# Explicit, recurrent, determinantal expressions of the $k$th power of formal
power series and applications to the generalized Bernoulli numbers

Said Zriaa and Mohammed Mouçouf  Said Zriaa and Mohammed Mouçouf, University
Chouaib Doukkali. Department of Mathematics, Faculty of science Eljadida,
Morocco<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this work, the authors provide closed forms and recurrence expressions for
computing the $k$th power of the formal power series, some of them in terms of
a determinant of some matrices. As a consequence, we obtain the reciprocal of
the unit of any formal power series. We apply these results to the generalized
Bernoulli numbers and Bernoulli numbers, we derive new closed-form expressions
and some recursive relations of these famous numbers. In addition, we present
several identities in determinant form. Using these results, an elegant
generalization of a well known identity of Euler is presented. We also note
some connections between the Stirling numbers of the second kind and the
generalized Bernoulli numbers.

###### Key words and phrases:

Formal power series,Toeplitz-Hessenberg matrix,Trudi’s formula,Euler’s
formula,Combinatorial identity,The Bernoulli numbers,Stirling numbers,Closed
forms,Determinantal expression

###### 2020 Mathematics Subject Classification:

Primary:13F25,15A15,05A19,11B68; secondary:11C20,11Y55,11B73

##  1\. Formal power series

An $n\times n$ Toeplitz-Hessenberg matrix over a unitary commutative ring is
the transpose of the following matrix

$TH_{n}(a_{0},a_{1},\cdots,a_{n})=\begin{pmatrix}a_{1}&a_{2}&\cdots&\cdots&\cdots&a_{n}\\\
a_{0}&\ddots&\ddots&&&\vdots\\\ 0&\ddots&\ddots&\ddots&&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\vdots&&\ddots&\ddots&\ddots&a_{2}\\\
0&\cdots&\cdots&0&a_{0}&a_{1}\end{pmatrix}$

where we assume $a_{0}\neq 0$. This class of matrices has significance in both
pure and applied mathematics. They are encountered in various applications
(see, e.g.,[3, 5, 12, 13]). The determinant of Toeplitz-Hessenberg matrices
has realized significant importance due to their frequent use in several
applications. Using elementary properties of determinants, it is clear that
for all $n\geq 1$:

$\det(TH_{n})=\sum_{i=1}^{n}(-1)^{i-1}a_{0}^{i-1}a_{i}\det(TH_{n-i})\,\,\
\mbox{with}\,\,\ \det(TH_{0})=1.$

In addition to the last recursive formula, there exists an explicit formula
for computing $\det(TH_{n})$, known in the literature as Trudi’s formula [16],
it states that for any positive integer $n$

$\det(TH_{n})=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=l\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}l!\frac{a_{1}^{k_{1}}a_{2}^{k_{2}}\cdots
a_{n}^{k_{n}}}{k_{1}!k_{2}!\cdots k_{n}!}(-a_{0})^{n-l}\,\,\ \mbox{with}\,\,\
k_{1},k_{2},\cdots,k_{n}\geq 0.$

another equivalent formula is

$\det(TH_{n})=\sum_{k=1}^{n}(-a_{0})^{n-k}\sum_{\begin{subarray}{c}i_{1}+i_{2}+\cdots+i_{k}=n\end{subarray}}a_{i_{1}}a_{i_{2}}\cdots
a_{i_{k}}\,\,\ \mbox{with}\,\,\ i_{1},i_{2},\cdots,i_{k}\geq 1.$

Note that when $a_{0}=1$, the formula of Trudi is known as Brioschi’s formula
[16].
Several combinatorial identities involving multinomial coefficients can be
generated using Trudi’s formula. In our results, we may use this identity to
produce various new algebraic identities.
The theory of formal power series still an exciting subject. Therefore, the
formal power series is an essential topic in diverse areas of mathematics due
to their natural appearance in various applications (see [7, 8, 17, 20, 22]
and references therein). They are a powerful tool in the theoretical computer
sciences. The formal power series are closely related to number theory and
many others topics such that special functions and combinatorics (for more
details see [9, 17, 21] and the references cited therein). The foundations of
this important theory were laid by Lagrange, Moivre, L. Euler. For a developed
systematic theory and a rigorous foundation of formal power series, the
elegant expository work of Niven [17] is recommended.
There are several concrete situations that make the formal power series a
useful tool in number theory. In additive number theory, the representation of
some functions by power series is more convenient. For example, the study of
number-partitions using the formal power series provides ingenious results.
The famous mathematician Euler began the theory of partitions in 1748 and he
discovered the ordinary generating function of the well known partition
function $p(n)$ by means of the q-series. G. E. Andrews [1] stated Euler’s
result as

$1+\sum_{n=1}^{\infty}p(n)q^{n}=\prod_{n=1}^{\infty}(1-q^{n})^{-1}$

The pentagonal number theorem is one of Euler’s most profound results, it
relates the product and the power series representations of Euler’s function

$1+\sum_{n=1}^{\infty}a_{n}q^{n}=\prod_{n=1}^{\infty}(1-q^{n})$

where

$\displaystyle a_{n}=\left\\{\begin{array}[]{lll}1,&\text{if}&n=m(3m+1)/2,\,\
m\in\mathbb{Z}\,\ \text{even},\\\ -1,&\text{if}&n=m(3m+1)/2,\,\
m\in\mathbb{Z}\,\ \text{odd},\\\ 0,&\text{otherwise.}\end{array}\right.$

Here the generalized pentagonal numbers are the exponents of nonzero terms.
On the other hand, it is clear that

$\bigg{(}1+\sum_{n=1}^{\infty}(-1)^{n}\bigg{\\{}x^{\frac{3n^{2}-n}{2}}+x^{\frac{3n^{2}+n}{2}}\bigg{\\}}\bigg{)}\bigg{(}1+\sum_{n=1}^{\infty}p(n)x^{n}\bigg{)}=1$

Also by a simple manipulation of formal power series, we can easily obtain
that

$p(m)=\sum_{n=1}^{\infty}(-1)^{n+1}\bigg{\\{}p\bigg{(}m-(3n^{2}-n)/2\bigg{)}+p\bigg{(}m-(3n^{2}+n)/2\bigg{)}\bigg{\\}}$

This identity leads to an efficient method of calculating the partition
function $p(n)$. We note that this formula is known in the large literature as
Euler’s pentagonal number recurrence.
It should be mentioned that by calculating the reciprocal of the ordinary
generating function of $p(n)$, we can express the function $p(n)$ in
determinant formulas. More precisely, using one of our results we have

$p(n)=(-1)^{n}\left|\begin{array}[]{cccccc}a_{1}&a_{2}&\cdots&\cdots&\cdots&a_{n}\\\
1&\ddots&\ddots&&&\vdots\\\ 0&\ddots&\ddots&\ddots&&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\vdots&&\ddots&\ddots&\ddots&a_{2}\\\
0&\cdots&\cdots&0&1&a_{1}\end{array}\right|.$

and

$a_{n}=(-1)^{n}\left|\begin{array}[]{cccccc}p(1)&p(2)&\cdots&\cdots&\cdots&p(n)\\\
1&\ddots&\ddots&&&\vdots\\\ 0&\ddots&\ddots&\ddots&&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\vdots&&\ddots&\ddots&\ddots&p(2)\\\
0&\cdots&\cdots&0&1&p(1)\end{array}\right|.$

Now let $S=a_{0}+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$ be any
formal power series over a unitary commutative ring $R$. For any nonegative
integer $k$, let us write the $k$th power of $S$ as

$S^{k}=a_{0}(k)+a_{1}(k)X+a_{2}(k)X^{2}+\cdots=\sum_{i\geq 0}a_{i}(k)X^{i}$

where the coefficients $a_{m}(k)$ can be calculated recursively by

$\displaystyle\left\\{\begin{array}[]{lll}a_{0}(k)&=&a_{0}^{k},\\\
a_{m}(k)&=&\displaystyle\frac{1}{ma_{0}}\sum_{i=1}^{m}(ik-m+i)a_{i}a_{m-i}(k),\quad
m\geq 1.\end{array}\right.$ (1.4)

This identity is one of the most elegant and oldest formulas used to calculate
the $k$th powers of a given power formal series. It is usually known as the
J.C.P. Miller formula, for example, it appears in the well known book of P.
Henrici [11], but actually, the history of this intersting identity is very
old. In [9] H. W. Gould attributed this formula to Euler and provides
extensive history and though discussion for this beautiful recurrence
relation.
The relation (1.4) provides an efficient way to determine the coefficients
$a_{m}(k)$. However, this method is valid only in the case where $m$ and
$a_{0}$ are units in the ring $R$. The goal here is to propose a new general
approach, which requires no assumption, for computing the $k$th power of
formal power series.
Throughout this section, $R$ will be used to denote a commutative unitary
ring.
Let $k$ and $r$ be two integers and consider the linear map defined by

$[.]_{k}^{r}:\begin{array}[]{rcl}R[X]&\longrightarrow&R[X]\\\
X^{i}&\longmapsto&X^{k-(r-i)}\binom{k}{r-i}\\\ \end{array}$

for $k\leq p$, we use the convention that

$X^{k-p}\binom{k}{p}=\delta_{k,p}$ (1.5)

Then using an important result of [14], we have the following theorem which
gives a closed-form expression of the $k$th power of formal power series.

