finite bias

Figure 7: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for a $1$D chain ($N=300$) in
presence of disorder ($W=0.5$) for the same parameter values used in Fig.
4(f): $U_{i}=2$, $V=0.2$ (linear potential profile) and $M=4$.

condition. For a finite bias, the localization of energy levels always starts
from the band edges and the width of the localized energy zones can be
controlled by the imposed electric field. Obviously, for strong enough
electric field almost all the energy levels are localized and the extended
energy regions disappear, so that in this particular case metal-insulator (MI)
transition will no longer be observed. This localization phenomenon in
presence of an external electric field has already been established in the
literature, but the central issue of our present investigation - the interplay
between the Hubbard interaction strength, the superlattice configuration and
the electric field has not been addressed earlier.

To explore it, we present in Fig. 4 the results of a traditional Hubbard chain
where all sites are interacting (left column) together with the results of a
superlattice geometry where four non-interacting atoms are placed between two
interacting atoms (right column). The results are shown for three different
values of the voltage bias, taking a linear bias drop along the $1$D chain.
For the chain where all sites are interacting a single energy gap only appears
at the band centre, while in the superlattice geometry, depending on the unit
cell configuration, multiple energy gaps are generated which are clearly
visible from the ADOS spectra. Therefore, in a superlattice geometry, in
presence of external electric field associated with bias voltage $V$ between
two electrodes, zero transmission ($T=0$) energy regions exist, and are
separated by regions of extended states compared to the traditional Hubbard
chain, and, it leads to the possibility of getting an MI-like transition at
multiple energies.

The total number of energy sub-bands in a superlattice geometry for a

Figure 8: (Color online). Transmission probability $T$ (red color) and ADOS
(green color) as a function of energy $E$ for a $1$D chain ($N=300$) with no
disorder ($W=0$). The model parameters are $U=2$, $M=4$ and $V=0.2$, with the
bias potential profile taken as the green curve given of Fig. 2.

particular energy range generated in the ADOS profile strongly depends on
structural details i.e., the number $M$ of non-interacting atoms between two
interacting lattice sites. This is shown in Figs. 5 and 6 which show the ADOS
and the transmission probability for two models that are identical in all
details (see caption to Fig. 5) except that $M=5$ in Fig. 5 and $M=6$ in Fig.
6. These structures show more mobility edge phenomena, that is crossovers
between fully opaque and a transmitting zone, than in the corresponding case
of Fig. 4(f), suggesting a design concept based on such superlattice
structures as a switching devices at multiple energies.

The robustness of the observed behavior can be examined by its sensitivity to
the presence of disorder. Figure 7 displays the ADOS spectrum and the total
transmission probability for a $1$D chain in presence of diagonal disorder
affected by choosing $\epsilon_{i}^{0}$ from a uniform distribution of width
$W=0.5$ ($-0.25$ to $+0.5$). An average over $50$ disorder configurations is
presented. The resulting ADOS and transmission show similar qualitative
features, with sharp transitions between localized and extended spectral
regions as seen above for the ordered cases. Note that the presence of
disorder alone can cause state localization. For strong enough disorder almost
all energy levels get localized, even for such a finite size $1$D chain. In
this limit such crossover behavior will no longer be observed.

In the calculations presented so far we have assumed a linear drop of the
electrostatic potential along the chain. Figures 8 and 9 show results obtained
for an identical chain length with other potential profiles that are
characteristics of stronger screening.

Figure 9: (Color online). Same as Fig. 8, with the electrostatic potential
profile given by the blue curve in Fig. 2.

We see that the localized region gradually decreases with increasing flatness
of the potential profile in the interior of the conducting bridge. If the
potential drop takes place only at the chain-to-electrode interfaces i.e.,
when the potential profile becomes almost flat along the chain the width of
the localized region almost vanishes and the metal-insulator transition is not
observed, as was the case for the zero bias limit.

Finally, we point out that by locating the Fermi energy in appropriate places
of the sub-bands, the system can be used as a p-type or an n-type
semiconductor. For example, let us imagine, at absolute zero temperature, the
Fermi level is fixed in the localized region which is very close to the fully
transmitting zone (right hand side). In this case, the left sub-bands up to
the Fermi level are completely filled with electrons. Now, if the energy gap
between the Fermi level, pinned in the localized region, and the bottom of the
transmitting region (right hand side) is small enough for electrons to hop,
then the system will behave as an n-type semiconductor. On the other hand, by
reverting the situation we can generate a p-type semiconductor where electrons
hop from a filled transmitting zone (valence band) to unoccupied localized
zone (conduction band) generating holes in the valence band.

