# Some identities on $\lambda$-analogues of $r$-Stirling numbers of the second
kind

DAE SAN KIM  Department of Mathematics, Sogang University, Seoul 121-742,
Republic of Korea<EMAIL_ADDRESS>,  Hyekyung Kim  Department Of
Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of
Korea<EMAIL_ADDRESS>and  Taekyun Kim  Department of Mathematics, Kwangwoon
University, Seoul 139-701, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

Recently, the $\lambda$-analogues of $r$-Stirling numbers of the first kind
were studied by Kim-Kim. The aim of this paper is to introduce the
$\lambda$-analogues of $r$-Stirling numbers of the second kind and to
investigate some properties, recurrence relations and certain identities on
those numbers. We also introduce the $\lambda$-analogues of Whitney-type
$r$-Stirling numbers of the second and derive similar results to the case of
the $\lambda$-analogues of $r$-Stirling numbers of the second kind. In
addition, we consider the $\lambda$-analogues of Dowling polynomials and
deduce a Dobinski-like formula.

###### Key words and phrases:

$\lambda$-anlogues of $r$-Stirling numbers of the second; $\lambda$-analogues
of Whitney-type $r$-Stirling numbers of the second; $\lambda$-analogues of
Dowling polynomials

###### 2010 Mathematics Subject Classification:

11B73; 11B83

* are corresponding authors

##  1\. Introduction

Carlitz [3] initiated a study of the degenerate Bernoulli and Euler
polynomials and numbers, which are degenerate versions of the Bernoulli and
Euler polynomials and numbers. In recent years, studying degenerate versions
of special numbers and polynomials regained interests of some mathematicians.
They have been explored with various tools and many fascinating results have
been revealed. It is remakable that this quest for degenerate versions is not
just restricted to polynomials but also extended to transcendental functions,
like gamma functions. In addition, it also led to the introduction of
$\lambda$-umbral calculus and $\lambda$-Sheffer sequences.

The degenerate Stirling numbers of the first kind and of the second kind,
which are degenerate versions of the Stirling numbers of the first kind and of
the second kind, appear frequently when we study degenerate versions of some
special numbers and polynomials. They arise naturally when we replace the
powers of $x$ by the generalized falling factorial polynomials
$(x)_{k,\lambda}$ in the defining equations of the Stirling numbers of both
kinds (see (1)), while the $\lambda$-analogues of Stirling numbers of the
first kind and of the second kind appear when we replace the falling
factorials by the generalized falling factorials.

The aim of this paper is to introduce the $\lambda$-analogues of $r$-Stirling
numbers of the second kind and to investigate some properties, recurrence
relations and certain identities on those numbers. We also introduce the
$\lambda$-analogues of Whitney-type $r$-Stirling numbers of the second and
derive similar results to the case of the $\lambda$-analogues of $r$-Stirling
numbers of the second kind. In addition, we consider the $\lambda$-analogues
of Dowling polynomials, which are a natural extension of the analogues of
Whitney-type Stirling numbers of the second kind, and deduce a Dobinski-like
formula.

The outline of this paper is as follows. In Section 1, we recall the
generalized falling factorial sequence, the degenerate exponential functions,
the Stirling numbers of both kinds, the $\lambda$-analogues of $r$-Stirling
numbers of the first kind and the $\lambda$-analogues of unsigned $r$-Stirling
numbers of the first kind. In Section 2, we introduce the $\lambda$-analogues
of $r$-Stirling numbers of the second as the coefficients appearing when
powers of $x+r$ are expressed in terms of the degenerate falling factorial
sequence. In the special case of $r=0$, we get the $\lambda$-analogues of
Stirling numbers of the second kind. In Theorem 1, we obtain the generating
function of the $\lambda$-analogues of $r$-Stirling numbers of the second. We
express those numbers in terms of the forward difference operator in Theorem
2. In Theorems 3 and 6, we find an expression of the analogues of $r$-Stirling
numbers of the second kind in terms of the analogues of Stirling numbers of
the second kind and vice versa. In Theorems 4 and 5, we get recurrence
relations for the analogues of $r$-Stirling numbers of the second kind. In
Theorem 7, we derive an identity connecting the analogues of $r$-Stirling
numbers of the second kind and some values of the higher order Bernoulli
polynomials. In Section 3, introduced are the $\lambda$-analogues of Whitney-
type $r$-Stirling numbers of the second. In case of $r=1$, we get the
$\lambda$-analogues of Whitney-type Stirling numbers of the second. The
generating function of those numbers are derived in Theorem 12. In Theorem 13,
we obtain an identity relating the $\lambda$-analogues of Whitney-type
$r$-Stirling numbers of the second, the $\lambda$-analogues of Whitney-type
Stirling numbers of the second and some values of higher order Bernoulli
numbers. We introduce the $\lambda$-analogues of Dowling polynomials, and
deduce the generating function and Dobinski-like formula in Theorems 10 and
11, respectively. In the rest of this section, we recall the facts that are
needed in this paper.

Throughout this paper, let $\lambda$ be any nonzero real number. The
generalized falling factorial sequence is defined by

(1)
$(x)_{0,\lambda}=1,\quad(x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),\quad(n\geq
1),\quad(\mathrm{see}\ [2,6,8,9,11]).$

It is known that the degenerate exponential functions are defined by

(2) $e_{\lambda}^{x}(t)=(1+\lambda
t)^{\frac{x}{\lambda}}=\sum_{n=0}^{\infty}(x)_{n,\lambda}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[3,7,10]).$

When $x=1$, we use the notation as $e_{\lambda}(t)=e_{\lambda}^{1}(t)$.

