# Faulhaber’s Formula, Odd Bernoulli Numbers, and the Method of Partial
Sums††thanks: This work is licensed under the CC BY 4.0, a Creative Commons
Attribution License.

Ryan Zielinski
<EMAIL_ADDRESS>
(4 August 2022)

###### Abstract

Let “Faulhaber’s formula” refer to an expression for the sum of powers of
integers written with terms in n(n+1)/2. Initially, the author used
Faulhaber’s formula to explain why odd Bernoulli numbers are equal to zero.
Next, Cereceda gave alternate proofs of that result and then proved the
converse, if odd Bernoulli numbers are equal to zero then we can derive
Faulhaber’s formula. Here, the original author will give a new proof of the
converse using the method of partial sums and mathematical induction.

##  1 Motivation

If we knew nothing of the history of the problem and tried to discover for
ourselves a general expression for

$\sum_{k=1}^{n}k^{m}=1^{m}+2^{m}+\cdots+n^{m},$

where $n,m$ are positive integers, we might notice there appear to be two ways
to write such sums. For example,

$\displaystyle\sum k=$
$\displaystyle\frac{n(n+1)}{2}=\frac{1}{2}n^{2}+\frac{1}{2}n,$ (1)
$\displaystyle\sum k^{2}=$
$\displaystyle\frac{2n+1}{3}\cdot\frac{n(n+1)}{2}=\frac{1}{3}n^{3}+\frac{1}{2}n^{2}+\frac{1}{6}n,$
$\displaystyle\sum k^{3}=$
$\displaystyle\left(\frac{n(n+1)}{2}\right)^{2}=\frac{1}{4}n^{4}+\frac{1}{2}n^{3}+\frac{1}{4}n^{2}.$

The next two cases are

$\displaystyle\sum k^{4}=$
$\displaystyle\frac{1}{5}\left[6\cdot\frac{n(n+1)}{2}-1\right]\cdot\sum
k^{2}=\frac{1}{5}n^{5}+\frac{1}{2}n^{4}+\frac{1}{3}n^{3}-\frac{1}{30}n,$ (2)
$\displaystyle\sum k^{5}=$
$\displaystyle\frac{1}{3}\left[4\cdot\frac{n(n+1)}{2}-1\right]\left(\sum
k\right)^{2}=\frac{1}{6}n^{6}+\frac{1}{2}n^{5}+\frac{5}{12}n^{4}-\frac{1}{12}n^{2}.$

(When no confusion will arise, we will abbreviate $\sum_{k=1}^{n}k^{m}$ by
$\sum k^{m}$.)

We can write each sum using terms of $\frac{n(n+1)}{2}$ or $n$. Of course, if
we have the former then we always can expand it into the latter. Do we always
have the former?

At a later point, reading up on the matter we would learn that writing an
expression for $\sum k^{m}$ using terms in $n$ is associated with the name of
Jakob Bernoulli (1654-1705), and writing the same expression using terms in
$\frac{n(n+1)}{2}$ is associated with that of Johann Faulhaber (1580-1635).
Bernoulli’s contribution has long overshadowed Faulhaber’s, but now we know
the two are linked inextricably.

##  2 Background

In order to write an expression for $\sum k^{m}$ in $n$, we introduce the
Bernoulli numbers. Set $B_{0}=1$ and define $B_{n}$ by

$\sum_{k=0}^{n}\binom{n+1}{k}B_{k}=0,$

where $n\geq 1$. Then we can write

$\sum
k^{m}=\frac{1}{m+1}\sum_{j=0}^{m}\left(-1\right)^{j}\binom{m+1}{j}B_{j}n^{m+1-j}.$
(3)

For a sum in $\frac{n(n+1)}{2}$, for even powers we have

$\sum
k^{2m}=\left[c_{0}\left(\frac{n(n+1)}{2}\right)^{m-1}+\cdots+c_{m-2}\cdot\frac{n(n+1)}{2}+c_{m-1}\right]\cdot\sum
k^{2},$ (4)

and for odd powers we have

$\sum
k^{2m+1}=\left[a_{0}\left(\frac{n(n+1)}{2}\right)^{m-1}+\cdots+a_{m-2}\cdot\frac{n(n+1)}{2}+a_{m-1}\right]\left(\sum
k\right)^{2},$ (5)

where the $c_{i},a_{i}$ are rational numbers and $m\geq 1$. We will refer to
these two expressions as “Faulhaber’s formula.”

