# Some identities of fully degenerate Dowling and fully degenerate Bell
polynomials arising from $\lambda$-umbral calculus

Yuankui Ma  School of Science, Xi’an Technological University, Xi’an, 710021,
Shaanxi, P. R. China<EMAIL_ADDRESS>,  Taekyun Kim  School of Science,
Xi’an Technological University, Xi’an, 710021, Shaanxi, P. R. China
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of
Korea<EMAIL_ADDRESS>,  Hyunseok Lee  Department of Mathematics, Kwangwoon
University, Seoul 139-701, Republic of Korea<EMAIL_ADDRESS>and  Dae
San Kim  Department of Mathematics, Sogang University, Seoul 121-742, Republic
of Korea<EMAIL_ADDRESS>
###### Abstract.

Recently, Kim-Kim introduced the $\lambda$-umbral calculus, in which the
$\lambda$-Sheffer sequences occupy the central position. In this paper, we
introduce the fully degenerate Bell and the fully degenerate Dowling
polynomials, and investigate some properties and identities relating to those
polynomials with the help of $\lambda$-umbral calculus. Here we note that the
fully degenerate Bell poynomials and the fully degenerate Dowling polynomials
are respectively degenerate versions of the Bell polynomials and the Dowling
polynomials, of which the latters are the natural extension of the Whitney
numbers of the second kind.

###### Key words and phrases:

degenerate Whitney numbers of the second kind; fully degenerate Bell
polynomials; fully degenerate Dowling polynomials

###### 2010 Mathematics Subject Classification:

11B73; 11B83; 05A19; 05A40

##  1\. Introduction

A finite lattice $L$ is geometric if it is a finite semimodular lattice which
is also atomic. For details on lattices, one refers to [17]. Dowling
constructed an important finite geometric lattice $Q_{n}(G)$ of rank $n$ (see
[3,Theorem 3]) which satisfies the chain condition (see [3 ,Theorem 1]), out
of a finite set of $n$ elements and a finite group $G$ of order $m$. This is
called Dowling lattice of rank $n$ over a finite group of order $m$. We let
the interested reader refer to [3] for the details on the construction of
$Q_{n}(G)$ and its many fascinating properties.
For a finite geometric lattice $L$ of rank $n$, Dowling [3] defined the
Whitney numbers $V_{L}(n,k)$ of the first kind and the Whitney numbers
$W_{L}(n,k)$ of the second kind. In particular, if $L$ is the Dowling lattice
$Q_{n}(G)$ of rank $n$ over a finite group $G$ of order $m$, then the Whitney
numbers of the first kind $V_{Q_{n}(G)}(n,k)$ and the Whitney numbers of the
second kind $W_{Q_{n}(G)}(n,k)$ are respectively denoted by $V_{m}(n,k)$ and
$W_{m}(n,k)$. These notations are justified, since the Whitney numbers of both
kinds depend only on the order $m$ of $G$. In Corollary 6.1 of [3], it was
shown that, for any fixed group $G$ of order $m$, the Whitney numbers
$V_{m}(n,k)$ and $W_{m}(n,k)$ satisfy the following Stirling number-like
relations:

(1) $\displaystyle(mx+1)^{n}=\sum_{k=0}^{n}W_{m}(n,k)m^{k}(x)_{k},$ (2)
$\displaystyle
m^{n}(x)_{n}=\sum_{k=0}^{n}V_{m}(n,k)(mx+1)^{k},\quad(\mathrm{see}\ [3]).$

where $(x)_{0}=1$, $(x)_{n}=x(x-1)\cdots(x-n+1),\quad(n\geq 1)$.
From (1) and (2), it is evident that the Whitney numbers satisfy the
orthogonality realtions.

Let $r\in\mathbb{N}$. Then, as further generalizations of Whitney numbers of
both kinds, the $r$-Whitney numbers of the first kind $V_{m}^{(r)}(n,k)$ and
those of the second $W_{m}^{(r)}(n,k)$ are respectively defined by

(3) $m^{n}(x)_{n}=\sum_{k=0}^{n}V_{m}^{(r)}(n,k)(mx+r)^{k},$

and

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

We remark that $r$-Whitney numbers of both kinds and their applications were
studied by several authors (see [14]). Combinatorial interpretations for the
$r$-Whitney numbers of both kinds were found by Gyimesi-Nyul in [4]. We state
them here for the convenience of the reader.