For $n\geq 0$, the Stirling numbers of the first kind are defined by

(3) $(x)_{n}=\sum_{k=0}^{n}S_{1}(n,k)x^{k},\quad(\mathrm{see}\
[1,2,4,7,12,13]),$

where the falling factorial sequence is given by

$(x)_{0}=1,\quad(x)_{n}=x(x-1)\cdots(x-n+1),\quad(n\geq 1).$

As the inversion formula of (3), the Stirling numbers of the second kind are
defined as

(4) $x^{n}=\sum_{k=0}^{n}{n\brace k}(x)_{k},\quad(\mathrm{see}\ [4,9,12]).$

The $\lambda$-analogues of Stirling numbers of the first kind are defined by

(5) $(x)_{n,\lambda}=\sum_{k=0}^{n}S_{1,\lambda}(n,k)x^{k},\quad(n\geq
0),\quad(\mathrm{see}\ [9]).$

For $r\in\mathbb{N}\cup\\{0\\}$, the $\lambda$-analogues of $r$-Stirling
numbers of the first kind are defined by

(6)
$(x+r)_{n,\lambda}=\sum_{k=0}^{n}S_{1,\lambda}^{(r)}(n,k)x^{k},\quad(\mathrm{see}\
[9]).$

Also, the $\lambda$-analogues of unsigned $r$-Stirling numbers of the first
kind are given by

(7) $\langle x+r\rangle_{n,\lambda}=\sum_{k=0}^{n}{n+r\brack
k+r}_{r,\lambda}x^{k},\quad(n\geq 0),\quad(\mathrm{see}\ [9]).$

Note that $\displaystyle\lim_{\lambda\rightarrow 1}{n+r\brack
k+r}_{r,\lambda}={n+r\brack k+r}_{r}\displaystyle$ are the ordinary unsigned
$r$-Stirling numbers of the first kind.

##  2\. $\lambda$-analogues of $r$-Stirling numbers of the second kind

First, we consider the $\lambda$-analogues of Stirling numbers of the second
kind as the inversion formula of (5), which are defined by

(8) $x^{n}=\sum_{k=0}^{n}{n\brace k}_{\lambda}(x)_{k,\lambda},\quad(n\geq 0).$

From (8), we note that

(9) $\displaystyle e^{xt}=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}x^{n}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\sum_{k=0}^{n}{n\brace
k}_{\lambda}(x)_{k,\lambda}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{(}\sum_{n=k}^{\infty}{n\brace
k}_{\lambda}\frac{t^{n}}{n!}\bigg{)}(x)_{k,\lambda}.$

On the other hand, by (1), we get

(10) $\displaystyle e^{xt}$ $\displaystyle=\big{(}e^{\lambda
t}-1+1\big{)}^{\frac{x}{\lambda}}=\sum_{k=0}^{\infty}\binom{\frac{x}{\lambda}}{k}(e^{\lambda
t}-1)^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\lambda^{-k}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}(x)_{k,\lambda}.$

From (9) and (10), we note that

(11) $\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}=\sum_{n=k}^{\infty}{n\brace k}_{\lambda}\frac{t^{n}}{n!}.$

Note that $\displaystyle\lim_{\lambda\rightarrow 0}{n\brace
k}_{\lambda}={n\brace k}\displaystyle$ are the Stirling numbers of the second
kind which are defined by

$x^{n}=\sum_{k=0}^{n}{n\brace k}(x)_{k},\quad(n\geq 0).$

For $r\in\mathbb{N}\cup\\{0\\}$, we consider the $\lambda$-analogues of
$r$-Stirling numbers of the second kind as the inversion formula of (6) which
are defined by

(12) $(x+r)^{n}=\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(x)_{k,\lambda},\quad(n\geq 0).$

From (12), we note that

(13) $\displaystyle e^{(x+r)t}$
$\displaystyle=\sum_{n=0}^{\infty}(x+r)^{n}\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(x)_{k,\lambda}\bigg{)}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{(}\sum_{n=k}^{\infty}{n+r\brace
k+r}_{r,\lambda}\frac{t^{n}}{n!}\bigg{)}(x)_{k,\lambda}.$

On the other hand, by (1), we get

(14) $\displaystyle e^{(x+r)t}$ $\displaystyle=e^{rt}e^{xt}=e^{rt}(e^{\lambda
t}-1+1)^{\frac{x}{\lambda}}$
$\displaystyle=e^{rt}\sum_{k=0}^{\infty}\binom{\frac{x}{\lambda}}{k}(e^{\lambda
t}-1)^{k}=\sum_{k=0}^{\infty}\frac{1}{k}(e^{\lambda
t}-1)^{k}e^{rt}\frac{1}{\lambda^{k}}(x)_{k,\lambda}.$

From (13) and (14), we note that

(15) $\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}e^{rt}=\sum_{n=k}^{\infty}{n+r\brace
k+r}_{r,\lambda}\frac{t^{n}}{n!}.$

######  Theorem 1.

The generating function of the $\lambda$-analogues of $r$-Stirling numbers of
the second kind is given by

$\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}e^{rt}=\sum_{n=k}^{\infty}{n+r\brace
k+r}_{r,\lambda}\frac{t^{n}}{n!},\quad(k\geq 0).$

Note that $\displaystyle\lim_{\lambda\rightarrow 1}{n+r\brace
k+r}_{r,\lambda}={n+r\brace k+r}_{r}\displaystyle$ are the ordinary
$r$-Stirling numbers of the second kind which are defined by

$(x+r)^{n}=\sum_{k=0}^{n}{n+r\brace k+r}_{r}(x)_{k},\quad(n\geq
0),\quad(\mathrm{see}\ [\ ]).$

Let $\triangle$ be the difference operator with

$\triangle f(x)=f(x+1)-f(x).$

Then we have

(16) $\triangle^{k}f(x)=\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}f(x+l).$

From Theorem 1, we note that

(17) $\displaystyle\sum_{n=k}^{\infty}{n+r\brace
k+r}_{r,\lambda}\frac{t^{n}}{n!}$
$\displaystyle=\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}e^{rt}$
$\displaystyle=\frac{1}{\lambda^{k}}\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}e^{(l\lambda+r)t}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\frac{1}{\lambda^{k}}\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}(l\lambda+r)^{n}\bigg{)}\frac{t^{n}}{n!}.$