######  Theorem 1.1.

Let $\\{a_{n}\\}_{n\in\mathbb{N}}$ be a sequence of elements of $R$ such that
$a_{0}$ is invertible. For all integer $k$, the $k$th power of the following
formal power series

$S=a_{0}+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$

are given as follows:

$\displaystyle
S^{k}=[\mathcal{X}_{0}]_{k}^{0}(a_{0})+[\mathcal{X}_{1}]_{k}^{1}(a_{0})X+[\mathcal{X}_{2}]_{k}^{2}(a_{0})X^{2}+\cdots=\sum_{i\geq
0}[\mathcal{X}_{i}]_{k}^{i}(a_{0})X^{i}$ (1.6)

where $\mathcal{X}_{n}$ is the polynomial

$\delta(-X,a_{1},\ldots,a_{n})=\left|\begin{array}[]{cccccc}a_{1}&a_{2}&\cdots&\cdots&\cdots&a_{n}\\\
-X&\ddots&\ddots&&&\vdots\\\ 0&\ddots&\ddots&\ddots&&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\vdots&&\ddots&\ddots&\ddots&a_{2}\\\
0&\cdots&\cdots&0&-X&a_{1}\end{array}\right|.$ (1.7)

and $\mathcal{X}_{0}=1$. On the other hand, we have

$\delta(-X,a_{1},\ldots,a_{n})=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=l\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}l!\frac{a_{1}^{k_{1}}a_{2}^{k_{2}}\cdots
a_{n}^{k_{n}}}{k_{1}!k_{2}!\cdots k_{n}!}X^{n-l}$

more explicitly

$[\mathcal{X}_{n}]_{k}^{n}(a_{0})=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=l\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}k(k-1)\cdots(k-l+1)\frac{a_{1}^{k_{1}}a_{2}^{k_{2}}\cdots
a_{n}^{k_{n}}}{k_{1}!k_{2}!\cdots k_{n}!}a_{0}^{k-l}$

###### Proof.

It is easy to see that the canonical mapping

$\displaystyle[a_{0},a_{1},a_{2},\ldots]\longrightarrow
a_{0}+a_{1}X+a_{2}X^{2}+\cdots$

is an isomorphism from the ring of infinite semicirculant matrices onto the
ring of formal power series. Thus the result follows immediately from [14,
Theorem 2.5]. ∎

######  Remark 1.2.

It should be noted that if $a_{0}$ is not invertible the last theorem remains
valid for calculating the nonnegative kth power of the formal power series. In
this case, we just consider the following map

$[.]_{k}^{r}:\begin{array}[]{rcl}R[X]&\longrightarrow&R(X)\\\
X^{i}&\longmapsto&X^{k-(r-i)}\binom{k}{r-i}\\\ \end{array}$

As a consequence of Theorem (1.1), we recover a well-known formula attributed
in [11] to Wronski and in [16] to Trudi. This foumus formula can also be found
in [12, p.347] and [18, Lemma 2.4]. Due to its importance, there are various
different proofs in the literature. The interested reader is referred for
example to [11, 18, 19]. In the following, we provide a new and elegant proof
of Wronski’s formula using some important algebraic tools.

######  Theorem 1.3.

Let $\\{a_{n}\\}_{n\in\mathbb{N}}$ be a sequence of the elements of $R$ such
that $a_{0}$ is invertible. The inverse of the formal power series

$S=a_{0}+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$

is given by:

$S^{-1}=b_{0}+b_{1}X+b_{2}X^{2}+\cdots=\sum_{i\geq 0}b_{i}X^{i}$

where

$b_{n}=\dfrac{(-1)^{n}}{a_{0}^{n+1}}\left|\begin{array}[]{cccccc}a_{1}&a_{2}&\cdots&\cdots&\cdots&a_{n}\\\
a_{0}&\ddots&\ddots&&&\vdots\\\ 0&\ddots&\ddots&\ddots&&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\vdots&&\ddots&\ddots&\ddots&a_{2}\\\
0&\cdots&\cdots&0&a_{0}&a_{1}\end{array}\right|.$ (1.8)

Moreover $b_{n}$ can be explicitly expressed as

$b_{n}=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=p\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\binom{p}{k_{1},\ldots,k_{n}}(-1)^{p}a_{0}^{-1-p}a_{1}^{k_{1}}a_{2}^{k_{2}}\cdots
a_{n}^{k_{n}}$

###### Proof.

We first prove that for all polynomial $P(X)$, we have

$[P(X)]_{-1}^{r}(a_{0})=\dfrac{(-1)^{r}}{a_{0}^{r+1}}P(-a_{0})$

Now, it is clear that

$[X^{i}]_{-1}^{r}=X^{i-(r+1)}(-1)^{r+i}$

Let $P(X)=\sum_{i=0}^{s}c_{i}X^{i}$, then by linearity

$\displaystyle[P(X)]_{-1}^{r}(a_{0})$
$\displaystyle=\sum_{i=0}^{s}c_{i}[X^{i}]_{-1}^{r}(a_{0})$
$\displaystyle=\dfrac{(-1)^{r}}{a_{0}^{r+1}}\sum_{i=0}^{s}c_{i}(-a_{0})^{i}$
$\displaystyle=\dfrac{(-1)^{r}}{a_{0}^{r+1}}P(-a_{0})$

Consequently, we have

$\displaystyle b_{n}$ $\displaystyle=[\mathcal{X}_{n}]_{-1}^{n}(a_{0})$
$\displaystyle=\dfrac{(-1)^{n}}{a_{0}^{n+1}}\mathcal{X}_{n}(-a_{0})$
$\displaystyle=\dfrac{(-1)^{n}}{a_{0}^{n+1}}\delta(a_{0},a_{1},\ldots,a_{n})$

∎

A direct consequence of Theorem (1.3) is the following

######  Theorem 1.4.

$\Bigg{(}1-\sum_{n=1}^{k}a_{n}X^{n}\Bigg{)}^{-1}=1+\sum_{n\geq 1}b_{n}X^{n}$

where

$b_{n}=\sum_{i_{1}+2i_{2}+\cdots+ki_{k}=n}\frac{(i_{1}+i_{2}+\cdots+i_{k})!}{i_{1}!i_{2}!\cdots
i_{k}!}a_{1}^{i_{1}}a_{2}^{i_{2}}\cdots a_{k}^{i_{k}}$

As a consequence, we can provide the following corollary

######  Corollary 1.5.

For every positive integer $k\geq 1$ and every nonnegative integer $n$, we
have

$\binom{n+k-1}{n}=\sum_{\begin{subarray}{c}i_{1}+i_{2}+\cdots+i_{k}=p\\\
i_{1}+2i_{2}+\cdots+ki_{k}=n\end{subarray}}(-1)^{n-p}\binom{p}{i_{1},\ldots,i_{k}}\binom{k}{1}^{i_{1}}\binom{k}{2}^{i_{2}}\cdots\binom{k}{k}^{i_{k}}$

###### Proof.

It is easy to check that

$(1-X)^{-k}=1+\sum_{n\geq 1}\binom{n+k-1}{n}X^{n}$

On the other hand, we have

$(1-X)^{-k}=\bigg{(}1-\sum_{n=1}^{k}\binom{k}{n}(-1)^{n-1}X^{n}\bigg{)}^{-1}$

From this the result follows easily. ∎

The following theorem provides another way to obtain a closed-form expression
of the $k$th power of formal power series.

######  Theorem 1.6.

Let $\\{a_{n}\\}_{n\geq 0}$ be a sequence of elements of $R$. For all
nonnegative integers $k$, the $k$th power of the formal power series

$S=a_{0}+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$

are given as follows:

$\displaystyle
S^{k}=\boldsymbol{a}_{0}^{(k)}+\boldsymbol{a}_{1}^{(k)}X+\boldsymbol{a}_{2}^{(k)}X^{2}+\cdots=\sum_{i\geq
0}\boldsymbol{a}_{i}^{(k)}X^{i}$

where

$\displaystyle\left\\{\begin{array}[]{lll}\boldsymbol{a}_{0}^{(k)}&=&a_{0}^{k}\binom{k}{0},\\\
\\\
\boldsymbol{a}_{n}^{(k)}&=&\displaystyle\sum_{p=1}^{n}a_{0}^{k-p}\binom{k}{p}\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=p\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\frac{p!}{k_{1}!\ldots
k_{n}!}a_{1}^{k_{1}}\cdots a_{n}^{k_{n}},\quad n\geq 1.\end{array}\right.$
(1.12)

###### Proof.

Taking into account the canonical isomorphism

$\displaystyle[a_{0},a_{1},a_{2},\ldots]\longrightarrow
a_{0}+a_{1}X+a_{2}X^{2}+\cdots$

from the ring of infinite semicirculant matrices onto the ring of formal power
series and Theorem $3.1$ of [15], we easily obtain the result. ∎

The following theorem provides another elegant simple formula for computing
the $k$th power of any formal power series

######  Theorem 1.7.

Let $\\{a_{n}\\}_{n\geq 0}$ be a sequence of elements of $R$. For all
nonnegative integers $k$, the kth power of the formal power series

$S=a_{0}+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$

are given as follows:

$\displaystyle
S^{k}=\boldsymbol{a}_{0}^{(k)}+\boldsymbol{a}_{1}^{(k)}X+\boldsymbol{a}_{2}^{(k)}X^{2}+\cdots=\sum_{i\geq
0}\boldsymbol{a}_{i}^{(k)}X^{i}$

where the sequence $\\{\boldsymbol{a}_{n}^{(k)}\\}_{n\geq 0}$ is determined
by:

$\boldsymbol{a}_{n+1}^{(k)}=a_{n+1}\boldsymbol{\widehat{a}}_{0}^{(k)}+\cdots+a_{i+1}\boldsymbol{\widehat{a}}_{n-i}^{(k)}+\cdots+a_{1}\boldsymbol{\widehat{a}}_{n}^{(k)}$

and

$\displaystyle\left\\{\begin{array}[]{lll}\boldsymbol{\widehat{a}}_{0}^{(k)}&=&a_{0}^{k-1}\binom{k}{1},\\\
\\\
\boldsymbol{\widehat{a}}_{n}^{(k)}&=&\displaystyle\sum_{p=1}^{n}a_{0}^{k-p-1}\binom{k}{p+1}\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=p\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\frac{p!}{k_{1}!\ldots
k_{n}!}a_{1}^{k_{1}}\cdots a_{n}^{k_{n}},\quad n\geq 1.\end{array}\right.$
(1.16)

###### Proof.

By considering the canonical isomorphism between the ring of infinite
semicirculant matrices and the ring of formal power series and equations
$(13)$ and $(16)$ of [15], we easily obtain the desired result. ∎

We continue to provide important result for computing the $k$th power of any
formal power series. In the following, we present a new recursive relation.