##  IV Conclusion

To summarize, we have investigated in detail the two-terminal finite bias
electron transport in a $1$D superlattice structure composed of interacting
and non-interacting atoms. The electron-electron interaction is considered in
the Hubbard form, and the Hamiltonian is solved within a generalized HF
scheme. We numerically calculate two-terminal transport by using a Green’s
function formalism and analyze the results for some specific chain structures
characterized by different arrangements of the atomic sites in the chain. Our
analysis may be utilized in designing a tailor made switching device for
multiple values of Fermi energy (or, more practically, for different values of
a gating potential). The sensitivity of this switching action i.e., metal-to-
insulator transition and vice versa on the electric field variation has also
been discussed. Though the results presented in this article are worked out at
absolute zero temperature limit, the results should remain valid even at
finite temperatures ($\sim 300\,$K) since the broadening of the energy levels
of the superlattice structure due to its coupling with the metal electrodes is
much higher than that of the thermal broadening datta .

##  V Acknowledgment

The research of A.N. is supported by the Israel Science Foundation, the
Israel-US Binational Science Foundation, the European Research Council under
the European Union’s Seventh Framework Program (FP7/2007-2013; ERC Grant No.
226628) and the Israel-Niedersachsen Research Fund.

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# Towards Automatically Generating Release Notes using Extractive
Summarization Technique

Sristy Sumana Nath  Computer Science
University of Saskatchewan
Saskatoon, Saskatchewan
Email<EMAIL_ADDRESS>Banani Roy  Computer Science
University of Saskatchewan
Saskatoon, Saskatchewan
Email<EMAIL_ADDRESS>
###### Abstract

Release notes are admitted as an essential document by practitioners. They
contain the summary of the source code changes for the software releases, such
as issue fixes, added new features, and performance improvements. Manually
producing release notes is a time-consuming and challenging task. For that
reason, sometimes developers neglect to write release notes. For example, we
collect data from GitHub with over 1,900 releases, among them 37% of the
release notes are empty. We propose an automatic generate release notes
approach based on the commit messages and merge pull-request (PR) titles to
mitigate this problem. We implement one of the popular extractive text
summarization techniques, i.e., the TextRank algorithm. However, accurate
keyword extraction is a vital issue in text processing. The keyword matching
and topic extraction process of the TextRank algorithm ignores the semantic
similarity among texts. To improve the keyword extraction method, we integrate
the GloVe word embedding technique with TextRank. We develop a dataset with
1,213 release notes (after null filtering) and evaluate the generated release
notes through the ROUGE metric and human evaluation. We also compare the
performance of our technique with another popular extractive algorithm, latent
semantic analysis (LSA). Our evaluation results show that the improved
TextRank method outperforms LSA.

###### Index Terms:

Software release notes, Extractive text summarization, Software maintenance
and documentation

††footnotetext: DOI reference number: 10.18293/SEKE2021-119

##  I Introduction

Software release is a way to deliver the software package, with release notes,
of the stable version to the end-users. Release notes are essential
documentation in software development that contains a set of project
activities, e.g., issue fixes, improvements and new features that have been
implemented to a specific release [18]. Different stakeholders (e.g., software
development teams and external users) might benefit from release notes [12,
19]. For example, the development team members (project manager, team lead,
developer, tester) use them to learn what has changed in the source code to
solve issues or integrate new features [12]. Similarly, integrators, who are
using a library in their code, use the library release notes to decide whether
such a library should be upgraded to the latest release [19]. End-users and
clients read the release notes to decide whether it would be worthwhile to
upgrade to the latest software (e.g., application software or tool) version
[3]. Besides, release notes serve as valuable resources to generate software
documentation (e.g., software release report) submitted to the client.
Moreover, practitioners use release notes in requirements engineering,
software programming, software debugging and testing phase [2].

Generally, more than one developer is working on a single software project
[3]. Generating release notes of a project by an individual is challenging as
it is not feasible to know all fixed issues or the integrated new features of
the project. Therefore, primary release note producers collect information
from contributors to produce the release notes manually [5]. A few studies
have proposed the generation of release notes - some of them are semi-
automated [12], and some are automated techniques [19, 3] using manually-
defined templates. The automated technique ARENA generates release notes by
extracting source code change of Java projects and using some manually pre-
defined templates based on the source code change type [19] (example in Figure
1). The primary limitation of this method is that it fails if the source code
change or code statement is not matched with the pre-defined data, this
approach cannot generate a sentence. Besides, this technique is only helpful
for Java-based software. In an empirical study [5], 108 GitHub contributors
and 206 IT professionals were surveyed for production and usage of release
notes. The survey results reveal that none of the participants used ARENA for
producing release notes. Whereas the existing approach is not suitable to use;
therefore, this is one of the motivations of our study. On the other hand, the
semi-automated technique [12] filters good quality commit messages between two
releases and appends them all to suggest producing release notes. Appending
the commit messages of interrelated releases is not a feasible solution to
generate release notes because it generates a lengthy release note. The
audience wants to read precise release notes [5]. Some text management tools
(e.g., issue trackers) are used to keep track of changes for producing and
managing release notes, still need some considerable manual effort to create
the release notes from these sources [5].