Let us take $f(x)=\big{(}\frac{x}{\lambda}\big{)}^{n}$ in (8). Then we have

(18) $\displaystyle\triangle^{k}\bigg{(}\frac{r}{\lambda}\bigg{)}^{n}$
$\displaystyle=\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}\bigg{(}l+\frac{r}{\lambda}\bigg{)}^{n}$
$\displaystyle=\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}(\lambda
l+r)^{n}\lambda^{-n}.$

By (17) and (18), we get

(19) $\sum_{n=k}^{\infty}{n+r\brace
k+r}_{r,\lambda}\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\lambda^{n-k}\frac{1}{k!}\triangle^{k}\bigg{(}\frac{r}{\lambda}\bigg{)}^{n}\frac{t^{n}}{n!}.$

######  Theorem 2.

For $k\geq 0$, we have

$\lambda^{n-k}\frac{1}{k!}\triangle^{k}\bigg{(}\frac{r}{\lambda}\bigg{)}^{n}=\left\\{\begin{array}[]{ccc}{n+r\brace
k+r}_{r,\lambda},&\textrm{if $n\geq k$,}\\\ 0,&\textrm{if $0\leq
n<k$}.\end{array}\right.$

By Theorem 1, we get

(20) $\displaystyle\sum_{n=k}^{\infty}{n+r\brace
k+r}_{r,\lambda}\frac{t^{n}}{n!}$
$\displaystyle=\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}e^{rt}$ $\displaystyle=\sum_{l=k}^{\infty}{l\brace
k}_{\lambda}\frac{t^{l}}{l!}\sum_{m=0}^{\infty}r^{m}\frac{t^{m}}{m!}$
$\displaystyle=\sum_{n=k}^{\infty}\bigg{(}\sum_{l=k}^{n}{l\brace
k}_{\lambda}r^{n-l}\binom{n}{l}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (20), we obtain the
following theorem.

######  Theorem 3.

For $n,k\in\mathbb{Z}$ with $n\geq k\geq 0$, we have

${n+r\brace k+r}_{r,\lambda}=\sum_{l=k}^{n}\binom{n}{l}{l\brace
k}_{\lambda}r^{n-l}.$

For $n\geq 1$, we have

$\displaystyle(x+r)^{n+1}$
$\displaystyle=(x+r)^{n}(x+r)=\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(x+r)(x)_{k,\lambda}$ (21)
$\displaystyle=\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(x-k\lambda+k\lambda+r)(x)_{k,\lambda}.$
$\displaystyle=\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(x)_{k+1,\lambda}+\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(x)_{k,\lambda}(\lambda k+r)$
$\displaystyle=\sum_{k=1}^{n+1}{n+r\brace
k-1+r}_{r,\lambda}(x)_{k,\lambda}+\sum_{k=0}^{n}(\lambda k+r){n+r\brace
k+r}_{r,\lambda}(x)_{k,\lambda}$
$\displaystyle=\sum_{k=0}^{n+1}\bigg{(}{n+r\brace k-1+r}_{r,\lambda}+(\lambda
k+r){n+r\brace k+r}_{r,\lambda}\bigg{)}(x)_{k,\lambda}.$

On the other hand, by (12), we get

(22) $(x+r)^{n+1}=\sum_{k=0}^{n+1}{n+1+r\brace
k+r}_{r,\lambda}(x)_{k,\lambda}.$

Therefore, by (21) and (22), we obtain the following theorem.

######  Theorem 4.

For $n,k\in\mathbb{Z}$ with $n\geq k\geq 1$, we have

${n+1+r\brace k+r}_{r,\lambda}={n+r\brace k-1+r}_{r,\lambda}+(\lambda
k+r){n+r\brace k+r}_{r,\lambda}.$

Now, we observe that

(23) $\displaystyle\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}\frac{1}{\lambda^{m}}\frac{1}{m!}\big{(}e^{\lambda
t}-1\big{)}^{m}e^{rt}$
$\displaystyle=\frac{1}{\lambda^{k+m}}\frac{1}{(k+m)!}(e^{\lambda
t}-1)^{k+m}e^{rt}\frac{(k+m)!}{k!m!}$
$\displaystyle=\binom{k+m}{k}\sum_{n=m+k}^{\infty}{n+r\brace
k+m+r}_{r,\lambda}\frac{t^{n}}{n!}.$

On the other hand, by (15), we get

(24) $\displaystyle\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}\frac{1}{\lambda^{m}}\frac{1}{m!}(e^{\lambda t}-1)^{m}e^{rt}$
$\displaystyle=\sum_{l=k}^{\infty}{l\brace
k}_{\lambda}\frac{t^{l}}{l!}\sum_{j=m}^{\infty}{j+r\brace
m+r}_{r,\lambda}\frac{t^{j}}{j!}$
$\displaystyle=\sum_{n=m+k}^{\infty}\sum_{l=k}^{n-m}{l\brace
k}_{\lambda}{n-l+r\brace m+r}_{r,\lambda}\binom{n}{l}\frac{t^{n}}{n!}.$

Therefore, by (23) and (24), we obtain the following theorem.

######  Theorem 5.

For $m,n,k\geq 0$ with $n\geq m+k$, we have

$\binom{m+k}{k}{n+r\brace
k+m+r}_{r,\lambda}=\sum_{l=k}^{n-m}\binom{n}{l}{l\brace
k}_{\lambda}{n-l+r\brace m+r}_{r,\lambda}.$

From the definition of the $\lambda$-analogues of the Stirling numbers of the
second kind, we have

(25) $\displaystyle\sum_{n=k}^{\infty}{n\brace k}_{\lambda}\frac{t^{n}}{n!}$
$\displaystyle=\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{t}-1\big{)}^{k}=\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}e^{rt}e^{-rt}$ $\displaystyle=\sum_{l=k}^{\infty}{l+r\brace
k+r}_{r,\lambda}\frac{t^{l}}{l!}\sum_{m=0}^{\infty}(-r)^{m}\frac{t^{m}}{m!}$
$\displaystyle=\sum_{n=k}^{\infty}\bigg{(}\sum_{l=k}^{n}\binom{n}{l}{l+r\brace
k+r}_{r,\lambda}(-1)^{n-l}r^{n-l}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (25) we obtain the
following theorem.