######  Theorem 1.8.

Let $\\{a_{n}\\}_{n\geq 0}$ be a sequence of elements of $R$. For all
nonnegative integers $k$, the $k$th power of the formal power series

$S=a_{0}+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$

are given as follows:

$\displaystyle
S^{k}=\boldsymbol{a}_{0}^{(k)}+\boldsymbol{a}_{1}^{(k)}X+\boldsymbol{a}_{2}^{(k)}X^{2}+\cdots=\sum_{i\geq
0}\boldsymbol{a}_{i}^{(k)}X^{i}$

where the sequence $\\{\boldsymbol{a}_{n}^{(k)}\\}_{n\geq 0}$ is determined
recursively by:

$\boldsymbol{a}_{n+1}^{(k)}=a_{n+1}a_{0}^{k-1}\binom{k}{1}+\sum_{i=1}^{n}a_{i}\sum_{l=1}^{k-1}a_{0}^{l-1}\boldsymbol{a}_{n+1-i}^{(k-l)}$

###### Proof.

We have by Theorem(1.7)

$\boldsymbol{a}_{n+1}^{(k)}=a_{n+1}a_{0}^{k-1}\binom{k}{1}+\sum_{i=1}^{n}a_{i}\boldsymbol{\widehat{a}}_{n+1-i}^{(k)}$

Since

$\boldsymbol{\widehat{a}}_{n}^{(k)}=\sum_{p=1}^{n}a_{0}^{k-p-1}\binom{k}{p+1}\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=p\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\frac{p!}{k_{1}!\ldots
k_{n}!}a_{1}^{k_{1}}\cdots a_{n}^{k_{n}}$

Then we have

$\boldsymbol{\widehat{a}}_{n}^{(k)}=\boldsymbol{a}_{n}^{(k-1)}+a_{0}\boldsymbol{\widehat{a}}_{n}^{(k-1)}$

Applying repeatedly the same procedure, we obtain

$\boldsymbol{\widehat{a}}_{n}^{(k)}=\sum_{l=1}^{k-1}a_{0}^{l-1}\boldsymbol{a}_{n}^{(k-l)}+a_{0}^{k-1}\boldsymbol{\widehat{a}}_{n}^{(0)}$

and since $\boldsymbol{\widehat{a}}_{n}^{(0)}=0$, we immediately have

$\boldsymbol{a}_{n+1}^{(k)}=a_{n+1}a_{0}^{k-1}\binom{k}{1}+\sum_{i=1}^{n}a_{i}\sum_{l=1}^{k-1}a_{0}^{l-1}\boldsymbol{a}_{n+1-i}^{(k-l)}$

as claimed. ∎

Another interesting result is given in the following theorem

######  Theorem 1.9.

Let $\\{a_{n}\\}_{n\geq 1}$ be a sequence of elements of $R$ and $k$ be an
integer. Let

$S=1+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$

and

$\displaystyle
S^{k}=1+\boldsymbol{b}_{1}^{(k)}X+\boldsymbol{b}_{2}^{(k)}X^{2}+\cdots=1+\sum_{i\geq
1}\boldsymbol{b}_{i}^{(k)}X^{i}$

Then the sequence $\\{\boldsymbol{b}_{n}^{(k)}\\}_{n\geq 1}$ satisfies the
following relations:

$\boldsymbol{b}_{n}^{(k)}=\sum_{j=0}^{n}\binom{k}{j}\binom{n-k}{n-j}\boldsymbol{b}_{n}^{(j)}$

and

$\binom{n+k}{n}\boldsymbol{b}_{n}^{(k)}=\sum_{j=0}^{n}\binom{n+k}{n+j}\binom{n+j}{n}\binom{n-k}{n-j}\boldsymbol{b}_{n}^{(j)}$

###### Proof.

It is clear that

$S^{k}=\sum_{i\geq 0}\binom{k}{i}(S-1)^{i}=\sum_{i\geq
0}\binom{k}{i}\sum_{j=0}^{i}\binom{i}{j}(-1)^{i-j}S^{j}$

If $[X^{n}]S$ denotes the coefficient of $X^{n}$ in the power series $S$. Then
it is not difficult to check that

$[X^{n}]S^{k}=\sum_{i=0}^{n}\binom{k}{i}\sum_{j=0}^{i}\binom{i}{j}(-1)^{i-j}[X^{n}]S^{j}$

Thus

$\displaystyle\boldsymbol{b}_{n}^{(k)}=\sum_{i=0}^{n}\binom{k}{i}\sum_{j=0}^{i}\binom{i}{j}(-1)^{i-j}\boldsymbol{b}_{n}^{(j)}=\sum_{j=0}^{n}\sum_{i=j}^{n}\binom{k}{i}\binom{i}{j}(-1)^{i-j}\boldsymbol{b}_{n}^{(j)}$

Now, we can write

$\sum_{i=j}^{n}\binom{k}{i}\binom{i}{j}(-1)^{i-j}=\binom{k}{j}\sum_{i=j}^{n}\binom{k-j}{i-j}(-1)^{i-j}$

Using some basic tools, we easily see that

$\sum_{i=j}^{n}\binom{k-j}{i-j}(-1)^{i-j}=\binom{k-j-1}{n-j}(-1)^{n-j}=\binom{n-k}{n-j}$

Then

$\boldsymbol{b}_{n}^{(k)}=\sum_{j=0}^{n}\binom{k}{j}\binom{n-k}{n-j}\boldsymbol{b}_{n}^{(j)}$

as claimed. By the first identity, we have

$\binom{n+k}{n}\boldsymbol{b}_{n}^{(k)}=\sum_{j=0}^{n}\binom{k}{j}\binom{n+k}{n}\binom{n-k}{n-j}\boldsymbol{b}_{n}^{(j)}$

Using the relation

$\binom{k}{j}\binom{n+k}{n}=\binom{n+j}{j}\binom{n+k}{n+j}$

we obtain the second desired formula. ∎

In a commutative unitary ring, it is useful to define the formal derivative of
any formal power series. To do this, if we let $S=\sum_{i\geq 0}a_{i}X^{i}$,
then the formal derivative of $S$ is defined by $S^{{}^{\prime}}=\sum_{i\geq
0}(i+1)a_{i}X^{i}$. For any formal power series $F$ and $G$ the product rule
holds, that is

$(FG)^{{}^{\prime}}=F^{{}^{\prime}}G+FG^{{}^{\prime}}$

To prove this statement, let $F=\sum_{i\geq 0}f_{i}X^{i}$ and $G=\sum_{i\geq
0}g_{i}X^{i}$. Then the derivative of the product $FG=\sum_{i\geq
0}\sum_{j\geq 0}f_{i}g_{j}X^{i+j}$ is

$\displaystyle(FG)^{{}^{\prime}}$ $\displaystyle=\sum_{i\geq 0}\sum_{j\geq
0}(i+j)f_{i}g_{j}X^{i+j-1}$ $\displaystyle=\sum_{i\geq 0}\sum_{j\geq
0}if_{i}g_{j}X^{i-1+j}+\sum_{i\geq 0}\sum_{j\geq 0}f_{i}jg_{j}X^{i+j-1}$
$\displaystyle=F^{{}^{\prime}}G+FG^{{}^{\prime}}$

Using the induction, we can easily prove that for all positive integer $k$

$(F^{k})^{{}^{\prime}}=kF^{{}^{\prime}}F^{k-1}$

If $S=\sum_{i\geq 0}a_{i}X^{i}$ is invertible, then for all integer $k$ it is
easy to check that the derivative of $S^{k}$ is

$(S^{k})^{{}^{\prime}}=kS^{{}^{\prime}}S^{k-1}$

This is a direct consequence of the fact that $S^{k}S^{-k}=1$ and the product
rule.
The theory of formal power series provides an elegant algebraic framework for
several situations. It encodes a great deal of algebraic manipulations often
used to produce stupendously simple proofs of some seemingly difficult
problems. We note that the study of the formal power series in a general
unitary commutative ring is quite difficult, especially in proving some
relations because we have few standard techniques. In contrast, it should be
noted that the formal derivative is an essential tool because we can use it to
provide several striking results. For example, the following result can be
deduced easily using this useful operation, but difficult to prove it using
standard calculus.