Figure 1: Generated Release Note by ARENA TABLE I: A Release Note in the
Laravel Project Release Note:
---
Add sanctum cookie endpoint to default cors paths
Modify the cache.php docblocks
Add stub handler
Closed auth correctly
Commit messages & merge PR titles:
Commit 1: Update CHANGELOG.md
Commit 2: Merge branch ‘8.x’ of github.com:laravel/laravel into 8.x
Commit 3: Modify the cache.php docblocks
Commit 4: add stub handler
Commit 5: closed auth correctly
Commit 6: add sanctum cookie endpoint to default cors paths
Commit 7: add auth line

Developing new tools could greatly help to improve release notes production
and usage [5]. As discussed above, there is a great lack of automatic release
notes generation tools, which motivates us to develop an automated release
notes generation technique. To work collaboratively, developers upload the
source code in social coding platforms, e.g., GitHub, and can track the
changes of the projects. A release is a collection of several commits, and
commits can be regular commits and merge commits. Table I represents a release
note of Laravel
project††https://github.com/laravel/laravel/releases/tag/v8.4.2 with the
related commit messages and merge pull-request titles. Therefore, commit
messages and merge pull-request titles are valuable artifacts to generate
automated release notes.

We prepare a new dataset that contains the release information (e.g., release
notes, release date) with their commit messages and merge pull-request titles
of 13 projects from GitHub. After data filtering and pre-processing, our
prepared dataset contains over 1,200 release data (detail described in Section
III). In our study, we regard the combination of commit messages and merge
pull-request (PR) titles as an input source and release notes as the summary
of the input sources. We apply the TextRank algorithm; however, traditional
TextRank algorithm ignores the semantic similarities. We integrate word
embedding technique GloVe (Global Vectors for Word Representations) [22] to
solve this problem and analyzed its effectiveness (described in Section V-C).

To evaluate the automatically generated release notes, we also implemented the
Latent Semantic Analysis (LSA) text summarization technique. We evaluated
automated release notes on the dataset using Recall-Oriented Understudy for
Gisting Evaluation (ROUGE) [14] and the F1-score obtained with ROUGE-1,
ROUGE-2, and ROUGE-L is 31.74%, 18.53%, and 26.90%, respectively. We also
conduct a human evaluation to assess the quality of the generated release
notes, which shows that our approach performs significantly better than the
LSA and can generate more high-quality release notes.

In summary, our contributions are:

  * •

We develop a dataset with over 1,200 releases and their commit messages and
merge pull-request titles from GitHub for release notes generation task.

  * •

We implement the TextRank algorithm and integrate the GloVe word embedding
technique to generate release notes for software release from their commit
messages.

  * •

We evaluate this approach on the dataset using the ROUGE metric and human
evaluation.

We describe the research questions and usage scenarios in Section II and data
collection process in Section III. We elaborate our proposed approach in
Section IV and present the evaluation results in Section V. Section VI and VII
discusses threats to validity and the related work. We conclude this paper
with some future plans in Section VIII.

##  II Study Design

Our study aims to answer the following two research questions:

RQ1: What is the vital information that needs to include in the release notes?
We investigate in the GitHub projects’ repositories to understand the contents
of the release note. We find that for the most cases developers prepare a list
of software changes of the current version from the last previous version. The
list of software changes comes from commits and merge pull requests (Table I).
For reason, we collect the commit messages and merge pull requests titles as a
input source to generate automated release notes (detail describe in Section
III).

RQ2: What is the efficient way to generate release notes? As per our previous
discussion, it is clear that no appropriate tool exists for automated release
notes generation in the software development practice. To produce concise
release notes, we apply text summarization techniques. We implemet the
TextRank summarization algorithm by integrating the GloVe model. Using the
GloVe model, we can resolve the semantic similarity issue of the traditional
TextRank algorithm (detail describe in Section IV). For evaluation, we have
used the evaluation metric ROUGE to assess the text quality of the automated
summary.

####  II-1 Usage Scenarios

The release notes may help developers and users to capture a summary of the
latest release. Our approach can help developers to write the good quality of
release notes. The usage scenarios are as follows:

  * •

It is challenging for developers to keep in mind all the changes for the
upcoming release. Therefore, our approach can assist developers in producing
release notes.

  * •

Our approach can help to replace the existing empty release notes in
GitHub††https://github.com/django/django/releases.