######  Theorem 6.

For $n,k\geq 0$ with $n\geq k$, we have

${n\brace k}_{\lambda}=\sum_{l=k}^{n}\binom{n}{l}{l+r\brace
k+r}_{r,\lambda}(-1)^{n-l}r^{n-l}.$

For $m\in\mathbb{N}$, the higher-order Bernoulli polynomials are defined by

(26)
$\bigg{(}\frac{t}{e^{t}-1}\bigg{)}^{m}e^{xt}=\sum_{n=0}^{\infty}B_{n}^{(m)}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[1,3,7]).$

From (26), we note that

(27) $\displaystyle\sum_{n=k}^{\infty}{n+r\brace
k+r}_{r,\lambda}\frac{t^{n}}{n!}=\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}e^{rt}$
$\displaystyle\quad=\frac{1}{t^{m}}\frac{1}{\lambda^{k+m}}\frac{(k+m)!}{k!}\frac{1}{(k+m)!}(e^{\lambda
t}-1)^{k+m}\bigg{(}\frac{\lambda t}{e^{\lambda t}-1}\bigg{)}^{m}e^{rt}$
$\displaystyle\quad=\binom{k+m}{k}\frac{m!}{t^{m}}\sum_{l=k+m}^{\infty}{l\brace
k+m}_{\lambda}\frac{t^{l}}{l!}\bigg{(}\sum_{j=0}^{\infty}B_{j}^{(m)}\frac{r}{\lambda}\bigg{)}\frac{\lambda^{j}t^{j}}{j!}$
$\displaystyle\quad=\binom{k+m}{k}\sum_{l=k}^{\infty}{l+m\brace
k+m}_{\lambda}\frac{m!l!}{(l+m)!}\frac{t^{l}}{l!}\sum_{j=0}^{\infty}B_{j}^{(m)}\bigg{(}\frac{r}{\lambda}\bigg{)}\frac{\lambda^{j}t^{j}}{j!}$
$\displaystyle\quad=\binom{k+m}{k}\sum_{n=k}^{\infty}\bigg{(}\sum_{l=k}^{n}\frac{\binom{n}{l}}{\binom{l+m}{l}}{l+m\brace
k+m}_{\lambda}\lambda^{n-l}B_{n-l}^{(m)}\bigg{(}\frac{r}{\lambda}\bigg{)}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (27), we obtain the
following theorem.

######  Theorem 7.

For $n,k\geq 0$ with $n\geq k$, we have

$\frac{{n+r\brace
k+r}_{r,\lambda}}{\binom{k+m}{k}}=\sum_{l=k}^{n}\frac{\binom{n}{l}}{\binom{l+m}{l}}{l+m\brace
k+m}_{\lambda}B_{n-l}^{(m)}\bigg{(}\frac{r}{\lambda}\bigg{)}\lambda^{n-l}.$

##  3\. Further Remarks

For $m,n\geq 0$, we define $\lambda$-analogues of the Whitney-type Stirling
numbers of the second kind as

(28)
$(mx+1)^{n}=\sum_{k=0}^{n}W_{m,\lambda}(n,k)m^{k}(x)_{k,\lambda},\quad(n\geq
0).$

From (28), we note that

(29) $\displaystyle e^{(mx+1)t}$
$\displaystyle=\sum_{m=0}^{\infty}(mx+1)^{n}\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\sum_{k=0}^{n}W_{m,\lambda}(n,k)m^{k}(x)_{k,\lambda}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{n=k}^{\infty}W_{m,\lambda}(n,k)\frac{t^{n}}{n!}m^{k}(x)_{k,\lambda}.$

On the other hand, by (11), we get

(30) $\displaystyle e^{(mx+1)t}$ $\displaystyle=e^{t}\big{(}e^{\lambda
mt}-1+1\big{)}^{\frac{x}{\lambda}}=e^{t}\sum_{k=0}^{\infty}\binom{\frac{x}{\lambda}}{k}(e^{\lambda
mt}-1)^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{(}\frac{1}{\lambda^{k}}\frac{1}{k!}\bigg{(}\frac{e^{\lambda
mt}-1}{m}\bigg{)}^{k}e^{t}\bigg{)}m^{k}(x)_{k,\lambda}.$

Therefore, by (29) and (30), we obtain the generating function
$W_{m,\lambda}(n,k),\ (n,k\geq 0)$.

######  Theorem 8.

For $k\geq 0$, we have

$\frac{1}{k!}\frac{1}{\lambda^{k}}\bigg{(}\frac{e^{\lambda
mt}-1}{m}\bigg{)}^{k}e^{t}=\sum_{n=k}^{\infty}W_{m,\lambda}(n,k)\frac{t^{n}}{n!}.$

From Theorem 8, we note that

(31) $\displaystyle\sum_{n=k}^{\infty}{n+1\brace
k+1}_{\lambda}\frac{t^{n}}{n!}=\frac{d}{dt}\bigg{\\{}\sum_{n=k}^{\infty}{n+1\brace
k+1}_{\lambda}\frac{t^{n+1}}{(n+1)!}\bigg{\\}}$
$\displaystyle=\frac{d}{dt}\bigg{(}\frac{1}{(k+1)!}\frac{1}{\lambda^{k+1}}(e^{\lambda
t}-1)^{k+1}\bigg{)}=\frac{1}{k!}\frac{1}{\lambda^{k}}(e^{\lambda
t}-1)^{k}e^{t}e^{(\lambda-1)t}$
$\displaystyle=\sum_{l=k}^{\infty}W_{1,\lambda}(l,k)\frac{t^{l}}{l!}\sum_{j=0}^{\infty}(\lambda-1)^{j}\frac{t^{j}}{j!}$
$\displaystyle=\sum_{n=k}^{\infty}\bigg{(}\sum_{l=k}^{n}W_{1,\lambda}(l,k)(\lambda-1)^{n-l}\binom{n}{l}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (31), we obtain the
following theorem.