######  Theorem 1.10.

Let $\\{a_{n}\\}_{n\in\mathbb{N}}$ be a sequence of elements of $R$ such that
$a_{0}$ is invertible. For all integer $k$, the $k$th power of the following
formal power series

$S=a_{0}+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$

are

$\displaystyle
S^{k}=\boldsymbol{a}_{0}^{k}+\boldsymbol{a}_{1}^{(k)}X+\boldsymbol{a}_{2}^{(k)}X^{2}+\cdots=\sum_{i\geq
0}\boldsymbol{a}_{i}^{(k)}X^{i}$

where the sequence $\\{\boldsymbol{a}_{n}^{(k)}\\}_{n\geq 1}$ satisfies the
following relation:

$n\boldsymbol{a}_{n}^{(k)}=k\sum_{i=1}^{n}ia_{i}\boldsymbol{a}_{n-i}^{(k-1)}.$

###### Proof.

The formal derivative of

$S^{k}=\boldsymbol{a}_{0}^{k}+\boldsymbol{a}_{1}^{(k)}X+\boldsymbol{a}_{2}^{(k)}X^{2}+\cdots=\sum_{n\geq
0}\boldsymbol{a}_{n}^{(k)}X^{n}$

gives

$k\bigg{(}\sum_{n\geq
0}\boldsymbol{a}_{n}^{(k-1)}X^{n}\bigg{)}\bigg{(}\sum_{n\geq
0}(n+1)a_{n+1}X^{n}\bigg{)}=\sum_{n\geq
0}(n+1)\boldsymbol{a}_{n+1}^{(k)}X^{n}$

This yields

$\sum_{n\geq
0}\bigg{(}\sum_{i=0}^{n}k(i+1)a_{i+1}\boldsymbol{a}_{n-i}^{(k-1)}\bigg{)}X^{n}=\sum_{n\geq
0}(n+1)\boldsymbol{a}_{n+1}^{(k)}X^{n}$

Equating the coefficients, we obtain

$\sum_{i=0}^{n}k(i+1)a_{i+1}\boldsymbol{a}_{n-i}^{(k-1)}=(n+1)\boldsymbol{a}_{n+1}^{(k)}$

Then the formula of this theorem follows. ∎

The following theorem is not new, it is Euler’s result introduced at the
beginning of this section. Let us prove this elegant result in a general
context, which is a commutative ring with unity.

######  Theorem 1.11.

Let $\\{a_{n}\\}_{n\in\mathbb{N}}$ be a sequence of elements of $R$ such that
$a_{0}$ is invertible. For all integer $k$, the $k$th power of the following
formal power series

$S=a_{0}+a_{1}X+a_{2}X^{2}+\cdots=\sum_{i\geq 0}a_{i}X^{i}$

are

$S^{k}=\boldsymbol{a}_{0}^{k}+\boldsymbol{a}_{1}^{(k)}X+\boldsymbol{a}_{2}^{(k)}X^{2}+\cdots=\sum_{i\geq
0}\boldsymbol{a}_{i}^{(k)}X^{i}$

where the sequence $\\{\boldsymbol{a}_{n}^{(k)}\\}_{n\geq 0}$ satisfies the
following relation:

$(n+1)a_{0}\boldsymbol{a}_{n+1}^{(k)}=\sum_{i=0}^{n}(k(i+1)-(n-i))a_{i+1}\boldsymbol{a}_{n-i}^{(k)},\,\,\
n\geq 0.$

###### Proof.

We have

$S^{k}=\boldsymbol{a}_{0}^{k}+\boldsymbol{a}_{1}^{(k)}X+\boldsymbol{a}_{2}^{(k)}X^{2}+\cdots=\sum_{n\geq
0}\boldsymbol{a}_{n}^{(k)}X^{n}$

The formal derivative gives

$kS^{k-1}S^{{}^{\prime}}=\sum_{n\geq 0}(n+1)\boldsymbol{a}_{n+1}^{(k)}X^{n}$

If we multiply both sides of this formula by $S$, we obtain

$kS^{k}S^{{}^{\prime}}=\bigg{(}\sum_{n\geq
0}(n+1)\boldsymbol{a}_{n+1}^{(k)}X^{n}\bigg{)}\bigg{(}\sum_{n\geq
0}a_{n}X^{n}\bigg{)}$

Thus

$kS^{k}S^{{}^{\prime}}-\bigg{(}\sum_{n\geq
0}(n+1)\boldsymbol{a}_{n+1}^{(k)}X^{n}\bigg{)}\bigg{(}\sum_{n\geq
1}a_{n}X^{n}\bigg{)}=a_{0}\sum_{n\geq 0}(n+1)\boldsymbol{a}_{n+1}^{(k)}X^{n}$

The expansion of this equation is

$k\bigg{(}\sum_{n\geq
0}\boldsymbol{a}_{n}^{(k)}X^{n}\bigg{)}\bigg{(}\sum_{n\geq
0}(n+1)a_{n+1}X^{n}\bigg{)}-\bigg{(}\sum_{n\geq
0}n\boldsymbol{a}_{n}^{(k)}X^{n}\bigg{)}\bigg{(}\sum_{n\geq
0}a_{n+1}X^{n}\bigg{)}=a_{0}\sum_{n\geq
0}(n+1)\boldsymbol{a}_{n+1}^{(k)}X^{n}$

Simple manipulation gives

$\sum_{n\geq
0}\bigg{(}\sum_{i=0}^{n}k(i+1)a_{i+1}\boldsymbol{a}_{n-i}^{(k)}-a_{i+1}(n-i)\boldsymbol{a}_{n-i}^{(k)}\bigg{)}X^{n}=\sum_{n\geq
0}(n+1)a_{0}\boldsymbol{a}_{n+1}^{(k)}X^{n}$

Therefore

$\sum_{n\geq
0}\bigg{(}\sum_{i=0}^{n}(k(i+1)-(n-i))a_{i+1}\boldsymbol{a}_{n-i}^{(k)}\bigg{)}X^{n}=\sum_{n\geq
0}(n+1)a_{0}\boldsymbol{a}_{n+1}^{(k)}X^{n}$

This gives the desired identity. ∎

##  2\. Applications to Bernoulli numbers and Stirling numbers of the second
kind

The Bernoulli numbers have numerous interesting applications in combinatorics
and number theory. These numbers have been extensively studied and
successfully applied, over the last two centuries, in pure and applied
mathematics. Therefore, a very large literature has been devoted to the study
of Bernoulli numbers. It is practically impossible to provide a nearly
complete bibliography on these well known numbers.
The Bernoulli numbers $B_{n}$ can be defined formally by [7]

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{-1}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}X^{n}$

In the vast literature on Bernoulli numbers, there are several results
concerning explicit representations of these numbers. For an excellent
readable summary and some extremely interesting historical vignettes, the
interested reader is referred to [10]. In addition, there exists a large
number of curious identities involving binomial convolutions of these numbers,
the most known being Euler’s formula

$\sum_{i=0}^{n}\binom{n}{i}B_{i}B_{n-i}=(1-n)B_{n}-nB_{n-1}\,\ \mbox{for}\,\
n\in\\{0,1,2,\ldots\\}$ (2.1)

which is equivalent to

$\sum_{i=2}^{n-2}\binom{n}{i}B_{i}B_{n-i}=-(n+1)B_{n}\,\ \mbox{for}\,\
n\in\\{4,5,6,\ldots\\}$

In mathematics, the Stirling numbers of the second kind arise in a variety of
combinatorics and number theory problems. These useful numbers were introduced
by the famous mathematician James Stirling. An excellent treatment of the
Stirling numbers of the second kind can be found in the well known research
monograph [6].
The Stirling numbers of the second kind $S(n,k)$ can be defined formally by
[4]

$\frac{1}{k!}\Bigg{(}\sum_{n\geq 1}\frac{1}{n!}X^{n}\Bigg{)}^{k}=\sum_{n\geq
k}\frac{1}{n!}S(n,k)X^{n}$

In many situations of number theory and calculus, we encounter the Bernoulli
numbers. For example, these famous numbers are helpful to express the power
sum $1^{m}+2^{m}+\cdots+n^{m}$ as

$1^{m}+2^{m}+\cdots+n^{m}=\frac{1}{m+1}\sum_{j=0}^{m}\binom{m+1}{j}(n+1)^{m-j+1}B_{j},\,\
m\geq 0,n\geq 0.$

This identity is the well known Bernoulli formula. It is important to remark
that the power sum can also be calculated in a direct way using the Stirling
numbers of the second kind.

$1^{m}+2^{m}+\cdots+n^{m}=\sum_{j=0}^{n}\binom{n+1}{j+1}j!S(m,j),\,\ m\geq
0,n\geq 0.$

The remarkable feature of the Bernoulli numbers is the fact that they appear
in the evaluation of the well known Riemann zeta function via the following
striking formulas

$\zeta(2m)=\sum_{n=1}^{\infty}\frac{1}{n^{2m}}=(-1)^{m-1}\frac{(2\pi)^{2m}}{2(2m)!}B_{{}^{2m}},$

and

$\zeta(-m)=-\frac{1}{m+1}B_{m+1}.$

where $m\in\\{1,2,3,\ldots\\}$. The intimate relationship between the zeta
function and the Bernoulli numbers led to the fact that these numbers have
several profound arithmetical properties.
For any integer $m$, the generalized Bernoulli numbers $B_{n}^{(m)}$ can be
defined by means of [7]

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{-m}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(m)}X^{n}$ (2.2)

The generalized Bernoulli numbers are a natural generalization of the
classical Bernoulli numbers. They have various interesting properties in
mathematics. Also, the generalized Bernoulli numbers have remarkable
arithmetic properties see e.g [2]. From the corresponding definitions, we
easily see that

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{n!}B_{n}X^{n}\Bigg{)}^{m}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(m)}X^{n}$ (2.3)

The results of the last section can be used to generate a large variety of
elegant formulas and determinant identities for well known and new algebraic
identities involving Bernoulli numbers and Stirling numbers of the second
kind.

######  Theorem 2.1.

For each integer $n\geq 0$, the Bernoulli numbers can be expressed as

$B_{n}=n!\sum_{\begin{subarray}{c}k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\frac{(k_{1}+k_{2}+\cdots+k_{n})!}{k_{1}!k_{2}!\cdots
k_{n}!}\Bigg{(}\frac{-1}{2!}\Bigg{)}^{k_{1}}\cdots\Bigg{(}\frac{-1}{(n+1)!}\Bigg{)}^{k_{n}}$

Moreover

$B_{n}=(-1)^{n}n!\left|\begin{array}[]{cccccc}\frac{1}{2!}&\frac{1}{3!}&\cdots&\cdots&\cdots&\frac{1}{(n+1)!}\vspace*{0.5pc}\\\
1&\ddots&\ddots&&&\vdots\vspace*{0.5pc}\\\
0&\ddots&\ddots&\ddots&&\vdots\vspace*{0.5pc}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\vspace*{0.5pc}\\\
\vdots&&\ddots&\ddots&\ddots&\frac{1}{3!}\vspace*{0.5pc}\\\
0&\cdots&\cdots&0&1&\frac{1}{2!}\end{array}\right|.$ (2.4)

###### Proof.

Follows easily using Theorem 1.3. ∎

The expression (2.4) is not new, it appeared for example in [12, 18].
Another application of Theorem 1.3 is given in the following result.