Figure 2: Sample Release Note

####  II-2 Problem Formulation

Inspired from the example shown in Table I, we observe the production of the
release notes as a text summarization task with the combination of the commit
messages and merge PR titles in the release as the article and the release
notes as the summary. Therefore, the problem is formulated as follows:

Given a source sequence $S_{seq}$ = $(x_{1},x_{2},...,x_{n})$, $n$th sentences
in source sequence. An extractive summarizer aims to produce a summary
$RN_{gen}$ by selecting $m$ sentences from $S_{seq}$, where $m$ is the length
of existing release notes $RN_{ref}$. We use a scoring function $f()$ to
generate a score for every sentence of $S_{seq}$ and then select top-ranked m
sentences to produce an automated summary.

##  III Dataset

###  III-A Data Preparation:

In our study, we need to develop a dataset of release notes with commit
messages and pull-request titles of the GitHub projects. To eliminate the
trivial projects, we defined three criteria for project selection from GitHub
projects: (i) the project is active (i.e., the repository of the project is
being updated); (ii) the number of release notes in the project is more than
40; and (iii) the project have more than 8,000 stars. Generally, GitHub
projects are classified into six domains [7]: (1) Application software, e.g.,
browsers, text editors; (2) System software, e.g., operating systems, (3) Web
libraries and frameworks, e.g., Web API, (4) Non-web libraries and frameworks,
e.g., android framework, (5) Software tools, like IDEs, and compilers; (6)
Documentation, like documentation, tutorials.

We extract 1,924 release data (see in Figure 2), e.g., release note, hash,
release data and tag name, of five domains except Documentation repositories
of 13 projects (among them 6 Java, 3 Python and 4 PHP projects). We find 711
(37%) release have empty release notes and develop a dataset with 1,213
release data. After that, we extracted the commit messages and merge pull-
request titles of the releases. In GitHub, the new commit is linked with the
previous commit using the parent, i.e., hash value (shown in Figure 3). We
extract all parent hash values between two releases and then collect the
commit messages and merge PR titles. Our study extracts the first sentences
from the commit messages because the first sentences often are the summaries
of the entire commit messages [11]. Similarly, merge PR commits have two
parents (shown in Figure 3(b)), one is previous commit and second is push PR
commit. In our study, we collect all parents hash value between two releases
and then extract the commit messages and merge PR titles using these hash
values. Figure 4 describes the process flow of data collection process. Data
is uploaded in this link††https://github.com/sristysumana/SEKE2021Paper119.

(a) Regular Commit

(b) Merge PR Commit

Figure 3: Commit Figure 4: Data Collection Process

###  III-B Data Pre-processing:

We conduct some text pre-processing techniques in the release notes and source
sequences to filter out noise from the text. First, we eliminate the empty
release notes. Then, we remove the HTML tags from the extracted release notes.
Then, we split the text into sentences using NLTK (Natural Language Toolkit
[6]) and delete the a) url, b) reference number (e.g., “#123”), c) signature,
e,g, signed-off-by or co-authored-by, d) ‘@name’, e) markdown headlines, e.g.,
###Added, by identifying through regular expressions. Then, we filter out some
trivial commit messages (for example, duplicate commit messages in same
releases, ‘merge pull request/branch’, ‘update .gitattributes’ and so on).

Figure 5: Overview of the methodology

##  IV Approach

Automatic text summarization process produces a summary by reducing the text
of a document. Text summarization approaches can be categorized into two
types, extractive and abstractive summary. Extractive summary is created by
extracting the phrases or sentences from the input sources and abstractive
summarization process produces new words by understanding the main content of
the input sources. We use extractive techniques in our study. Usually,
extractive summarization processes follow three steps to generate summary: (a)
intermediate representation of the document, (b) scoring sentences and (c)
strategies for selection of summary sentences. The procedure of our approach
is described in Figure 5.

###  IV-A TextRank Algorithm

TextRank is an unsupervised graph-based ranking algorithm, and it is widely
used in keyword extraction and text summarization. In general, graph-based
ranking algorithms decide the importance of a vertex within a graph based on
global information iteratively drawn from the entire graph. The core idea of
the TextRank algorithm is to split the whole document into sentences. These
sentences are used as nodes, and the weight between the nodes is used as
edges. The weight between two nodes is calculated by similarity function and
transform a similarity matrix among the sentences of the whole document. Then
a text graph is formed, and the score (importance or rank) is computed of each
vertex recursively until convergence.

Given, the source sequence, $D$, is set of $n$ sentences
$\\{S_{1},S_{2},...,S_{n}\\}$ using NLTK package [6], i.e.,
$D=\\{S_{1},S_{2},...,S_{n}\\}$. By using each $S_{i}$ in $D$ as the node and
the similarity between the nodes as the edge, a text graph $G=(D,E,W)$ can be
constructed, where $E\subseteq SXS$ is nonempty finite set of edge between the
nodes, $W$ is the weight set of edges, and $w_{ij}$ is the weight value of
edges between node $S_{i}$ and $S_{j}$. The similarity function (Jaccard
Similarity [17]) for $S_{i},S_{j}$ can be defined as:

$Sim(S_{i},S_{j})=\frac{|\\{t_{k}|t_{k}\epsilon S_{i}\&t_{k}\epsilon
S_{j}\\}|}{log(|S_{i}|)+log(|S_{j}|}$ (1)

According to the $Sim(S_{i},S_{j})$ value, an similarity matrix $SM_{n\times
n}$ is constructed between nodes. The weight of elements on the diagonal of
$SM_{n\times n}$ are all 1. Based on similarity matrix, the TextRank algorithm
generates the importance or score of vertex $S_{i}$ using following equation,

$S_{c}(S_{i})=(1-d)+d*\sum_{S_{j}\epsilon
In(S_{i})}\frac{w_{ji}}{\sum\limits_{S_{k}\epsilon
Out(S_{j})}w_{jk}}Sc(S_{j})$ (2)

Here, $S_{c}(S_{i})$ is the ranking score of the node $S_{i}$, $d$ is the
damping factor that represent the probability of the current node jumping to
any other node, and at the same time can enable the weight to be transferred
to the convergence stably. This co-efficient value can be set between 0 to 1,
however $d$ usually set to 0.85 [17, 21]. $In(S_{i})$ is the collection of all
nodes pointing to node $S_{i}$, $Out(S_{i})$ is the set of all nodes pointed
by node $S_{i}$. The sum of the right side in the equation (2) indicates the
contribution of each adjacent node to the node. The summed numerator $w_{ij}$
represents the degree of similarity between the two nodes $S_{i}$ and $S_{j}$
as well as the denominator is a weighted sum. $S_{c}(S_{j})$ represents the
weight value of node $S_{j}$ after the last iteration. After generating the
score of all sentences, the TextRank algorithm generates a summary by
selecting top-scored sentences.

TABLE II: ROUGE scores Approach | ROUGE 1 | ROUGE 2 | ROUGE-L
---|---|---|---
Recall |  Precision |  F1-Score |  Recall |  Precision |  F1-Score |  Recall |  Precision |  F1-Score
LSA |  28.65% |  30.28% |  29.44% |  14.89% |  14.35% |  14.61% |  23.54% |  25.45% |  24.45%
TextRank+GloVe |  30.23% |  33.42% |  31.74% |  17.29% |  19.97% |  18.53% |  24.29% |  30.15% |  26.90%

###  IV-B Text Vectorization

Text Vectorization is the process of transforming text into numerical
representation. Several ways are existing for the text vectorization, like,
bag-of-words (BoWs), term frequency-inverse document frequency (TF-IDF),
global vector for word representation (GloVe). The traditional TextRank
algorithm converts the document into BoWs vector representation, which is an
unordered collection of word counts [17] and the size of this vector is equal
to the number of words in the vocabulary. The main issue of the BoW models is
that if a sentence comes with the new words, then the vocabulary size would
increase as well as the length of the vectors would increase. For that reason,
the vector representation of the BoW will be a sparse matrix. Sparse
representations are harder to model both for computational reasons and
informational reasons as well as required a huge amount of training data. To
resolve this issue, the modified TextRank algorithm [16] integrates the TF-IDF
algorithm. TF-IDF model reflects the importance of a word to a document in a
collection or corpus. It can keep the relevant words score in the sentences
and also considers the different lengths of the sentence in the document. TF
measures the frequency of the word in each document in the corpus and IDF
calculates the weight of rare words across all documents in the corpus. The
formula of $TF-IDF(w_{i})$ is as follows:

$TF(w_{i})=\frac{f_{Sj}(w_{i})}{f_{Sj}(w)}$
$IDF(w_{i})=\log\frac{N_{s}}{f_{S}(w_{i})}$ $TF-IDF(w_{i})=TF(w_{i})\times
IDF(w_{i})$

where $f_{Sj}(w_{i})$ represents the number of repetition of the word $w_{i}$
in the sentence $s_{j}$, $f_{Sj}(w)$ is the total number of words $w$ in the
sentence $s_{j}$, $N_{s}$ represents the total number of sentences in the
corpus; $f_{S}(w_{i})$ represents the number of sentences containing the word
$w_{i}$.

Still, capturing semantic similarity between the sentences is the issue of the
TextRank algorithm. We observe that generating automated release notes needs
to consider semantic similarity among the input sources (e.g., commit
messages). Therefore, we integrate the GloVe word embedding model with the
TextRank algorithm, which can improve keyword extraction performance by
enhancing the semantic representations of documents. GloVe approach generates
vector representation by calculating the co-occurrence matrix of each word
based on the frequency of word and co-occurrence counts. It primarily gives
information about the frequency of two words $Wi$ and $Wj$ appear together in
the huge corpus. To store this information, GloVe model generates co-
occurrence matrix $X_{w}$, each entry of which corresponds to the number of
times word $j$ occurs in the context of word $i$. As the consequence:

$P_{{Wi}{Wj}}=P(Wj|Wi)=\frac{X_{{Wi}{Wj}}}{X_{Wi}}$

is the probability that word with index $j$ occurs in the context of word $i$.
This co-occurrence probability matrix gives a vector space with meaningful
sub-structures.