Regarding earlier work on the topic, Edwards [6, 7] took a matrix-based
approach and looked for recurrence relations amongst all such sums (4) and
(5). The coefficients $c_{i},a_{i}$ then followed as entries in the inverses
of such matrices. Gessel and Viennot [8, section 12] investigated sums of
powers and alternating sums of powers. Explicit expressions for both sets of
coefficients were derived and their combinatorial properties were discussed.
Knuth [12] looked at a lot of material. Perhaps the most important part was
examining Faulhaber’s original work and placing it into a modern perspective.

Now we begin with the present contribution, which aimed to unravel just what
allows for writing $\sum k^{m}$ with terms in $\frac{n(n+1)}{2}$. But, we
started in the opposite direction.

If we look at the expressions in (1) and (2), we notice a few powers of $n$
are missing. The reason is because odd Bernoulli numbers are equal to zero:
$B_{1}=-\frac{1}{2}$, but for all $m\geq 1$, $B_{2m+1}=0$. When we write such
sums using (3), the powers of $n$ which have odd Bernoulli numbers for
coefficients drop out.

There are established ways to prove such a result ([13, Chapters 1-2]). The
new insight of Zielinski in [15] was that, with the different expressions for
$\sum k^{m}$, we already have enough information to justify such an outcome.

If we write $\sum k^{2m+1}$ in the two forms of (3) and (5), the coefficients
for the terms in $n$ must agree. Since (5) contains a factor of $\left(\sum
k\right)^{2}=\frac{1}{4}n^{4}+\frac{1}{2}n^{3}+\frac{1}{4}n^{2}$, there is no
term of $n$. That means the last term of (3), $-B_{2m+1}\cdot n$, must be
equal to zero. In other words, $B_{2m+1}=0$ for all $m\geq 1$.111If we write
(4) in a manner analogous to (2.5) of [15], we can give the same type of
argument for even powers.

To prove the converse, Cereceda [5] chose a different line of attack and
introduced Bernoulli polynomials:

$B_{m}\left(x\right)=\sum_{j=0}^{m}\binom{m}{j}B_{j}x^{m-j},$ (6)

where $x$ is a real variable and $B_{m}\left(0\right)=B_{m}$. This allows an
expression for the sum of powers to be written as

$\sum k^{m}=\frac{1}{m+1}\left(B_{m+1}(n+1)-B_{m+1}\right).$ (7)

In the new context, the property $B_{2m+1}=0$ is related to Bernoulli
polynomials being evaluated at $x=\frac{1}{2}$. The notion of symmetry allows
such polynomials to be rewritten with terms of $\left(x-\frac{1}{2}\right)$,
which can be rewritten once more with terms of $\frac{x(x-1)}{2}$. Once this
is done, (7) leads immediately to Faulhaber’s formula. (Again, [5] contains
alternate proofs of the main result of [15].)

Together, the papers lead to a surprising revelation, one which has been a
long time in the making. Denote $\sum_{k=1}^{n}k^{m}$ by $S_{m}$. Then we have

######  Theorem 1 (Zielinski [15] and Cereceda [5]).

For positive integers $m$,

$B_{2m+1}=0\iff\begin{cases}S_{2m}=&S_{2}\cdot F_{2m}\left(S_{1}\right),\\\
S_{2m+1}=&S_{1}^{2}\cdot F_{2m+1}\left(S_{1}\right),\end{cases}$

where $F_{2m}\left(S_{1}\right)$ and $F_{2m+1}\left(S_{1}\right)$ are
polynomials in $S_{1}=\frac{n(n+1)}{2}$.

In this paper we will give a different proof of the converse. Our approach
will be based on what commonly is referred to as the method of partial sums:

$\sum_{k=1}^{n}k^{m+1}=\left(n+1\right)\sum_{k=1}^{n}k^{m}-\sum_{k=1}^{n}\sum_{l=1}^{k}l^{m}.$

We state it in a pointed fashion to illustrate that it serves as a generator
for sums of powers ([14, Section 2]).222This exact form of the method of
partial sums is said to go back 1,000 years to ibn al-Haytham (965-1039). Katz
[11] and Boudreaux [2] discuss some of the history. In this context, when we
use (3) to write expressions for $\sum\sum l^{m}$, the property $B_{2m+1}=0$
will cause the bulk of the terms to be of even or odd parity like $\sum
k^{m+1}$.

##  3 Main Result

We start with a proof of our method of partial sums.