######  Theorem 2.2.

Let $n$ be a positive integer, we have

$\frac{1}{(n+1)!}=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=p\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\binom{p}{k_{1},\ldots,k_{n}}(-1)^{p}\bigg{(}\frac{1}{1!}B_{1}\bigg{)}^{k_{1}}\bigg{(}\frac{1}{2!}B_{2}\bigg{)}^{k_{2}}\cdots\bigg{(}\frac{1}{n!}B_{n}\bigg{)}^{k_{n}}$

and

$\frac{(-1)^{n}}{(n+1)!}=\left|\begin{array}[]{cccccc}\frac{1}{1!}B_{1}&\frac{1}{2!}B_{2}&\cdots&\cdots&\cdots&\frac{1}{n!}B_{n}\vspace*{0.5pc}\\\
1&\ddots&\ddots&&&\vdots\vspace*{0.5pc}\\\
0&\ddots&\ddots&\ddots&&\vdots\vspace*{0.5pc}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\vspace*{0.5pc}\\\
\vdots&&\ddots&\ddots&\ddots&\frac{1}{2!}B_{2}\vspace*{0.5pc}\\\
0&\cdots&\cdots&0&1&\frac{1}{1!}B_{1}\end{array}\right|.$

###### Proof.

By definition, we have

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{n!}B_{n}X^{n}\bigg{)}^{-1}=1+\sum_{n\geq
1}\frac{1}{(n+1)!}X^{n}$

Taking into account Theorem 1.3, the identity of this theorem and its
determinant expression follows. ∎

######  Theorem 2.3.

Let $m$ be an integer and $n$ be a nonnegative integers. Then

$\frac{1}{n!}B_{n}^{(m)}=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=l\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}(-1)^{l}\binom{m+l-1}{l}\frac{l!}{k_{1}!k_{2}!\cdots
k_{n}!}\prod_{i=1}^{n}\Bigg{(}\frac{1}{(i+1)!}\Bigg{)}^{k_{i}}$

###### Proof.

Using Formula (2.2) and a direct application of Theorem (1.1) gives

$\frac{1}{n!}B_{n}^{(m)}=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=l\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\binom{-m}{l}l!\frac{1}{k_{1}!k_{2}!\cdots
k_{n}!}\prod_{i=1}^{n}\Bigg{(}\frac{1}{(i+1)!}\Bigg{)}^{k_{i}}$

Since $\binom{-m}{l}(-1)^{l}=\binom{m+l-1}{l}$, then we obtain the result. ∎

######  Theorem 2.4.

Let $m$ be an integer and $n$ be a nonnegative integer, then

$\frac{1}{n!}B_{n}^{(m)}=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=l\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\binom{m}{l}\frac{l!}{k_{1}!k_{2}!\cdots
k_{n}!}\prod_{i=1}^{n}\bigg{(}\frac{1}{i!}B_{i}\bigg{)}^{k_{i}}$

###### Proof.

Formula (2.2) tells us that

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{n!}B_{n}X^{n}\Bigg{)}^{m}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(m)}X^{n}$

and the result of Theorem (1.1) gives

$\frac{1}{n!}B_{n}^{(m)}=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=l\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\binom{m}{l}l!\frac{1}{k_{1}!k_{2}!\cdots
k_{n}!}\prod_{i=1}^{n}\bigg{(}\frac{1}{i!}B_{i}\bigg{)}^{k_{i}}$

and so the result is shown. ∎

######  Theorem 2.5.

Let $n,m$ be two positive integers, then

$\frac{(-1)^{n}}{n!}B_{n}^{(m)}=\left|\begin{array}[]{cccccc}\frac{1}{1!}B_{1}^{(-m)}&\frac{1}{2!}B_{2}^{(-m)}&\cdots&\cdots&\cdots&\frac{1}{n!}B_{n}^{(-m)}\vspace*{0.5pc}\\\
1&\ddots&\ddots&&&\vdots\vspace*{0.5pc}\\\
0&\ddots&\ddots&\ddots&&\vdots\vspace*{0.5pc}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\vspace*{0.5pc}\\\
\vdots&&\ddots&\ddots&\ddots&\frac{1}{2!}B_{2}^{(-m)}\vspace*{0.5pc}\\\
0&\cdots&\cdots&0&1&\frac{1}{1!}B_{1}^{(-m)}\end{array}\right|.$

where

$\frac{1}{n!}B_{n}^{(m)}=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=l\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}(-1)^{l}\frac{l!}{k_{1}!k_{2}!\cdots
k_{n}!}\bigg{(}\frac{1}{1!}B_{1}^{(-m)}\bigg{)}^{k_{1}}\bigg{(}\frac{1}{2!}B_{2}^{(-m)}\bigg{)}^{k_{2}}\cdots\bigg{(}\frac{1}{n!}B_{n}^{(-m)}\bigg{)}^{k_{n}}$

###### Proof.

By Formula (2.2), it is clear that

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{m}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(-m)}X^{n}$

Then we obtain the following identity

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{-m}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(m)}X^{n}$

this identity entails

$\Bigg{(}1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(-m)}X^{n}\Bigg{)}^{-1}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(m)}X^{n}$

Therefore the result follows directly from Theorem 1.3. ∎

######  Theorem 2.6.

Let $n$ and $k$ be two positive integer such that $n\geq k\geq 1$. We have the
following well known identity

$S(n,k)=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=k\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\frac{n!}{k_{1}!k_{2}!\cdots
k_{n}!}\Bigg{(}\frac{1}{1!}\Bigg{)}^{k_{1}}\Bigg{(}\frac{1}{2!}\Bigg{)}^{k_{2}}\cdots\Bigg{(}\frac{1}{n!}\Bigg{)}^{k_{n}}$

Accordingly, one has

$S(n+1,k+1)=\frac{1}{k+1}\sum_{i=1}^{n+1-k}\binom{n+1}{i}S(n+1-i,k)$

###### Proof.

Using Theorem 1.1, we have

$\Bigg{(}\sum_{n\geq 1}\frac{1}{n!}X^{n}\Bigg{)}^{k}=\sum_{n\geq
0}[\mathcal{X}_{n}]_{k}^{n}(0)X^{n}$

On the other hand, it is clear that

$[\mathcal{X}_{n}]_{k}^{n}(0)=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=k\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\frac{k!}{k_{1}!k_{2}!\cdots
k_{n}!}\Bigg{(}\frac{1}{1!}\Bigg{)}^{k_{1}}\Bigg{(}\frac{1}{2!}\Bigg{)}^{k_{2}}\cdots\Bigg{(}\frac{1}{n!}\Bigg{)}^{k_{n}}$

Note that if $n<k$, then $[\mathcal{X}_{n}]_{k}^{n}(0)=0$. Consequently,
equating the coefficients of $x^{n}$, we obtain

$\frac{1}{n!}S(n,k)=\sum_{\begin{subarray}{c}k_{1}+k_{2}+\cdots+k_{n}=k\\\
k_{1}+2k_{2}+\cdots+nk_{n}=n\end{subarray}}\frac{1}{k_{1}!k_{2}!\cdots
k_{n}!}\Bigg{(}\frac{1}{1!}\Bigg{)}^{k_{1}}\Bigg{(}\frac{1}{2!}\Bigg{)}^{k_{2}}\cdots\Bigg{(}\frac{1}{n!}\Bigg{)}^{k_{n}}$

Therefore, we get the first identity. The second identity follows directly
from Lemma $2.2$ of [15]. ∎

One of the most appealing aspects of the Stirling numbers of the second kind
is their connection with the generalized Bernoulli numbers

######  Theorem 2.7.

For all $k\geq 0$ and $n\geq 1$

$\sum_{i=0}^{n}\binom{n+k}{n-i}B_{n-i}^{(k)}S(i+k,k)=0$

In particular

$\sum_{i=0}^{n}\binom{n+1}{n-i}B_{n-i}=0$

###### Proof.

By definition, we have

$\frac{1}{k!}X^{k}\Bigg{(}1+\sum_{n\geq
1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{k}=\sum_{n\geq k}\frac{1}{n!}S(n,k)X^{n}$

It is not difficult to check

$\Bigg{\\{}1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(k)}X^{n}\Bigg{\\}}\Bigg{\\{}\sum_{n\geq
0}\frac{1}{(n+k)!}S(n+k,k)X^{n}\Bigg{\\}}=\frac{1}{k!}$

Therefore

$\sum_{n\geq
0}\Bigg{(}\sum_{i=0}^{n}\frac{1}{(n-i)!}B_{n-i}^{(k)}\frac{1}{(i+k)!}S(i+k,k)\Bigg{)}X^{n}=\frac{1}{k!}$

This completes the proof. ∎

######  Theorem 2.8.

For all $k\geq 0$ and $n\geq 1$

$\binom{n+k}{k}B_{n}^{(-k)}=S(n+k,k)$

and

$\sum_{i=0}^{n}\binom{n+k}{n-i}\binom{i+k}{i}B_{n-i}^{(k)}B_{i}^{(-k)}=0$

###### Proof.

Formula (2.2) tells us that for any integer $m$, the generalized Bernoulli
numbers can be defined formally by

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{m}=\sum_{n\geq
0}\frac{1}{n!}B_{n}^{(-m)}X^{n}$