##  V Evaluation Result & Discussion

To compare the quality of generated release notes, we implement another text
summarization technique named Latent Semantic Analysis (LSA) [20]. It is an
unsupervised approach, not required to train the model. LSA extracts hidden
semantic structures of words and sentences. An algebraic method, Singular
Value Decomposition (SVD), is used to find out the interrelations between
sentences and words.

TABLE III: Release Note Example I Source Sequence: • made the parameter of
flowable,observable.collect(collector) contravariant on t. • 3.x: fix map()
conditional chain causing npe • suppress undeliverable exception handling in
tests. • fixed image link and added java examples for connectable observable
operators. • 3.x update conditional-and-boolean-operators.md • updating
suppress undeliverable exception rule to have a named inner class instead of
an anonymous inner class. • edit dependency for gradle
---
Reference: • make the collector type of the collect operator contravariant in
t • fix map() conditional chain causing npe • suppress undeliverable exception
handling in tests
LSA: • suppress undeliverable exception handling in tests • 3.x update
conditional-and-boolean-operators.md • updating suppress undeliverable
exception to have a named inner class instead of an anonymous inner class
TextRank+GloVe: • 3.x: fix map() conditional chain causing npe • fixed image
link and added java examples for connectable observable operators • suppress
undeliverable exception handling in tests

###  V-A Evaluation Metric

We evaluate the automated release notes with the ROUGE metric [14], which has
been shown the quality of summarized text by high correlation with human
assessments. Specifically, we use ROUGE-N (N=1,2) and ROUGE-L, which are
widely used to evaluate text summarization systems. The score of ROUGE-N is
based on comparing n-grams in the generated and the original summary. The
recall, precision and F1-score for ROUGE-N are calculated as follows:

$Recall_{n}=\frac{Count_{gram_{n}}(RN_{ref},RN_{gen})}{Count_{gram_{n}}(RN_{ref})}$
$Precision_{n}=\frac{Count_{gram_{n}}(RN_{ref},RN_{gen})}{Count_{gram_{n}}(RN_{gen})}$
$F1_{n}=2\frac{Recall_{n}*Precision_{n}}{Recall_{n}+Precision_{n}}$

where, $Count_{gram_{n}}(RN_{ref},RN_{gen})$ calculates the number of
overlapping n-grams found in both the original and the generated text. The
precision, recall and F1-score for ROUGE-L are similar with those for ROUGE-N,
but instead of n-grams, they are calculated using the longest common
subsequences between generated descriptions and reference descriptions [14].
ROUGE is usually reported as a percentage value between 0 and 100. We obtained
ROUGE scores using the ‘rouge-score’ package [13] with Porter stemmer enabled.

###  V-B Experiment Settings

We use pandas, Python library, for data manipulation and analysis. In text
pre-processing phase, we use nltk library for tokenization and removing
stopwords as well as write some regular expressions in the code to identify
and delete the reference code, markdown, etc. from the text. In this study, we
generate extractive summary for producing automated release notes. Therefore,
we count the number of sentences (n) of reference text and set n is a
parameter in the summarization technique. In order that, the sentence length
of the reference and the generated release notes remain the same. For
implementation, we use several library like, gensim [23], sumy [4], sklearn
[8], networkx [10].

###  V-C Analysis and Discussion

####  V-C1 The Effectiveness of Our Approach

To investigate the effectiveness of our approach (i.e., TextRank+GloVE), we
evaluate the automated release notes with reference text on our dataset and
the results are shown in Table II. TextRank+GloVe model has a higher precision
and F1 score than the LSA model for each ROUGE metric. The improvements
compared with LSA in terms of the three F1 scores are 2.30, 3.91 and 2.44
points, respectively. These results indicate that compared with the two
models, our approach can capture the key points of a release more precisely.
The TextRank approach integrates the GloVe model, for that reason, this
technique can consider semantic similarity between two sentences. On the other
hand, sentence scoring and selection process of LSA depends on co-occurrence
of words. For example, LSA approach extract “suppress undeliverable exception
handling in tests” and “updating suppress undeliverable exception to have a
named inner class instead of an anonymous inner class.” in Table III.
TextRank+GloVe can handle this situation by generating co-occurrence matrix of
words based on the context.