######  Proposition 1.

For integers $n,m$, where $n\geq 1$ and $m\geq 0$,

$\sum_{k=1}^{n}k^{m+1}+\sum_{k=1}^{n}\sum_{l=1}^{k}l^{m}=\left(n+1\right)\sum_{k=1}^{n}k^{m}.$

###### Proof.

A way to prove $\frac{n(n+1)}{2}=\sum_{k=1}^{n}k$ is to write

$n(n+1)=2\left(1+2+\cdots+n\right)$

and then to interpret the left side as a rectangle of area $n(n+1)$ and the
right side as two pieces of $\left(1+2+\cdots+n\right)$ squares. We will do
something analogous in this case.

Let us write $n+1$ rows of

$\begin{array}[]{c}1^{m}+2^{m}+\cdots+\left(n-1\right)^{m}+n^{m}\\\
1^{m}+2^{m}+\cdots+\left(n-1\right)^{m}+n^{m}\\\ \vdots\\\
1^{m}+2^{m}+\cdots+\left(n-1\right)^{m}+n^{m}\\\
1^{m}+2^{m}+\cdots+\left(n-1\right)^{m}+n^{m}\end{array}$

and then divide the sum into

$\displaystyle 1^{m}+2^{m}+\cdots+\left(n-1\right)^{m}+n^{m}$ $\displaystyle
1^{m}$ $\displaystyle 2^{m}+\cdots+\left(n-1\right)^{m}+n^{m}$ $\displaystyle
1^{m}+2^{m}$ $\displaystyle+$ $\displaystyle\vdots$ $\displaystyle\vdots$
$\displaystyle\left(n-1\right)^{m}+n^{m}$ $\displaystyle
1^{m}+2^{m}+\cdots+\left(n-1\right)^{m}$ $\displaystyle n^{m}$ $\displaystyle
1^{m}+2^{m}+\cdots+\left(n-1\right)^{m}+n^{m}$

This gives us

$\left(n+1\right)\sum_{k=1}^{n}k^{m}=\sum_{k=1}^{n}\sum_{l=1}^{k}l^{m}+\sum_{k=1}^{n}k^{m+1}.$

∎

Before we prove the converse we state, without proof, a lemma which points out
critical, intermediate relationships.

######  Lemma 1.

For positive integers $n$,

$\displaystyle\left(n+\frac{1}{2}\right)\left(\sum k\right)^{2}=$
$\displaystyle\frac{3}{2}\cdot\frac{n(n+1)}{2}\cdot\sum k^{2},$
$\displaystyle\left(n+\frac{1}{2}\right)\sum k^{2}=$
$\displaystyle\left(\frac{4}{3}\cdot\frac{n(n+1)}{2}+\frac{1}{6}\right)\sum
k.$

A consequence of the second relationship is that $\left(\sum k^{2}\right)^{2}$
can be rewritten using only terms of $\sum k$, which then implies, through
(4), that $\left(\sum k^{2m}\right)^{2}$ can be rewritten in $\sum k$ as well
([1, Section 3]). We have

$\left(\sum k^{2}\right)^{2}=G\left(S_{1}\right)\;\textrm{and}\;\left(\sum
k^{2m}\right)^{2}=H\left(S_{1}\right),$ (8)

where $G\left(S_{1}\right)$ and $H\left(S_{1}\right)$ are polynomials in
$S_{1}=\frac{n(n+1)}{2}$.

Now we give a new proof of the converse.

######  Proposition 2.

If odd Bernoulli numbers are equal to zero, we can derive Faulhaber’s formula.

###### Proof.

We proceed by mathematical induction. Expression 3, with $B_{3}=0,$ allows us
to write

$\displaystyle\sum_{k=1}^{n}k^{2}=$
$\displaystyle\frac{1}{3}n^{3}+\frac{1}{2}n^{2}+\frac{1}{6}n=\frac{2n+1}{3}\cdot\frac{n(n+1)}{2}=1\cdot\sum
k^{2},$ $\displaystyle\sum_{k=1}^{n}k^{3}=$
$\displaystyle\frac{1}{4}n^{4}+\frac{1}{2}n^{3}+\frac{1}{4}n^{2}-0\cdot
n=\left(\frac{n(n+1)}{2}\right)^{2}=1\cdot\left(\sum k\right)^{2}.$

We will assume (4) and (5) are true for all $1,2,...,m$ and then establish the
case of $m+1$.