######  Theorem 9.

For $n,k\geq 0$ with $n\geq k$, we have

$\sum_{l=k}^{n}\binom{n}{l}W_{1,\lambda}(l,k)(\lambda-1)^{n-l}={n+1\brace
k+1}_{\lambda}.$

Now, we consider the $\lambda$-analogues of Dowling polynomials which are
defined by

(32) $d_{m,\lambda}(n,x)=\sum_{k=0}^{n}W_{m,\lambda}(n,k)x^{k},\quad(n\geq
0).$

Thus, by (32), we get

(33) $\displaystyle\sum_{n=0}^{\infty}d_{m,\lambda}(n,x)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}W_{m,\lambda}(n,k)x^{k}\frac{t^{n}}{n!}$
(34)
$\displaystyle=\sum_{k=0}^{\infty}x^{k}\sum_{n=k}^{\infty}W_{m,\lambda}(n,k)\frac{t^{n}}{n!}$
$\displaystyle=e^{t}\sum_{k=0}^{\infty}x^{k}\frac{1}{k!}\frac{1}{\lambda^{k}}\bigg{(}\frac{e^{\lambda
mt}-1}{m}\bigg{)}^{k}$ $\displaystyle=e^{t}e^{x(\frac{e^{\lambda
mt}-1}{\lambda m})}.$

######  Theorem 10.

For $m\in\mathbb{N}$, we have

$e^{t}e^{x(\frac{e^{\lambda mt}-1}{\lambda
m})}=\sum_{n=0}^{\infty}d_{m,\lambda}(n,x)\frac{t^{n}}{n!}.$

When $x=1$, $d_{m,\lambda}(n,1)=d_{m,\lambda}(n)$ are called the
$\lambda$-analogues of Dowling numbers.

We define the $\lambda$-analogues of Bell polynomials by

(35) $e^{\frac{x}{\lambda}(e^{\lambda
t}-1)}=\sum_{n=0}^{\infty}\phi_{n,\lambda}(x)\frac{t^{n}}{n!}.$

Thus, we easily get $\displaystyle\phi_{n,\lambda}(x)=\sum_{k=0}^{n}{n\brace
k}_{\lambda}x^{k},\ (n\geq 0)\displaystyle$. When $x=1,\
\phi_{n,\lambda}(1)=\phi_{n,\lambda}$ are called the $\lambda$-analogues of
Bell numbers.

From Theorem 10, we note that

(36) $\displaystyle\sum_{n=0}^{\infty}d_{m,\lambda}(n,x)\frac{t^{n}}{n!}$
$\displaystyle=e^{t}e^{x(\frac{e^{\lambda mt}-1}{\lambda
m})}=e^{-\frac{x}{\lambda m}}e^{t}e^{x\frac{e^{\lambda mt}}{\lambda m}}$
$\displaystyle=e^{-\frac{x}{\lambda m}}\sum_{k=0}^{\infty}\frac{e^{\lambda
mkt+t}}{\lambda^{k}m^{k}}\frac{x^{k}}{k!}$ $\displaystyle=e^{-\frac{x}{\lambda
m}}\sum_{k=0}^{\infty}\frac{x^{k}}{k!m^{k}\lambda^{k}}\sum_{n=0}^{\infty}(\lambda
mk+1)^{n}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}e^{-\frac{x}{\lambda
m}}\sum_{k=0}^{\infty}\frac{x^{k}}{k!m^{k}\lambda^{k}}(\lambda
mk+1)^{n}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (36), we obtain the
following Dobinski-like formula.

######  Theorem 11.

For $n\geq 0$, we have

$d_{m,\lambda}(n,x)=e^{-\frac{x}{\lambda
m}}\sum_{k=0}^{\infty}\frac{x^{k}}{k!m^{k}\lambda^{k}}(\lambda mk+1)^{n}.$

For $r\in\mathbb{N}$, $m,n\geq 0$, we consider the $\lambda$-analogues of the
Whitney- type $r$-Stirling numbers of the second kind defined by

(37)
$(mx+r)^{n}=\sum_{k=0}^{n}W_{m,\lambda}^{(r)}(n,k)m^{k}(x)_{k,\lambda},\quad(n\geq
0).$

From (37), we note that

(38) $\displaystyle e^{(mx+r)t}$
$\displaystyle=\sum_{n=0}^{\infty}(mx+r)^{n}\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}W_{m,\lambda}^{(r)}(n,k)m^{k}(x)_{k}\bigg{)}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{(}\sum_{n=k}^{\infty}W_{m,\lambda}^{(r)}(n,k)\frac{t^{n}}{n!}\bigg{)}m^{k}(x)_{k}.$

On the other hand, by (1), we get

(39) $\displaystyle e^{(mx+r)t}$ $\displaystyle=e^{rt}(e^{m\lambda
t}-1+1)^{\frac{x}{\lambda}}=\sum_{k=0}^{\infty}\binom{\frac{x}{\lambda}}{k}(e^{m\lambda
t}-1)^{k}e^{rt}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}\frac{1}{\lambda^{k}}(e^{m\lambda
t}-1)^{k}e^{rt}(x)_{k,\lambda}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{(}\frac{1}{k!}\frac{1}{\lambda^{k}}\bigg{(}\frac{e^{m\lambda
t}-1}{m}\bigg{)}^{k}e^{rt}\bigg{)}m^{k}(x)_{k,\lambda}.$

By (38) and (39), we get

(40) $\frac{1}{k!}\frac{1}{\lambda^{k}}\bigg{(}\frac{e^{\lambda
mt}-1}{m}\bigg{)}^{k}e^{rt}=\sum_{n=k}^{\infty}W_{m,\lambda}^{(r)}(n,k)\frac{t^{n}}{n!},$

where $k$ is a nonnegative integer.