####  V-C2 The Effects of Main Components

In this study, we generate release notes based on the TextRank approach by
integrating GloVe word embedding technique. We compare the automated release
notes between TextRank+GloVe and TextRank+TF-IDF with the reference release
notes (Table IV). TextRank+TF-IDF model extracts keywords based on the term
frequency (TF) and inverse document frequency (IDF) and generate sentence
score based on the high TF-IDF value. In release note example II (Table IV),
TextRank+TF-IDF model identifies error, missing, components as top keywords
and selects top three scored sentences which have these keywords more than the
other sentences. On the other hand, TextRank+GloVe extract better result than
the TextRank+TF-IDF model. Because GloVe is a very powerful word vector
learning technique which does not only rely on local context information of
words, but also incorporates global word co-occurrence to obtain word vectors.
For example, fix…replay and fix…switchmap are co-occurred in two sentences of
source sequence in Table IV. Therefore, our model selects good sentences than
the TextRank+TF-IDF. Table V shows the result by comparing the generated
release notes by TextRank+TF-IDF and TextRank+GloVe model.

TABLE IV: Release Note Example II Source Sequence: • 3.x: add error handling
section to observable.blockingfirst documentation • 3.x: add missing coverage,
update unused/inconsistent ops components • error occur when missing version
tag from single.concatmaps • add more tags those are missing previously •
update 2.x maintenance date, include 3.0 wiki • update readme.md • 3.x: fix
switchmap not canceling properly during onnext-cancel races • 3.x: fix
incorrect sync-fusion of switchmap and error management • 3.x: fix replay not
resetting when the connection is disposed • 3.x fix excess item retention in
the other replay components
---
Reference: • fix switchmap not canceling properly during onnext-cancel races.
• fix replay() not resetting when the connection is disposed. • add error
handling section to observable.blockingfirst documentation.
TextRank + TF-IDF: • 3.x: add missing coverage, update unused/inconsistent ops
components • error occur when missing version tag from single.concatmaps •
3.x: fix incorrect sync-fusion of switchmap and error management
TextRank + GloVe: • 3.x fix excess item retention in the other replay
components • 3.x: fix switchmap not canceling properly during onnext-cancel
races • 3.x: add error handling section to observable.blockingfirst
documentation
TABLE V: Effectiveness of the GloVe model Approach | ROUGE 1 | ROUGE 2 | ROUGE-L
---|---|---|---
Recall |  Precision |  F1-Score |  Recall |  Precision |  F1-Score |  Recall |  Precision |  F1-Score
TextRank+TF-IDF |  26.51% |  28.49% |  27.83% |  13.86% |  14.08% |  13.64% |  22.06% |  24.49% |  23.51%
TextRank+GloVe |  30.23% |  33.42% |  31.74% |  17.29% |  19.97% |  18.53% |  24.29% |  30.15% |  26.90%

####  V-C3 Where Does Our Approach Fail

We carefully inspected the release notes where our approach does not obtain
good F1 scores. We find that our approach usually performs badly because it
presents the top ranking sentences and cannot generate new sentences. The
common case is that the source sequence contains several bug fix commit
messages, however developers described ‘This release fixes few minor issues’
in the release notes to avoid a lengthy release notes. But our approach
extracts a high ranked sentences from the source sequence. This issue is very
common, hence it gets low ROUGE scores. Table VI represents a example where
our approach fails. Additionally, the length of the generated text is a
crucial matter for the poor score, because every sentence has a different word
size. In our implementation, we only consider the total number of sentences,
the ROUGE score would have been better if we had used the number of words to
generate the summaries. However, in the extractive summary techniques, the
word number cannot be used as a parameter. To reduce these limitations, we
will implement sequence-to-sequence model to produce high-quality release
notes in future and train the model.

TABLE VI: Release Note Example III  Source Sequence:
---
fixes failed to disable slave database and fixes unit test errors
fix oracle connection getschema()
fix shadow order
fix test cases and update encrypt strategy name
fix metric image path
fix integration test for select_with_case_expression
Reference:
this release fixes few minor issues
TextRank+GolVe:
fixes failed to disable slave database and fixes unit test errors

###  V-D Human Evaluation

In this section, we invite 16 software engineers and get response from 13
participants to assess the quality of the generated release notes by LSA and
TextRank algorithms. Automatic evaluation metric, ROUGE, calculates the
textual similarity between the existing and the generated release notes, while
the human study can evaluate the semantic similarity between them.

####  V-D1 Procedure

We also conduct a human evaluation to investigate our approach’s
effectiveness. We invited 13 human evaluators to assess the quality of the
automated release notes. All of them are software developers with 3-8 years of
experience and use GitHub. We randomly select 10 release notes from the
dataset and design an online survey. For each release, we show its original
release notes followed by the two automated release notes generated by the
TextRank+GloVe and LSA approaches to the evaluators. The two generated release
notes are randomly ordered. Participants also have no idea about how these
approaches work, so they cannot figure out which description is generated by
which approach. Every evaluator provided a score from 1 to 7 to measure the
semantic similarity between the generated release notes and the reference. The
higher score means the existing release notes are closer to the automated
summaries.