For $m+1$, Proposition 1 tells us

$\sum_{k=1}^{n}k^{2m+2}=\left(n+1\right)\sum_{k=1}^{n}k^{2m+1}-\sum_{k=1}^{n}\sum_{l=1}^{k}l^{2m+1}.$
(9)

We want to write $\sum l^{2m+1}$ in $k$. Expression (3) tells us

$\sum_{l=1}^{k}l^{2m+1}=\frac{1}{2m+2}\sum_{j=0}^{2m+1}\left(-1\right)^{j}\binom{2m+2}{j}B_{j}k^{2m+2-j}.$

Since we are assuming $B_{3}=B_{5}=\cdots=0$, we can rewrite the double sum as

$\displaystyle\sum_{k=1}^{n}\sum_{l=1}^{k}l^{2m+1}$ $\displaystyle=$
$\displaystyle\frac{1}{2m+2}\sum_{k=1}^{n}\left(1\cdot
B_{0}k^{2m+2}-\binom{2m+2}{1}B_{1}k^{2m+1}\right.$
$\displaystyle\left.+\binom{2m+2}{2}B_{2}k^{2m}+\cdots+\binom{2m+2}{2m}B_{2m}k^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2m+2}\sum_{k=1}^{n}k^{2m+2}+\frac{1}{2}\sum_{k=1}^{n}k^{2m+1}+b_{2m}\sum_{k=1}^{n}k^{2m}+\cdots+b_{2}\sum_{k=1}^{n}k^{2},$

where $b_{2m},\ldots,b_{2}$ are rational numbers which do not interest us. Now
we can rewrite (9) as

$\displaystyle\sum k^{2m+2}$ $\displaystyle=$
$\displaystyle\left(n+1\right)\sum k^{2m+1}-\frac{1}{2m+2}\sum
k^{2m+2}-\frac{1}{2}\sum k^{2m+1}$ (10) $\displaystyle-\left(b_{2m}\sum
k^{2m}+\cdots+b_{2}\sum k^{2}\right).$

By the inductive hypothesis for $\sum k^{2m}$, we can rewrite the sum in the
parentheses as

$b_{2m}\sum k^{2}\cdot F_{2m}+\cdots+b_{2}\sum k^{2}\cdot F_{2},$

where $F_{2m},\ldots,F_{2}$ are polynomials in $\frac{n(n+1)}{2}$. Together,
this is just $F\cdot\sum k^{2}$ for another such polynomial $F$. Now (10)
becomes

$\frac{2m+3}{2m+2}\sum k^{2m+2}=\left(n+\frac{1}{2}\right)\sum
k^{2m+1}-F\cdot\sum k^{2}.$ (11)

For the next step of the proof, first we invoke the inductive hypothesis for
$\sum k^{2m+1}.$ This allows us to rewrite the right side of (11) as

$\left(n+\frac{1}{2}\right)\left(\sum k\right)^{2}G_{2m+1}-F\cdot\sum k^{2},$
(12)

where $G_{2m+1}$ is a polynomial in $\frac{n(n+1)}{2}$. Then we use the lemma
to rewrite the left side of (12) as

$\frac{3}{2}\cdot\frac{n(n+1)}{2}\cdot\sum k^{2}\cdot G_{2m+1}=G\cdot\sum
k^{2},$

where $G$ is another polynomial in $\frac{n(n+1)}{2}$. The final form of (11)
becomes

$\displaystyle\sum k^{2m+2}$ $\displaystyle=$
$\displaystyle\frac{2m+2}{2m+3}\left(G\cdot\sum k^{2}-F\cdot\sum
k^{2}\right)=H\cdot\sum k^{2},$

where $H$ is a polynomial in $\frac{n(n+1)}{2}$.

The proof for $\sum k^{2m+3}$ proceeds along the same lines. We only wish to
point out an important difference when rewriting the double sum using
Bernoulli numbers. Starting with

$\sum_{k=1}^{n}k^{2m+3}=\left(n+1\right)\sum_{k=1}^{n}k^{2m+2}-\sum_{k=1}^{n}\sum_{l=1}^{k}l^{2m+2},$

the expression analogous to (11) will be

$\frac{2m+4}{2m+3}\sum k^{2m+3}=\left(n+\frac{1}{2}\right)\sum
k^{2m+2}-B_{2m+2}\sum k-F\cdot\left(\sum k\right)^{2}.$ (13)

We need to eliminate the term of $-B_{2m+2}\sum k$, which we do as follows.

The coefficient of $B_{2m+2}$ comes out of writing $\sum l^{2m+2}$ according
to (3). If we write the same expression using (4), which we just established,
we get

$\sum k^{2m+2}=\left[G+c_{m}\right]\sum
k^{2}=\left[G+c_{m}\right]\cdot\frac{2n^{3}+3n^{2}+n}{6},$

where $G$ is a polynomial in $\frac{n(n+1)}{2}$, of degree of at least one,
and $c_{m}$ is a rational number. The coefficient for the term of $n$ is
$\frac{c_{m}}{6}$. Since both coefficients must agree, we have
$\frac{c_{m}}{6}=B_{2m+2}$.