Therefore, by (40), we obtain the following theorem.

######  Theorem 12.

For $k\geq 0$, we have

$\frac{1}{k!}\frac{1}{\lambda^{k}}\bigg{(}\frac{e^{\lambda
mt}-1}{m}\bigg{)}^{k}e^{rt}=\sum_{n=k}^{\infty}W_{m,\lambda}^{(r)}(n,k)\frac{t^{n}}{n!}.$

From (15) and Theorem 11, we have

$W_{1,\lambda}^{(r)}(n,k)={n+r\brace k+r}_{r,\lambda},\
W_{1,\lambda}^{(0)}(n,k)={n\brace k}_{\lambda},\ \mathrm{and}\
W_{m,\lambda}^{(1)}(n,k)=W_{m,\lambda}(n,k),\quad(n,k\geq 0).$

Now, we observe that

(41)
$\displaystyle\sum_{n=k}^{\infty}W_{m,\lambda}^{(r)}(n,k)\frac{t^{n}}{n!}=\frac{1}{k!}\frac{1}{\lambda^{k}}\bigg{(}\frac{e^{\lambda
mt}-1}{m}\bigg{)}^{k}e^{rt}$
$\displaystyle=\frac{1}{k!}\frac{1}{\lambda^{k+\alpha}}\bigg{(}\frac{e^{\lambda
mt}-1}{m}\bigg{)}^{k+\alpha}e^{t}\frac{1}{t^{\alpha}}\bigg{(}\frac{\lambda
mt}{e^{\lambda mt}-1}\bigg{)}^{\alpha}e^{(r-1)t}$
$\displaystyle=\frac{(k+\alpha)!}{k!}\frac{1}{t^{\alpha}}\frac{1}{(k+\alpha)!}\frac{1}{\lambda^{k+\alpha}}\bigg{(}\frac{e^{\lambda
mt}-1}{m}\bigg{)}^{k+\alpha}e^{t}\sum_{j=0}^{\infty}B_{j}^{(\alpha)}\bigg{(}\frac{r-1}{m\lambda}\bigg{)}\lambda^{j}m^{j}\frac{t^{j}}{j!}$
$\displaystyle=\frac{\alpha!}{t^{\alpha}}\binom{k+\alpha}{k}\sum_{l=k+\alpha}^{\infty}W_{m,\lambda}(l,k+\alpha)\frac{t^{l}}{l!}\sum_{j=0}^{\infty}B_{j}^{(\alpha)}\bigg{(}\frac{r-j}{m\lambda}\bigg{)}\lambda^{j}m^{j}\frac{t^{j}}{j!}$
$\displaystyle=\binom{k+\alpha}{k}\bigg{(}\sum_{l=k}^{\infty}\frac{W_{m,\lambda}(l+\alpha,k+\alpha)}{\binom{l+\alpha}{l}}\frac{t^{l}}{l!}\bigg{)}\sum_{j=0}^{\infty}B_{j}^{(\alpha)}\bigg{(}\frac{r-j}{m\lambda}\bigg{)}\lambda^{j}m^{j}\frac{t^{j}}{j!}$
$\displaystyle=\binom{k+\alpha}{k}\sum_{n=k}^{\infty}\bigg{(}\sum_{l=k}^{n}\frac{\binom{n}{l}}{\binom{l+\alpha}{l}}W_{m,\lambda}(l+\alpha,k+\alpha)B_{n-l}^{(\alpha)}\bigg{(}\frac{r-n+l}{m\lambda}\bigg{)}\lambda^{n-l}m^{n-l}\bigg{)}\frac{t^{n}}{n!},$

where $\alpha$ is a positive integer.

Comparing the coefficients on both sides of (41), we have the following
theorem.

######  Theorem 13.

For $n,k\in\mathbb{N}\cup\\{0\\}$ and $\alpha\in\mathbb{N}$, we have

$\binom{k+\alpha}{k}^{-1}W_{m,\lambda}^{(r)}(n,k)=\sum_{l=k}^{n}\frac{\binom{n}{l}}{\binom{l+\alpha}{l}}W_{m,\lambda}(l+\alpha,k+\alpha)B_{n-l}^{(\alpha)}\bigg{(}\frac{r-n+l}{m\lambda}\bigg{)}\lambda^{n-l}m^{n-l}.$

##  4\. Conclusion

In this paper, we introduced the $\lambda$-analogues of $r$-Stirling numbers
of the second which appear as the coefficients when powers of $x+r$ are
expressed in terms of the degenerate falling factorial sequence. We obtained
some properties, recurrence relations and certain identities on such numbers.
We also introduced the $\lambda$-analogues of Whitney-type $r$-Stirling
numbers of the second and derived similar results to the case of the
$\lambda$-analogues of $r$-Stirling numbers of the second. Furthermore, the
$\lambda$-analogues of Dowling polynomials were introduced as a natural
extension of the analogues of Whitney-type Stirling numbers of the second kind
and a Dobinski-like formula for them was deduced.

It is one of our future projects to continue to study analogues of some
special numbers and polynomials and to find their applications in physics,
science and engineering.

Funding This work was supported by the Basic Science Research Program, the
National Research Foundation of Korea,(NRF-2021R1F1A1050151).