On the other hand, we have

$\frac{1}{k!}X^{k}\Bigg{(}1+\sum_{n\geq
1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{k}=\sum_{n\geq k}\frac{1}{n!}S(n,k)X^{n}$

Therefore

$\sum_{n\geq 0}\frac{1}{k!n!}B_{n}^{(-k)}X^{n+k}=\sum_{n\geq
0}\frac{1}{(n+k)!}S(n+k,k)X^{n+k}$

This completes the proof of the first statement. The second statement is a
consequence of Theorem(2.7) ∎

######  Theorem 2.9.

For all $k\geq 1$ and $n\geq 1$, we have

$B_{n}^{(k)}=\binom{k}{1}B_{n}+\sum_{i=1}^{n-1}\binom{n}{i}B_{i}\sum_{l=1}^{k-1}B_{n-i}^{(k-l)}$

and

$B_{n}^{(-k)}=\binom{k}{1}\frac{1}{n+1}+\sum_{i=1}^{n-1}\frac{1}{i+1}\binom{n}{i}\sum_{l=1}^{k-1}B_{n-i}^{(-k-l)}$

###### Proof.

The first identity follows easily from Theorem(1.8) and Formula (2.3). To
prove the second identity, we first applied Theorem(1.8) to Formula (2.2) and
we obtain

$\frac{1}{n!}B_{n}^{(-k)}=\binom{k}{1}\frac{1}{(n+1)!}+\sum_{i=1}^{n-1}\frac{1}{(i+1)!}\sum_{l=1}^{k-1}\frac{1}{(n-i)!}B_{n-i}^{(-k-l)}$

which implies

$B_{n}^{(-k)}=\binom{k}{1}\frac{1}{n+1}+\sum_{i=1}^{n-1}\frac{1}{(i+1)i!}\frac{n!}{(n-i)!}\sum_{l=1}^{k-1}B_{n-i}^{(-k-l)}$

Therefore

$B_{n}^{(-k)}=\binom{k}{1}\frac{1}{n+1}+\sum_{i=1}^{n-1}\frac{1}{i+1}\binom{n}{i}\sum_{l=1}^{k-1}B_{n-i}^{(-k-l)}$

By the above, this theorem is proved. ∎

The following result is an immediate consequence of Theorem (1.9).

######  Theorem 2.10.

Let $k$ be an integer. Then the generalized Bernoulli numbers satisfy the
following two formulas

$B_{n}^{(k)}=\sum_{j=0}^{n}\binom{k}{j}\binom{n-k}{n-j}B_{n}^{(j)},\,\ n\geq
0,$

and

$\binom{n+k}{n}B_{n}^{(k)}=\sum_{j=0}^{n}\binom{n+k}{n+j}\binom{n+j}{n}\binom{n-k}{n-j}B_{n}^{(j)},\,\
n\geq 0.$

######  Theorem 2.11.

Let $k$ be an integer. Then the generalized Bernoulli numbers satisfy the
following

$B_{n+1}^{(k)}=k\sum_{i=0}^{n}\binom{n}{i}B_{i+1}B_{n-i}^{(k-1)}.$

and

$B_{n+1}^{(-k)}=k\sum_{i=0}^{n}\binom{n}{i}\frac{1}{i+2}B_{n-i}^{(-k-1)}.$

###### Proof.

This result is a direct application Theorem (1.10) and the following relations

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{n!}B_{n}X^{n}\Bigg{)}^{k}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(k)}X^{n}$

and

$\Bigg{(}1+\sum_{n\geq 1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{k}=1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(-k)}X^{n}$

∎

######  Theorem 2.12.

Let $k$ be an integer. Then the generalized Bernoulli numbers satisfy the
following

$B_{n+1}^{(k)}=\sum_{i=0}^{n}\bigg{(}k\binom{n}{i}-\binom{n}{i+1}\bigg{)}B_{i+1}B_{n-i}^{(k)}.$

$B_{n+1}^{(-k)}=\sum_{i=0}^{n}\bigg{(}k\binom{n}{i}-\binom{n}{i+1}\bigg{)}\frac{1}{i+2}B_{n-i}^{(-k)}.$

###### Proof.

Taking into account Theorem (1.11) and Formula (2.3), we have the following
relation

$\frac{1}{n!}B_{n+1}^{(k)}=\sum_{i=0}^{n}(k(i+1)-(n-i))\frac{1}{(i+1)!}B_{i+1}\frac{1}{(n-i)!}B_{n-i}^{(k)}.$

which implies

$B_{n+1}^{(k)}=\sum_{i=0}^{n}\bigg{(}\frac{kn!(i+1)}{(i+1)!(n-i)!}-\frac{n!(n-i)}{(i+1)!(n-i)!}\bigg{)}B_{i+1}B_{n-i}^{(k)}.$

and

$B_{n+1}^{(k)}=\sum_{i=0}^{n}\bigg{(}\frac{kn!}{i!(n-i)!}-\frac{n!}{(i+1)!(n-i-1)!}\bigg{)}B_{i+1}B_{n-i}^{(k)}.$

this is just the first statement. The proof of the second statement can be
verified by proceeding as in the proof of the first identity with the aid of
Formula (2.2). ∎

In the following result, we provide a more interesting and potentially useful
recursive formula for the generalized Bernoulli numbers. This recurrence
formula allows us to compute these numbers efficiently. Also, this recursive
expression can be used to generate interesting useful identities. For example,
we use it to provide a generalization of a known identity due to Euler.

######  Theorem 2.13.

Let $k$ be an integer, not equal to $0$. We have for all positive integer
$n\geq 1$

$B_{n}^{(k+1)}=\bigg{(}1-\frac{n}{k}\bigg{)}B_{n}^{(k)}-nB_{n-1}^{(k)}.$

###### Proof.

Let

$b_{k}(X)=\Bigg{(}1+\sum_{n\geq 1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{-k}$

Taking the derivative, we obtain

$b_{k}^{{}^{\prime}}(X)=-k\Bigg{(}1+\sum_{n\geq
1}\frac{1}{(n+1)!}X^{n}\Bigg{)}^{-k-1}\Bigg{(}\sum_{n\geq
0}\frac{n+1}{(n+2)!}X^{n}\Bigg{)}$

Thus

$Xb_{k}^{{}^{\prime}}(X)=-kb_{k+1}(X)\Bigg{(}\sum_{n\geq
1}(\frac{1}{n!}-\frac{1}{(n+1)!})X^{n}\Bigg{)}$

and hence

$\displaystyle Xb_{k}^{{}^{\prime}}(X)$
$\displaystyle=-kXb_{k+1}(X)\Bigg{(}\sum_{n\geq
1}\frac{1}{n!}X^{n-1}\Bigg{)}+kb_{k+1}(X)\Bigg{(}\sum_{n\geq
1}\frac{1}{(n+1)!}X^{n}\Bigg{)}$
$\displaystyle=-kXb_{k+1}(X)\Bigg{(}1+\sum_{n\geq
1}\frac{1}{(n+1)!}X^{n}\Bigg{)}+kb_{k+1}(X)\Bigg{(}-1+1\sum_{n\geq
1}\frac{1}{(n+1)!}X^{n}\Bigg{)}$
$\displaystyle=-kXb_{k}(X)-kb_{k+1}(X)+kb_{k}(X)$

Therefore

$kb_{k+1}(X)=k(1-X)b_{k}(X)-Xb_{k}^{{}^{\prime}}(X)$

We know that

$b_{k}(X)=1+\sum_{n\geq 1}\frac{1}{n!}B_{n}^{(k)}X^{n}$

It follows that

$k\Bigg{(}1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(k+1)}X^{n}\Bigg{)}=k(1-X)\Bigg{(}1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(k)}X^{n}\Bigg{)}-X\Bigg{(}\sum_{n\geq
0}\frac{1}{n!}B_{n+1}^{(k)}X^{n}\Bigg{)}$

By comparing the coefficients, we easily obtain

$kB_{n}^{(k+1)}=kB_{n}^{(k)}-knB_{n-1}^{(k)}-nB_{n}^{(k)}\,\ \mbox{for}\,\
n\geq 1$

and

$B_{n}^{(k+1)}=\bigg{(}1-\frac{n}{k}\bigg{)}B_{n}^{(k)}-nB_{n-1}^{(k)}\,\
\mbox{for}\,\ n\geq 1.$

as claimed. ∎

It may be of great interest to provide a natural generalization of Euler’s
identity (2.1) cited at the beginning of this section. The purpose of the
following result is such generalization identity.