When we invoke the lemma we get

$\displaystyle\left(n+\frac{1}{2}\right)\left[G+c_{m}\right]\sum k^{2}$
$\displaystyle=$
$\displaystyle\left[G+c_{m}\right]\left(\frac{4}{3}\cdot\frac{n(n+1)}{2}+\frac{1}{6}\right)\sum
k$ $\displaystyle=$
$\displaystyle\left[G+c_{m}\right]\left(\frac{4}{3}\left(\sum
k\right)^{2}+\frac{1}{6}\sum k\right)$ $\displaystyle=$
$\displaystyle\left[G+c_{m}\right]\cdot\frac{4}{3}\left(\sum
k\right)^{2}+G\cdot\frac{1}{6}\sum k+\frac{c_{m}}{6}\sum k.$

Since the polynomial $G$ does not have a constant term, we can simplify the
right side to

$G^{{}^{\prime}}\cdot\left(\sum k\right)^{2}+\frac{c_{m}}{6}\sum k,$ (14)

where $G^{{}^{\prime}}$ is another polynomial in $\frac{n(n+1)}{2}$. Now the
term of $\frac{c_{m}}{6}\sum k$ cancels with that of $-B_{2m+2}\sum k$, and
(13) and (14) become

$\frac{2m+4}{2m+3}\sum k^{2m+3}=G^{{}^{\prime}}\cdot\left(\sum
k\right)^{2}-F\cdot\left(\sum k\right)^{2},$

from which the desired result follows. ∎

##  4 Concluding Remarks

###  4.1 Sums in $\left(n+\frac{1}{2}\right)$

In the preceding pages there have been hints that there is a third way to
write an expression for the sum of powers of integers, with terms in
$\left(n+\frac{1}{2}\right)$. This in fact is true, and can be approached in a
number of different ways (see the work of Beardon [1], Burrows and Talbot [3],
Cereceda [4], and Hersh [10]). Since we have developed the theory of writing
such sums with terms in $\frac{n(n+1)}{2}$, we will proceed in that direction.

Let us start with even powers. We have the identity

$\left(n+\frac{1}{2}\right)^{2}=2\cdot\frac{n(n+1)}{2}+\frac{1}{4},$ (15)

which allows us to write

$\displaystyle\sum k$
$\displaystyle=\frac{n(n+1)}{2}=\frac{1}{2}\left(n+\frac{1}{2}\right)^{2}-\frac{1}{8},$
(16) $\displaystyle\sum k^{2}$
$\displaystyle=\frac{2n+1}{3}\cdot\frac{n(n+1)}{2}=\left(n+\frac{1}{2}\right)\left(\frac{1}{3}\left(n+\frac{1}{2}\right)^{2}-\frac{1}{12}\right).$

Using the shorthand $N=n+\frac{1}{2}$ we can rewrite (4) as

$\displaystyle\sum k^{2m}$ $\displaystyle=$
$\displaystyle\left[c_{0}\left(\frac{1}{2}N^{2}-\frac{1}{8}\right)^{m-1}+\cdots+c_{m-1}\right]\cdot
N\left(\frac{1}{3}N^{2}-\frac{1}{12}\right)$ (17) $\displaystyle=$
$\displaystyle\left[c_{0}^{{}^{\prime}}N^{2\left(m-1\right)}+\cdots+c_{m-2}^{{}^{\prime}}N^{2}+c_{m-1}^{{}^{\prime}}\right]\cdot
N\left(\frac{1}{3}N^{2}-\frac{1}{12}\right)$ $\displaystyle=$
$\displaystyle\left[c_{0}^{{}^{\prime\prime}}N^{2\left(m-1\right)+2}+\cdots+c_{m-2}^{{}^{\prime\prime}}N^{2+2}+c_{m-1}^{{}^{\prime\prime}}N^{2}+c_{m}^{{}^{\prime\prime}}\right]\cdot
N$ $\displaystyle=$ $\displaystyle
d_{0}\left(n+\frac{1}{2}\right)^{2m+1}+\cdots+d_{m-1}\left(n+\frac{1}{2}\right)^{3}+d_{m}\left(n+\frac{1}{2}\right),$

which is an odd polynomial in $\left(n+\frac{1}{2}\right)$ with rational
coefficients $d_{i}$. For odd powers we can rewrite (5) as