## References

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ifaamas [AAMAS ’23]Proc. of the 22nd International Conference on Autonomous
Agents and Multiagent Systems (AAMAS 2023)May 29 – June 2, 2023 London, United
KingdomA. Ricci, W. Yeoh, N. Agmon, B. An (eds.) 2023 2023 Farrell and Fisher
are both supported by the Royal Academy of Engineering.

# Developing Multi-Agent Systems with
Degrees of Neuro-Symbolic Integration
[A Position Paper]

Louise Dennis Marie Farrell Michael Fisher111Corresponding author:
<EMAIL_ADDRESS>
Department of Computer Science, University of Manchester, UK

###### Abstract.

In this short position paper we highlight our ongoing work on verifiable
heterogeneous multi-agent systems and, in particular, the complex (and often
non-functional) issues that impact the choice of structure within each agent.

##  1\. Introduction

A traditional way to develop multi-agent systems is to take some overall goal
for the system, say $G$, and decompose this into a set of tasks (which could
be any of actions, updates, sub-goals, etc) Dunin-Keplicz and Verbrugge
(2010). These tasks are then undertaken by a set of agents, for simplicity one
task per agent. In devising symbolic teams, each agent is symbolic,
discharging its task appropriately (see Figure 1 (left)) giving us a
collection of behaviour specifications (usually in symbolic logic) for the
agents which can then be combined appropriately to provide the overall
description in the form of a symbolic logic formula.

Figure 1. Decomposing goals into collations of symbolic (left) and neural
(right) agents

An alternative approach would be to decompose our collective goal into a set
of ‘neural’ agents; see Figure 1 (right). These agents could be any of (deep)
learning components, adaptive control components, optimisation components, etc
Chen et al. (2020). Again, tasks would be implemented with these agents and
the result would be combined to give an overall outcome. The difference here
is that the behaviour of an individual ‘neural’ component is described using
some stochastic notation, either logical or differential equations.

In reality, however, our multi-agent system is unlikely to be wholly symbolic
or wholly neural and so we will get a more heterogeneous multi-agent system
such as in Figure 2.

Figure 2. Decomposing goals into heterogeneous collations of agents

##  2\. Position

The particular issue that we are concerned with here is how, and why, we
choose one particular type of agent over another when constructing our agent
team. Symbolic and ‘neural’ agents generally have quite distinct
properties222Not unlike Kahneman’s ‘fast’ and ‘slow’ categorisations Kahneman
(2012)

  * •

neural — fast, able to cope with vast amounts of data/input,
opaque/stochastic, …

  * •

symbolic — logical, transparent, explainable, verifiable, much slower, may be
overwhelmed by data, …

So, our aim is to capture, in the goal specification G, key aspects that need
to be considered/achieved relating to this goal. Specifically:

  1. (1)

Speed — Although this might relate to the Multi-Agent System (MAS) as a whole,
it is more likely to refer to how quickly a particular task (a decision,
recognition, or action) should be completed;

  2. (2)

Transparency — Again, while this might refer to the whole MAS, it will more
usually relate to specific decisions and actions, exposing exactly what the
agents do and why they do it.

Note: This transparency then forms a basis for both explainability and
verifiability Winfield et al. (2021).

  3. (3)

Accuracy — This refers to decisions that are taken on the basis of potentially
inaccurate measurements (e.g. sensor readings, images, etc.). Accuracy-related
properties will likely be captured using probabilistic reasoning and/or
statistical measures.

Consequently, in specifying the system goal, G, the above aspects must be
highlighted. Then, as we decompose this goal into tasks for specific agents,
we can assess how well each agent not only can achieve its task but can
satisfy the additional speed, transparency, and accuracy requirements. This
will surely affect how the MAS is composed and, in many practical cases, we
will develop a heterogeneous multi-agent system as in Figure 2.

Each agent will be selected to satisfy its speed, transparency, or accuracy
requirement and each will have a description of its behaviour. Typically,
symbolic agents will have logical descriptions while ‘neural’ agents will have
stochastic descriptions (though this may not always be the case).

  * Point 1: we highlight the speed/transparency/accuracy requirements in the MAS development process, attempting to ensure that the most appropriate style of agent is utilised.

While the above approach can provide us with a description of the distinct
agent recommendations, we also need to check that these requirements are
actually being achieved. We do this by also adding _runtime monitors_ to check
each agent’s behaviour and, in particular, whether it is matching the
speed/transparency/accuracy requirements. We check speed, in the obvious way
by timing progress, and check transparency by interrogating the agent
structures to ensure that “what” is done and “why” are explicit.

  * Point 2: since agents may well act sub-optimally, we utilise runtime verification Falcone et al. (2013) to assess how well each agent is actually matching its speed/transparency/accuracy requirements.

##  3\. Examples

### Autonomous Vehicle.

We consider an autonomous vehicle to actually be a MAS, comprising a wide
range of communicating components. We want sensing and recognition to be as
fast as possible and so speed is the main criterion here. Although we may veer
towards symbolic agents if we require some explanation of recognition, it is
much more likely that these aspects will be implemented as some learning
component. On the other hand, the decision-making component(s) that handle the
high-level decisions that a human driver/pilot/operator used to make will very
likely be symbolic agent(s). This facilitates verification, for example
against human-level rules, and explanations, for example to authorities or
driver who is taking control Webster et al. (2014) .Lower level actions will
undoubtedly be implemented in some adaptive or ‘neural’ control component.

An interesting element concerns obstacle avoidance. If this is required to be
very quick, for example in an emergency situation, then symbolic agents will
be too slow. However, if the vehicle has some time to “think” then a symbolic
agent making a reasoned and explainable decision might be more appropriate. It
might well be that both aspects are in the system, with the level of urgency
dictating which agent to invoke.

### Tele-Operated Robot.

In this situation, speed, accuracy and compliance with the human operator’s
commands will be the most important aspects. Consequently, very few of the
agents need be symbolic. Perhaps the few that are will be concerned with
situation awareness (for example, explaining to the distant operator what is
happening in the robot’s context) or fault diagnosis (for example, explaining
what the robot believes has occurred).