######  Theorem 2.14.

Let $k$ be an integer, not equal to $0$, and $n$ a nonnegative integer. The
generalized Bernoulli numbers and the Bernoulli numbers can be related by the
following relation

$\sum_{i=0}^{n}\binom{n}{i}B_{i}B_{n-i}^{(k)}=\bigg{(}1-\frac{n}{k}\bigg{)}B_{n}^{(k)}-nB_{n-1}^{(k)},\,\
n\geq 0$

or equivalently

$\sum_{i=2}^{n-2}\binom{n}{i}B_{i}B_{n-i}^{(k)}=-\frac{n}{k}B_{n}^{(k)}-\frac{1}{2}nB_{n-1}^{(k)}+\frac{1}{2}knB_{n-1}-B_{n},\,\
n\geq 4.$

###### Proof.

We start this proof by writing the following obvious identity

$\Bigg{(}1+\sum_{n\geq
1}\frac{1}{n!}B_{n}^{(k+1)}X^{n}\Bigg{)}=\Bigg{(}1+\sum_{n\geq
1}\frac{1}{n!}B_{n}X^{n}\Bigg{)}\Bigg{(}\sum_{n\geq
0}\frac{1}{n!}B_{n}^{(k)}X^{n}\Bigg{)}$

from this, we can derive

$B_{n}^{(k+1)}=\sum_{i=0}^{n}\binom{n}{i}B_{i}B_{n-i}^{(k)}$

On the other hand by Theorem (2.13), we get

$\sum_{i=0}^{n}\binom{n}{i}B_{i}B_{n-i}^{(k)}=\bigg{(}1-\frac{n}{k}\bigg{)}B_{n}^{(k)}-nB_{n-1}^{(k)}$

Therefore we reach the desired formula. ∎

## References

  * [1] G. E. Andrews, Euler’s pentagonal number theorem, Math. Magazine, 56, 279–284 (1983).
  * [2] L. Carlitz. Some properties of the Nörlund polynomial $B_{n}^{(x)}$, Math. Nachr. 33, 297-311 (1967).
  * [3] X. W. Chang, M. J. Gander, S. Karaa, Asymptotic properties of the QR factorization of banded Hessenberg–Toeplitz matrices. Numer. Linear Algebra Appl. 12 (2005), 659–682.
  * [4] C. A. Charalambides. Enumerative combinatorics. Chapman and Hall/CRC, 2002.
  * [5] H. Chen, Bernoulli numbers via determinants, Internat. J. Math. Ed. Sci. Tech. 34, 291–297 (2003).
  * [6] L. Comtet. Advanced combinatorics: the art of finite and infinite expansions. Reidel, Dordrecht 1974.
  * [7] G. Everest, A. Van Der Poorten, I. Shparlinski, and T. Ward. Recurrence sequences, American Mathematical Society, Mathematical Surveys and Monographs 104, 2003.
  * [8] I. M. Gessel. Lagrange inversion. J. Combin. Theory Ser. A, 144, 212–249 (2016).
  * [9] H. W. Gould. Coefficient identities for powers of Taylor and Dirichlet series. Amer. Math. Monthly. 81, 3-14 (1974).
  * [10] H. W. Gould. Explicit formulas for Bernoulli numbers. Amer. Math. Monthly. 79, 44-51 (1972).
  * [11] P. Henrici. Applied and computational complex analysis. Vol. 1, Wiley, New York, 1974.
  * [12] A. Inselberg, On determinants of Toeplitz-Hessenberg matrices arising in power series, J. Math. Anal. Appl. 63 (1978), 347–353.
  * [13] A. J. E. M. Janssen, Asymptotics of the Perron-Frobenius eigenvalue of nonnegative Hessenberg-Toeplitz matrices, IEEE Trans. Inf. Theor., 35 (1989), 1340–1344.
  * [14] M. Mouçouf. A closed-form expression for the kth power of semicirculant and r-circulant matrices. arXiv:2006.16198[math.RA](2020).
  * [15] M. Mouçouf. Arbitrary positive powers of semicirculant and $r$-circulant matrices. Linear and multilinear algebra. DOI:10.1080/03081087.2021.1968329 (2021).
  * [16] T. Muir. The theory of determinants in the historical order of development. Four volumes, Dover Publications, New York, 1960.
  * [17] I. Niven. Formal power series. Amer. Math. Monthly, 76, 871-889 (1969).
  * [18] F. Qi and R. J. Chapman. Two closed forms for the Bernoulli polynomials. J. Number Theory 159, 89–100 (2016).
  * [19] F. Qi. On signs of certain Toeplitz-Hessenberg determinants whose elements involve Bernoulli numbers. Contrib. Discrete Math. 18 (2023), no. 1, in press.
  * [20] J. Riordan. Combinatorial identities, Wiley, New York, 1968.
  * [21] B. Sambale, An invitation to formal power series. Jahresber. Dtsch. Math.-Ver. 125, 3-69 (2023).
  * [22] D. Zeitlin. On convoluted numbers and sums. Amer. Math. Monthly, 74, 235-246 (1967).


11institutetext: Politecnico di Torino, Torino Italy
11email<EMAIL_ADDRESS>
22institutetext: University of Texas at San Antonio, San Antonio, TX USA
22email<EMAIL_ADDRESS>
# Estimating Earthquake Magnitude in
Sentinel-1 Imagery via Ranking

Daniele Rege Cambrin equal contribution11 0000-0002-5067-2118
Isaac Corley* 22 0000-0002-9273-7303
Paolo Garza 11 0000-0002-1263-7522
Peyman Najafirad 22 0000-0001-9671-577X

###### Abstract

Earthquakes are commonly estimated using physical seismic stations, however,
due to the installation requirements and costs of these stations, global
coverage quickly becomes impractical. An efficient and lower-cost alternative
is to develop machine learning models to globally monitor earth observation
data to pinpoint regions impacted by these natural disasters. However, due to
the small amount of historically recorded earthquakes, this becomes a low-data
regime problem requiring algorithmic improvements to achieve peak performance
when learning to regress earthquake magnitude. In this paper, we propose to
pose the estimation of earthquake magnitudes as a metric-learning problem,
training models to not only estimate earthquake magnitude from Sentinel-1
satellite imagery but to additionally rank pairwise samples. Our experiments
show at max a $30\%+$ improvement in MAE over prior regression-only based
methods, particularly transformer-based architectures. Code, experimental
results, and model checkpoints are openly available at
github.com/isaaccorley/earthquake-monitoring-by-ranking.

###### Keywords:

Remote Sensing Earthquakes Disaster Management Magnitude Estimation

##  1 Introduction

(a) Magnitude 5.14

(b) Magnitude 5.20

(c) Magnitude 5.58

(d) Magnitude 5.61

(e) Magnitude 6.09

(f) Magnitude 6.48

(g) Magnitude 6.54

(h) Magnitude 6.56

(i) Magnitude 6.69

Figure 1: QuakeSet samples varied by earthquake magnitude. Each sample
contains a pair of pre and post earthquake event Sentinel-1 (SAR) imagery
containing 2 bands (VV & VH). The samples in this figure are plotted as false
color images (VV, VH, VV/VH) along with their magnitudes.

Earthquakes are one of the most devastating natural disasters, causing
significant loss of life, property, and economic damage [9]. Accurate and
timely detection and magnitude estimation of earthquakes are crucial for
effective disaster response and mitigation. Traditionally, this task relies on
an extensive network of physical seismic stations [16]. However, the
installation and maintenance of these stations are resource-intensive and
logistically challenging, limiting their global coverage.

With the advent of satellite technology, earth observation data has become
increasingly accessible [19], providing a promising alternative for monitoring
seismic activity. Leveraging this data, machine learning models can be
developed to detect and estimate the magnitude of earthquakes on a global
scale [4]. Despite this potential, the application suffers from the limited
amount of historical earthquake data combined with satellite data, which poses
a low-data regime problem that could limit the performance of regression
models.

To overcome this challenge, we propose a novel approach that redefines the
estimation of earthquake magnitudes as a metric-learning problem. Instead of
solely focusing on regression, our method trains models to not only predict
earthquake magnitudes from Sentinel-1 satellite imagery, but also to rank
pairwise samples based on their relative magnitudes. This dual-objective
training enhances the model’s ability to generalize from limited data,
improving its accuracy and robustness.

Our contributions are twofold and can be summarized as follows:

  1. 1.

We formulate earthquake magnitude estimation as a metric-learning problem,
incorporating both regression and ranking objectives to facilitate learning.

  2. 2.

We apply this solution to several state-of-the-art convolutional and
transformer based models, demonstrating a significant improvement in Mean
Absolute Error (MAE) of over 30%.

##  2 Related Works

### Remote Sensing with SAR imagery

Synthetic Aperture Radar (SAR) remote sensing has emerged as a critical
technology for Earth observation, offering unique capabilities to capture
high-resolution images under all-weather conditions, day or night. SAR imaging
techniques have significantly improved, enhancing the data’s quality and
applicability. Due to its effectiveness under many weather conditions, SAR
imagery was a choice for topographic mapping [1] and detecting ground
deformation with high precision [20]. In agriculture, the use of SAR to
monitor crop growth and soil moisture levels proves to be critical for
precision farming [17]. In hazard management, it proved effective in many use
cases like analyzing cyclones [15], floods [3], and wildfires [2].

### Machine Learning in Seismology

Interest in leveraging machine learning and deep learning techniques in
seismology is growing [5]. Machine learning methods are mainly employed for
real-time earthquake detection through seismic waves (P-waves and S-waves).
Additionally, these techniques help distinguish between earthquakes and
microtremors [12], and are employed in phase picking [16, 22]. Current
approaches include the use of convolutional neural networks [11, 22],
generative adversarial networks [12], and transformers [16] directly on
seismic wave data. While Sentinel-1 imagery has been explored for earthquake
analysis, it traditionally involved manual analysis rather than automated
systems [8]. However, recent research by Cambrin et al. [4] introduced the
QuakeSet dataset, which uses bi-temporal Sentinel-1 imagery for earthquake
detection, demonstrating the potential for automated solutions with SAR
imagery.

##  3 Methodology

### Magnitude Estimation

The magnitude estimation task can be formulated as follows. We have a set of
time-series $\mathcal{T}$, where each time-series $T_{s}\in\mathcal{T}$ is
composed of $N$ images, with $N=2$, of size $W\times H\times C$ related to the
same spatial area $S$ at different timesteps $\\{T_{1},\dots,T_{N}\\}$ and a
ground truth value $G_{t}\in[0...M_{m}]$ (where $M_{m}$ is the maximum value
for the given magnitude scale). Given a training dataset $D_{tr}$, composed of
a set of pairs $(T_{s},G_{t})$, we train a machine learning model $M$ to
regress the magnitude $G_{t}$. Given a test dataset $D_{ts}$, composed of a
set of $T_{s}$, we can regress $G_{t}$ for each sample $T_{s}$ using $M$.