$\displaystyle\sum k^{2m+1}$ $\displaystyle=$
$\displaystyle\left[a_{0}\left(\frac{1}{2}N^{2}-\frac{1}{8}\right)^{m-1}+\cdots+a_{m-1}\right]\left(\frac{1}{2}N^{2}-\frac{1}{8}\right)^{2}$
(18) $\displaystyle=$
$\displaystyle\left[a_{0}^{{}^{\prime}}N^{2\left(m-1\right)}+\cdots+a_{m-2}^{{}^{\prime}}N^{2}+a_{m-1}^{{}^{\prime}}\right]\left(\frac{1}{4}N^{4}-\frac{1}{8}N^{2}+\frac{1}{64}\right)$
$\displaystyle=$ $\displaystyle
e_{0}\left(n+\frac{1}{2}\right)^{2m+2}+\cdots+e_{m}\left(n+\frac{1}{2}\right)^{2}+e_{m+1},$

which is an even polynomial in $\left(n+\frac{1}{2}\right)$ with rational
coefficients $e_{i}$.

It is possible to proceed in the opposite direction: start with (17) and (18)
and then derive (4) and (5). However, some care needs to be taken with the
constant terms. Concerning the coefficients $d_{i},e_{i}$, we can find
explicit values in terms of Bernoulli numbers:

$\displaystyle d_{i}$
$\displaystyle=\frac{1}{2\left(m-i\right)+1}\binom{2m}{2i}B_{2i}\left(\frac{1}{2}\right),$
(19) $\displaystyle e_{i}$
$\displaystyle=\frac{1}{2\left(m-i\right)+2}\binom{2m+1}{2i}B_{2i}\left(\frac{1}{2}\right),$

where $0\leq i\leq m$. We evaluate the Bernoulli polynomial
$B_{r}\left(x\right)$ using

$B_{r}\left(\frac{1}{2}\right)=\left(2^{1-r}-1\right)B_{r}.$ (20)

For the lone coefficient of $e_{m+1}$ we have

$e_{m+1}=-\sum_{i=0}^{m}\frac{e_{i}}{4^{m-i+1}}.$ (21)

(These results were derived in a slightly different form in [4].)

###  4.2 Partial sums for Bernoulli polynomials

Since the expressions for sums of powers of integers have the form of
polynomials, it is common to interpret them as such. Previously we used the
notation $S_{m}=\sum_{k=1}^{n}k^{m}$. We introduce $S_{m}\left(x\right)$ to
designate the polynomial of degree $m+1$ in the real variable $x$ that is
obtained by replacing $n$ by $x$ in (3).

In [9, Theorem 2.2], He and Ricci derived the following expression:

$B_{m}\left(x\right)=\left(x-\frac{1}{2}\right)B_{m-1}\left(x\right)-\frac{1}{m}\sum_{r=0}^{m-2}\binom{m}{r}B_{m-r}B_{r}\left(x\right),$
(22)

where $m\geq 1$, and $B_{m}\left(x\right)$ refers to the Bernoulli polynomials
defined in (6). It followed as a corollary to more general results on Appell
polynomials which were derived using differential operators. Here we will use
the simpler approach of partial sums to derive an expression for
$S_{m}\left(x\right)$ which is equivalent to (22).

Starting with Proposition 1, we can arrive at

$S_{2m+2}=\frac{1}{2m+3}\left(\left(2m+2\right)\left(n+\frac{1}{2}\right)S_{2m+1}-\sum_{r=1}^{2m}\binom{2m+2}{r}B_{2m+2-r}S_{r}\right),$

which is analogous to (11). (Note: we do not assume $B_{2m+1}=0$ and we write
(3) without the alternating signs.) Likewise, an expression analogous to (13)
is

$S_{2m+3}=\frac{1}{2m+4}\left(\left(2m+3\right)\left(n+\frac{1}{2}\right)S_{2m+2}-\sum_{r=1}^{2m+1}\binom{2m+3}{r}B_{2m+3-r}S_{r}\right).$

Putting them together, we get

$S_{m}=\frac{1}{m+1}\left(m\left(n+\frac{1}{2}\right)S_{m-1}-\sum_{r=1}^{m-2}\binom{m}{r}B_{m-r}S_{r}\right),$

where $m\geq 2$, with the understanding that for $m=2$, the sum on the right
side is equal to zero. Generalizing to polynomials in $x$, we arrive at the
final result of

$S_{m}\left(x\right)=\frac{1}{m+1}\left(m\left(x+\frac{1}{2}\right)S_{m-1}\left(x\right)-\sum_{r=1}^{m-2}\binom{m}{r}B_{m-r}S_{r}\left(x\right)\right).$
(23)

Even though we will not give the details of the proof, we point out that (22)
and (23) can be shown to be equivalent to each other.