####  V-D2 Result

Each release note obtains thirteen scores from 13 evaluators. We calculate the
score and the Figure 6 shows the final score of human evaluation. Each bar
represents the obtained scores by the approaches in a specific score interval.
For example, the right-most bar shows that 8 participants provide seven score
to our approach. We notice that the generated release notes of TextRank+GloVe
get more 4 to 7 score because it can extract most relevant sentences than the
LSA. But we also notice that TextRank+GloVe generated release notes get the
score between 1 and 2. The reason may be that we do not train our model based
on the existing release notes and hence sometimes may fail to generate high
score of important sentences from source sequence.

$1$$2$$3$$4$$5$$6$$7$$5$$10$$15$$20$$25$score#ResponseLSATextRank+GloVe

Figure 6: Score of Human Evaluation

##  VI Threat to Validity

One threat to validity is that our dataset is built from Java, Python and PHP
projects in GitHub repositories. Therefore, it may not represent all
programming languages. Besides, our approach takes commit messages and merge
pull-request titles as input; hence can also be applied to projects of other
programming languages.

Another threat to validity is that the non-summary information, such as
acknowledgment of contributors, installation command for the new release, in
release notes may affect the effectiveness of our approach. Release notes are
free-form text, and we cannot guarantee their quality and content. We try to
mitigate this threat by using a set of heuristic rules and manual analysis to
filter out non-summary information when pre-processing. However, it is hard to
process the patterns of all non-summary information. For future work, we will
focus on data pre-processing for further improvements.

Another threat is we cannot compare the generated release notes from ARENA and
our approach because ARENA extracts all issues from JIRA to generate release
notes for Java projects and generate lengthy release notes. Moreover, from
survey study, no participants have adopted ARENA for producing release notes
[5].

There is also a threat related to our human evaluation. We cannot guarantee
that each score assigned to every release note is fair. To mitigate this
threat, we invite the professional developers to our survey who have
experience in GitHub and producing release notes.

##  VII Related Study

Several empirical studies focused on the usage of release notes. Abebe et al.
[1] identify nine important factors (e.g., issue priority and type) in
explaining the likehood of an issue being included in release notes. Tingting
et al. [5] analyzed 32,425 release notes and classified the content into eight
categories, e.g., issues fixed, new features, and internal changes. Among
them, issues fixed (79.3% of the release notes) and new features (55.1% of the
release notes) are the most documented categories. In GitHub, developers push
commits or send pull requests in a separate branch to resolve the issues. For
that reason, we propose an automated release notes generation technique based
on the commit messages and merge PR titles.

Some other studies aimed to generate automated release notes. Moreno et al.
[19] propose ARENA tool to produce release notes for Java projects. ARENA
extract the source code changes from GitHub and collect issues from JIRA using
issue-commit linker. Then it prepare a list of issues and generate change
description using predefined text templates. The example is shown in Figure 1.
Similarly, Ali et al. applied the same technique to generate automated RNs for
Node.js projects [3]. Klepper et al. [12] proposed a semi-automatic approach
by collecting information from the build server, issue tracker and version
control system. Then, release manager can edit the this list before
publishing. However, we implemented full-automatic approach to produce release
notes and pre-defined text templates is not required in our study.

For automated software documentation, researchers studies different types of
software artifacts, such as commit messages generation [9, 11] and automated
pull-requests description [15]. For example, Liu et al. [15] propose an
automatic approach to generate PR descriptions based on the commit messages
and the source code comments. On the other hand, release combines plenty of
commits and pull requests. Our work aims to generate release notes from commit
messages and pull-request titles using an extractive method.

##  VIII Conclusion & Future Work

This paper aims to generate release notes for software releases automatically.
In our work, we apply the extractive text summarization technique, i.e.,
TextRank, to produce automated release notes by selecting top-ranking
sentences from their commit messages and merge PR titles. The main novelty of
this work is that we do not use any pre-defined template like ARENA [19]. Our
approach is language-independent; oppositely, ARENA is used for Java projects.
Moreover, we integrate GloVe to convert the text into embeddings, which helps
to extract keywords than the traditional approach. We evaluate the generated
release notes using the ROUGE metric and conduct a human evaluation to check
the effectiveness of the automated release note generation technique.

In future, we plan to apply a sequence-to-sequence model to improve the
quality of release notes. We also plan to improve our approach by involving
additional related software artifacts as input. For example, by taking git
diff, relevant content from bug reports as input and text summarization models
may be able to infer the implementation details and the motivation of a
software release. Moreover, we plan to improve the structure of release notes.
We will categorize, e.g., bug fixes, improvements, new features, the content
of release notes for better representation to release note users.

## Acknowledgment

This research is supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC), and by two Canada First Research Excellence Fund
(CFREF) grants coordinated by the Global Institute for Food Security (GIFS)
and the Global Institute for Water Security (GIWS).

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