### Margin Ranking Loss

We employed Margin Ranking loss $L_{MR}$ to encourage the model to distinguish
between high-magnitude samples and lower-magnitude ones. The loss is expressed
in the following way:

$\displaystyle L_{MR}(x_{1},x_{2},y)=max(0,-y*(x_{1}-x_{2})+m)$ (1)
$\displaystyle y_{i}=\begin{cases}1&\text{if }x_{1i}>=x_{2i}\\\ -1&\text{if
}x_{1i}<x_{2i}\end{cases}$ (2)

where $x_{1}$ is a vector of predicted magnitudes for a set $S_{1}$ of images,
$x_{2}$ are the predicted magnitudes for a set $S_{2}$ and $m$ is the margin.
$y$ is a vector (of elements $y_{i}$) which assume values $1$ or $-1$ based on
the ranking between samples from $x_{1}$ and $x_{2}$. The final loss is
expressed as the sum of the Mean Squared Error (MSE) loss, $L_{MSE}$, and
Margin Ranking loss, $L_{MR}$:

$L=L_{MSE}+L_{MR}$ (3)

##  4 Experiments

### Models

To have a more complete overview of the possible architectures, we train a
mixture of convolutional (ConvNeXt v2[21] and MobileNet[10]) and transformer-
based (MobileViT v2[14] and ViT-tiny[7]) models. This selection is further
motivated to establish if model size is important in estimating earthquake
magnitude.

### Experimental Settings

Our experiments utilize the TorchGeo [18] implementation of the QuakeSet
dataset [4] which contains 3,327 pairs of 2-channel Sentinel-1 imagery (VV and
VH polarizations) of size $512\times 512$ sampled at different timesteps,
$T_{1}$ and $T_{2}$, before and after an earthquake, respectively.

The models were trained with the AdamW optimizer [13] for 10 epochs and a
batch size of 16. We employed a linear learning rate scheduler with a warmup
of 10% of the total training steps and a peak learning rate of $\alpha=1e-4$.
All models are initialized using ImageNet [6] pretrained weights. We
preprocess the raw Sentinel-1 imagery by converting to decibels and use random
horizontal and vertical flip augmentations with a probability of 0.5. We set
ranking loss margin $m$ to 0.02. Each image pair was concatenated along the
channel axis to get samples of size $4\times 512\times 512$.

### Results

Table 1: Results on the QuakeSet dataset for Earthquake Magnitude Regression with ($\checkmark$) and without margin ranking loss. Adding a margin ranking objective significantly improves the performance across all architectures. Model | # Params $\downarrow$ | MFLOPs $\downarrow$ | $L_{MR}$ | MAE $\downarrow$
---|---|---|---|---
ConvNeXtv2 - Atto | 3.4M | 373.08 |  | 0.2577
$\checkmark$ | 0.2287
MobileNet-S | 1.5M | 44.10 |  | 0.4540
$\checkmark$ | 0.3678
MobileNet-L | 4.2M | 155.33 |  | 0.5465
$\checkmark$ | 0.5414
MobileViTv2-100 | 4.4M | 964.60 |  | 0.3539
$\checkmark$ | 0.3383
ViT-T | 5.7M | 1541.56 |  | 0.3150
$\checkmark$ | 0.2613

We analyzed the models trained with $L_{MR}$ compared to models trained only
with $L_{MSE}$ to understand the contribution of the additional term. We
compare performance using the Mean Absolute Error (MAE) between the estimated
and ground truth magnitudes as well as the FLOPS computed on an Intel(R)
Core(TM) i9-10980XE CPU.

In Table 1, we can see this approach proves to be effective in any case, both
with CNNs and ViTs. ConvNeXt provides the best results overall, particularly
employing $L_{MR}$, followed by ViT. Although, it is still worse than ConvNeXt
without $L_{MR}$. ViT attention increases resource consumption, which makes
the network less competitive than a convolutional neural network.

MobileViT, thanks to the hybrid approach, provides good results with less
FLOPS than ViT. Although this solution is still not competitive with
convolutional solutions. MobileNet-Small (Mobilenet-S) proves to be a
competitive solution by achieving a low MAE and the least amount of FLOPS. In
comparison, MobileNet-Large (MobileNet-L) instead tends to overfit, obtaining
the worst performance and margin loss, $L_{MR}$.

##  5 Conclusion

In this paper, we modified the task of learning to regress earthquake
magnitudes from satellite imagery as a ranking problem. Our method benefits
from requiring no additional data other than the existing magnitude labels,
proving that the rankings between the predictions inside a batch can be
employed to better add distance between samples with different magnitudes.

## References

  * [1] Copernicus dem (2022). https://doi.org/10.5270/esa-c5d3d65, http://dx.doi.org/10.5270/ESA-c5d3d65
  * [2] Ban, Y., Zhang, P., Nascetti, A., Bevington, A.R., Wulder, M.A.: Near real-time wildfire progression monitoring with sentinel-1 sar time series and deep learning. Scientific Reports 10(1) (Jan 2020). https://doi.org/10.1038/s41598-019-56967-x, http://dx.doi.org/10.1038/s41598-019-56967-x
  * [3] Bonafilia, D., Tellman, B., Anderson, T., Issenberg, E.: Sen1floods11: A georeferenced dataset to train and test deep learning flood algorithms for sentinel-1. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops (June 2020)
  * [4] Cambrin, D.R., Garza, P.: Quakeset: A dataset and low-resource models to monitor earthquakes through sentinel-1. Proceedings of the 21th International Conference on Information Systems for Crisis Response and Management (In Press)
  * [5] Chandrakumar, C., Prasanna, R., Stephens, M., Tan, M.L., Holden, C., Punchihewa, A., Becker, J.S., Jeong, S., Ravishan, D.: Algorithms for detecting p-waves and earthquake magnitude estimation: Initial literature review findings (2023)
  * [6] Deng, J., Dong, W., Socher, R., Li, L.J., Li, K., Fei-Fei, L.: Imagenet: A large-scale hierarchical image database. In: 2009 IEEE conference on computer vision and pattern recognition. pp. 248–255. Ieee (2009)
  * [7] Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., Houlsby, N.: An image is worth 16x16 words: Transformers for image recognition at scale. In: International Conference on Learning Representations (2021), https://openreview.net/forum?id=YicbFdNTTy
  * [8] Funning, G.J., Garcia, A.: A systematic study of earthquake detectability using sentinel-1 interferometric wide-swath data. Geophysical Journal International 216(1), 332–349 (2019)
  * [9] Hodder, A.P.W.: The cost of earthquakes. Journal of Geoscience Education 45(3), 225–228 (May 1997). https://doi.org/10.5408/1089-9995-45.3.225, http://dx.doi.org/10.5408/1089-9995-45.3.225
  * [10] Howard, A., Sandler, M., Chu, G., Chen, L.C., Chen, B., Tan, M., Wang, W., Zhu, Y., Pang, R., Vasudevan, V., et al.: Searching for mobilenetv3. In: Proceedings of the IEEE/CVF international conference on computer vision. pp. 1314–1324 (2019)
  * [11] Khan, I., Kwon, Y.W.: P-detector: Real-time p-wave detection in a seismic waveform recorded on a low-cost mems accelerometer using deep learning. IEEE Geoscience and Remote Sensing Letters 19, 1–5 (2022)
  * [12] Liu, H., Li, S., Song, J.: Discrimination between earthquake p waves and microtremors via a generative adversarial network. Bulletin of the Seismological Society of America 112(2), 669–679 (2022)
  * [13] Loshchilov, I., Hutter, F., et al.: Fixing weight decay regularization in adam. arXiv preprint arXiv:1711.05101 5 (2017)
  * [14] Mehta, S., Rastegari, M.: Separable self-attention for mobile vision transformers (2022)
  * [15] Mouche, A., Soulat, F., Potin, P., Martino, L.: Sentinel-1 contribution to tropical cyclones observations at high resolution. In: IGARSS 2019 - 2019 IEEE International Geoscience and Remote Sensing Symposium. pp. 5472–5475 (2019). https://doi.org/10.1109/IGARSS.2019.8898584
  * [16] Mousavi, S.M., Ellsworth, W.L., Zhu, W., Chuang, L.Y., Beroza, G.C.: Earthquake transformer—an attentive deep-learning model for simultaneous earthquake detection and phase picking. Nature communications 11(1), 3952 (2020)
  * [17] Sishodia, R.P., Ray, R.L., Singh, S.K.: Applications of remote sensing in precision agriculture: A review. Remote Sensing 12(19), 3136 (Sep 2020). https://doi.org/10.3390/rs12193136, http://dx.doi.org/10.3390/rs12193136
  * [18] Stewart, A.J., Robinson, C., Corley, I.A., Ortiz, A., Ferres, J.M.L., Banerjee, A.: Torchgeo: deep learning with geospatial data. In: Proceedings of the 30th international conference on advances in geographic information systems. pp. 1–12 (2022)
  * [19] Torres, R., Snoeij, P., Geudtner, D., Bibby, D., Davidson, M., Attema, E., Potin, P., Rommen, B., Floury, N., Brown, M., et al.: Gmes sentinel-1 mission. Remote sensing of environment 120, 9–24 (2012)
  * [20] Wang, L., Marzahn, P., Bernier, M., Ludwig, R.: Sentinel-1 insar measurements of deformation over discontinuous permafrost terrain, northern quebec, canada. Remote Sensing of Environment 248, 111965 (Oct 2020). https://doi.org/10.1016/j.rse.2020.111965, http://dx.doi.org/10.1016/j.rse.2020.111965
  * [21] Woo, S., Debnath, S., Hu, R., Chen, X., Liu, Z., Kweon, I.S., Xie, S.: Convnext v2: Co-designing and scaling convnets with masked autoencoders. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 16133–16142 (2023)
  * [22] Zhu, W., Tai, K.S., Mousavi, S.M., Bailis, P., Beroza, G.C.: An end-to-end earthquake detection method for joint phase picking and association using deep learning. Journal of Geophysical Research: Solid Earth 127(3), e2021JB023283 (2022)