## Acknowledgements

The author thanks the referee for a careful reading of the paper, for
suggesting the topics in the concluding section, and for pointing out
additional references, in particular, the paper by He and Ricci.

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both lexical diversity and semantic fidelity (label-alignment), the messages
were first generated by the seq2seq model to be conditional on the class names
and then these messages were fidelity-checked by the model itself. This
approach was found to generate messages with superior lexical diversity and
semantic fidelity compared to existing baselines. ISA was then proposed as an
optimised version of STA that includes an iterative mechanism and duplication
mechanism to further improve the quality of generated messages. The results
demonstrated that ISA outperformed STA in both lexical diversity and semantic
fidelity and achieved better performance in few-shot crisis message
categorisation. In addition, the experiments revealed that both STA and ISA
exhibited strong generalisation capabilities across other domains such as
emotion and topic classification. Overall, the major contributions of this
chapter include the development of two state-of-the-art augmentation
approaches for crisis few-shot learning and insights into how to improve the
quality of augmented crisis messages in terms of lexical diversity and
semantic fidelity to enhance performance in few-shot crisis message
categorisation.

While crisis few-shot learning addresses the need for emergency response
systems to handle crisis message categorisation when there are new predefined
classes and limited annotated data for emerging events, it still requires
costly annotation when there is a large number of predefined classes and does
not account for unequal class distributions, which are common in real-world
scenarios. In Chapter 7, the research explored crisis zero-shot learning,
which addresses the challenge of categorising crisis messages without
depending on any annotated data and the findings of this research have been
published in [143]. To address this challenge, the research proposed a novel
approach called P-ZSC that uses pseudo-labelled data for zero-shot crisis
message categorisation. P-ZSC obtained the pseudo-labelled data set by
matching an unlabelled corpus of the target event with the label vocabularies
of new classes, allowing for the categorisation of crisis messages into new
classes without the need for labelled data. The results of comparing P-ZSC to
baselines showed that it achieved state-of-the-art performance not only in the
crisis categorisation domain but also demonstrated strong generalisation
capabilities across other domains such as emotion detection and topic
classification. It was also found that the pseudo-labelled data produced by
P-ZSC was as effective as hundreds of manually annotated samples in both
crisis categorisation and non-crisis classification tasks, indicating the
possibility of saving a significant amount of time and effort when training a
model with P-ZSC for these tasks. When comparing P-ZSC to a fully-supervised
BERT model, the results show that a gap still exists in terms of the
performance of P-ZSC. However, the fully-supervised scenario is artificial
because it relies on pre-annotated data (i.e. whole training dataset) in a
crisis situation, which is not practical as a crisis emerges. Due to time
constraints and the workload involved, making a dataset like this is not
trivial. Hence, a zero-shot approach like P-ZSC is a crucial area of research
because it can be realistically deployed in a real crisis situation where
users can set up labels to reflect what they are looking for and then start
matching messages almost immediately. While P-ZSC does not fully solve this
problem, it makes a significant improvement to the state of the art.

In summary, this thesis makes good contributions to the field of crisis
informatics by addressing the challenge of limited annotated data availability
for categorising crisis messages on social media during emerging events. The
approaches presented in this thesis have been developed in close collaboration
with real-world scenarios and have demonstrated state-of-the-art performance
in experiments. These approaches have the potential to be applied in practice
to provide timely and effective humanitarian aid responses.

###  8.2 Future directions

This research has made progress in improving the categorisation of social
media crisis messages in low data settings. However, there is still much work
to be done to fully solve this research problem. There are several key areas
of future work that should be considered in order to continue advancing in
this area. The following identifies limitations of this research and presents
a list of future work to consider.

  * •

In the many-to-many adaptation work presented in this thesis, the MTL-based
ensemble approach achieved the best performance to date. However, this
approach is not very efficient because it involves combining multiple MTL
models for crisis message categorisation. Inference efficiency is a critical
concern during crises, when messages may come in rapidly, and hence it is an
area of future work to optimise the efficiency of the ensemble approach.
Techniques such as distillation, pruning, or compression of categorisation
models could potentially improve the speed at which messages can be processed,
leading to more efficient and timely emergency responses.