### Social/Healthcare Teams.

These very complex teams comprise a wide range of agents: some are human
(healthcare specialists, etc); some are low-level sensors (detecting movement,
temperature, etc); and many are AI-based agents somewhere in between. We could
have:

  * •

simple cleaning robots, where accuracy (and safety) is central;

  * •

health monitoring software, where recognising problems (quickly, accurately)
is important but so is explaining to, and interacting with, humans in the
environment or team (transparency);

  * •

social robots, able to have complex dialogue with humans and able to explain
and expand on topics (transparency);

  * and so on.

In many cases, the strong reliability/verification of the agent, often based
on its transparency, will be important.

### Cybersecurity:

In this case, a cybersecurity system may use symbolic representations of known
attack patterns (based on experience and/or threat analysis) and neural
networks to detect previously unknown attacks by analyzing network traffic and
system logs. It is especially difficult to distinguish an attack from a
software error/failure Cardoso et al. (2022) so the complex computations
performed here by the neural components is beneficial. Here, transparency and
speed are particularly important: speed to identify attacks quickly so that
mitigations might be invoked and transparency to provide reasoning behind why
some behaviours were identified as potential attacks. The latter is useful to
build up a knowledge base of why and how previously unidentified attacks
occur. A similar approach could be used in other fields such as natural
language processing.

##  4\. Summary

Decomposing an overall goal into a set of agents that together can achieve
this goal, is well understood. We want to expose additional constraints,
particularly those of speed, accuracy, and transparency, and then use these in
the decomposition/development process to help us select the variety of agent
that is most suitable.

We recognise that, in realistic scenarios, “things will go wrong” and so we
need to provide dynamic checks that the agents are living up to their
requirements. We do this by inserting a range of runtime monitors to assess
both the speed and transparency of the agents.

The problems related to this involve

  * •

describing speed, accuracy, and transparency in a flexible but precise way,

  * •

developing effective, but not intrusive, runtime monitors to assess these, and

  * •

extending towards other properties such as flexibility, being able to cope
with vast amounts of data, etc.

Then, within these complexities we aim to ensure strong verification where
appropriate, so supporting the assurance and trustworthiness of complex
autonomous teams Dennis and Fisher (2023); Bourbouh et al. (2021); Fisher et
al. (2021)

## References

  * (1)
  * Bourbouh et al. (2021) Hamza Bourbouh, Marie Farrell, Anastasia Mavridou, Irfan Sljivo, Guillaume Brat, Louise A. Dennis, and Michael Fisher. 2021\.  Integrating Formal Verification and Assurance: An Inspection Rover Case Study. In _Proc. 13th International NASA Formal Methods Symposium (NFM)_ _(Lecture Notes in Computer Science, Vol. 12673)_. Springer, 53–71.  https://doi.org/10.1007/978-3-030-76384-8_4
  * Cardoso et al. (2022) Rafael C. Cardoso, Angelo Ferrando, and Michael Fisher. 2022\.  Extending Attack-Fault Trees with Runtime Verification. In _Proceedings Fourth International Workshop on Formal Methods for Autonomous Systems (FMAS) and Fourth International Workshop on Automated and verifiable Software sYstem DEvelopment (ASYDE)_ _(EPTCS, Vol. 371)_. 193–207.  https://doi.org/10.4204/EPTCS.371.14
  * Chen et al. (2020) Jiaxi Chen, Junmin Li, and Xinxin Yuan. 2020.  Global Fuzzy Adaptive Consensus Control of Unknown Nonlinear Multiagent Systems.  _IEEE Trans. Fuzzy Syst._ 28, 3 (2020), 510–522.  https://doi.org/10.1109/TFUZZ.2019.2908771
  * Dennis and Fisher (2023) Louise A. Dennis and Michael Fisher. 2023.  _Verifiable Autonomous Systems: Using Rational Agents to Provide Assurance about Decisions Made by Machines_.  Cambridge University Press.  ISBN 9781108755023.
  * Dunin-Keplicz and Verbrugge (2010) Barbara Dunin-Keplicz and Rineke Verbrugge. 2010.  _Teamwork in Multi-Agent Systems: Formal Approach_.  Wiley.
  * Falcone et al. (2013) Yliès Falcone, Klaus Havelund, and Giles Reger. 2013\.  A Tutorial on Runtime Verification.  In _Engineering Dependable Software Systems_. IOS Press, 141–175.
  * Fisher et al. (2021) Michael Fisher, Viviana Mascardi, Kristin Yvonne Rozier, Bernd-Holger Schlingloff, Michael Winikoff, and Neil Yorke-Smith. 2021\.  Towards a Framework for Certification of Reliable Autonomous Systems.  _Autonomous Agents and MultiAgent Systems_ 35, 1 (2021), 8\.  https://doi.org/10.1007/s10458-020-09487-2
  * Kahneman (2012) Daniel Kahneman. 2012\.  _Thinking, Fast and Slow_.  Penguin.
  * Webster et al. (2014) Matt P. Webster, Neil Cameron, Michael Fisher, and Mike Jump. 2014.  Generating Certification Evidence for Autonomous Unmanned Aircraft Using Model Checking and Simulation.  _Journal of Aerospace Information Systems_ 11, 5 (2014), 258–279.
  * Winfield et al. (2021) Alan F. T. Winfield, Serena Booth, Louise A. Dennis, Takashi Egawa, Helen F. Hastie, Naomi Jacobs, Roderick I. Muttram, Joanna I. Olszewska, Fahimeh Rajabiyazdi, Andreas Theodorou, Mark A. Underwood, Robert H. Wortham, and Eleanor Nell Watson. 2021.  IEEE P7001: A Proposed Standard on Transparency.  _Frontiers Robotics AI_ 8 (2021), 665729.  https://doi.org/10.3389/frobt.2021.665729