  * •

In crisis few-shot learning, STA and ISA use human-designed templates to
ensure lexical diversity and semantic fidelity for the generated messages.
This makes it challenging to determine which templates are the most effective
for optimal performance. As a result, future work could focus on automating
the process of finding the optimal set of templates for STA and ISA.

  * •

In P-ZSC, although the pseudo-labelled data represents high quality in some
aspects, there remains some room to improve it. For example, the existing
matching mechanism simply assigns a label to a message when they are locally
matched with each other with high confidence. Hence, the future work can seek
to explore better solutions such as global semantic meaning matching for
obtaining better quality of pseudo-labelled data.

  * •

It is necessary to develop a dynamic system that can adapt to different levels
of data availability. While this research has presented state-of-the-art
approaches for crisis domain adaptation, few-shot, and zero-shot learning, it
is important to combine these approaches into a system that can handle both
high and low levels of data availability, or that can adapt to changing data
availability over time. This includes integrating approaches for crisis
message categorisation depending on the amount of available data. For
instance, in situations where few labeled data and larger quantities of
unlabeled data for emerging events are available, it would be valuable to
explore how the zero-shot P-ZSC approach could be used in a semi-supervised
manner in conjunction with the augmentation-based few-shot approaches (STA and
ISA) to further improve crisis message categorisation.

  * •

The proposed approaches used in this research focus on the raw text content of
crisis messages. It is important to incorporate additional sources of
information beyond the text of the messages themselves. This could include
incorporating images, locations, or knowledge graphs to provide additional
context and help improve the accuracy of categorisation.

  * •

This research so far has just handled English crisis messages. It is important
to extend this work to other languages, as crisis messages are often written
in a variety of languages. Cross-lingual crisis message categorisation is
therefore a critical area of future work.

  * •

The proposed approaches operate on the assumption that the crisis messages
being learned from, and being classified, are credible and reliable sources of
information. Verifying the credibility of crisis messages is also critical, as
false or misleading information can cause confusion and hinder effective
response. Hence, it would be helpful to add a credibility-checking component
before feeding the messages to the categorisation system.

  * •

This research has made strides in decreasing the need for labelled data in the
categorisation of real-world social media crisis messages. However, it still
requires some labelled and/or unlabelled data for emerging events. One
potential area of future work is to optimise large-scale language models like
GPT-3 and ChatGPT, which are trained on vast amounts of text data. If these
models can be made more efficient, they have the potential to be used for
crisis message categorisation of any event without the need for additional
training on domain-specific data.

Overall, these are key areas of future work that are important not only for
this specific research, but also for the broader community working on social
media crisis message categorisation in the face of low data availability.

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##  Appendix A Appendix

###  A.1 Information types of TREC-IS

| Category |  Description
---|---|---
Actionable | Request-GoodsServices |  The user is asking for a particular service or physical good
Request-SearchAndRescue |  The user is requesting a rescue (for themselves or others)
Report-NewSubEvent |  The user is reporting a new occurence that public safety officers need to respond to
Report-ServiceAvailable |  The user is reporting that they or someone else is providing a service
CallToAction-MovePeople |  The user is asking people to leave an area or go to another area
Report-EmergingThreats |  The user is reporting a potential problem that may cause future loss of life or damage
Non-actionable | CallToAction-Volunteer |  The user is asking people to volunteer to help the response effort
CallToAction-Donations |  The user is asking people to donate goods/money
Report-Weather |  The user is providing a weather report (current or forcast)
Report-Location |  The post contains information about the user or observation location
Request-InformationWanted |  The user is requesting information
Report-FirstPartyObservation |  The user is giving an eye-witness account
Report-ThirdPartyObservation |  The user is reporting a information that they recieved from someone else
Report-MultimediaShare |  The user is sharing images or video
Report-Factoid |  The user is relating some facts, typically numerical
Report-Official |  An official report by a government or public safety representative
Report-News |  The post is a news report providing/linking to current/continious coverage of the event
Report-CleanUp |  A report of the clean up after the event
Report-Hashtags |  Reporting which hashtags correspond to each event
Report-OriginalEvent |  A report of the original event occuring.
Other-ContextualInformation |  The post contains contextual information that can help understand the event, but is not about the event itself
Other-Advice |  The user is providing some advice to the public
Other-Sentiment |  The post is expressing some sentiment about the event
Other-Discussion |  Users are discussing the event
Other-Irrelevant |  The post is unrelated to the event or contains no information
Table A.1: Actionable and non-actionable information types in the TREC-IS
dataset [85]