Let $0\leq k\leq n,\,r\geq 0,\,n+r\geq 1$, and let $m\geq 1.$ Then the
$r$-Whitney number of the first kind $V_{m}^{(r)}(n,k)$ is equal to the number
of coloured permutations in $S_{n+r}$ which are the product of $k+r$ disjoint
cycles such that
$\bullet$ the distinguished elements $1,\dots,r$ belong to distinct cycles,
$\bullet$ the smallest elements of the cycles are not coloured,
$\bullet$ an element in a cycle containing a distinguished element is not
coloured if there are no smaller numbers on the arc from the distinguished
element to this element,
$\bullet$ the remaining elements are coloured with $m$ colours.
Further, $V_{m}^{(0)}(0,0)$ is defined as $V_{m}^{(0)}(0,0)=1.$

Let $0\leq k\leq n,\,r\geq 0,\,n+r\geq 1$, and let $m\geq 1.$ Then the
$r$-Whitney number of the second kind $W_{m}^{(r)}(n,k)$ is equal to the
number of coloured partitions of $\left\\{1,\dots,n+r\right\\}$ into $k+r$
nonempty subsets such that
$\bullet$ the distinguished elements $1,\dots,r$ belong to distinct blocks,
$\bullet$ the smallest elements of the blocks are not coloured,
$\bullet$ elements in blocks containing a distinguished element are not
coloured,
$\bullet$ the remaining elements are coloured with $m$ colours.
Further, $W_{m}^{(0)}(0,0)$ is defined as $W_{m}^{(0)}(0,0)=1$.

Carlitz initiated the study of degenerate versions of some special numbers and
polynomials in [1]. It regained the interests of some mathematicians and
yielded many interesting results. For some of these, one refers to [5–9, 11
and references therein]. Here we would like to mention only one thing. In [7],
we were led to introduce the notions of degenerate Sheffer sequences and
$\lambda$-Sheffer sequences, starting from the question that what if we
replace the usual exponential function appearing in the generating functions
of Sheffer sequences by the degenerate exponential function as in (11).

The aim of this paper is to introduce the fully degenerate Bell and the fully
degenerate Dowling polynomials and investigate some properties and identities
for them with the help of $\lambda$-umbral calculus.

In more detail, we find an explicit expression and Dobinski-like formula for
the fully degenerate Bell polynomials, and the generating function for the
fully degenerate Dowling polynomials. Then, by using the $\lambda$-umbral
calculus, we derive a formula expressing any polynomial in terms of the fully
degenerate Bell polynomials. Then, by applying this formula to the degenerate
Bernoulli polynomials, the $\lambda$-falling factorial polynomials and the
degenerate poly-Bell polynomials we represent them in terms of the fully
degenerate Bell polynomials. In addition, we express the degenerate Bell
polynomials in terms of the degenerate bernoulli polynomials of the second
kind. Then we deduce a formula representing any polynomial in terms of the
fully degenerate Dowling polynomials. Applying this formula to the degenerate
Bernoulli polynomials and the $\lambda$-falling factorial polynomials, we
express them in terms of the fully degenerate Dowling polynomials. Further, we
express the fully degenerate Dowling polynomials in terms of the the fully
degenerate Bell polynomials.

##  2\. Preliminaries

In this section, we briefly go over the $\lambda$-umbral calculus, for the
details of which we let the reader refer to [7]. By the way, [15] is an
excellent reference for the usual umbral calculus which corresponds to
$0$-umbral calculus. And then we recall some special polynomials and numbers
that are needed in this paper.

Let $\mathbb{C}$ be the field of complex numbers. Let

$\mathcal{F}=\bigg{\\{}f(t)=m_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}\ \bigg{|}\
a_{k}\in\mathbb{C}\bigg{\\}},$

and let

$\mathbb{P}=\mathbb{C}[x]=\bigg{\\{}\sum_{i=0}^{\infty}a_{i}x^{i}\ \bigg{|}\
\textrm{$a_{i}\in\mathbb{C}$\ with $a_{i}=0$ for all but finite number of
$i$}\bigg{\\}}.$

Let $\mathbb{P}^{*}$ be the vector space of all linear functionals on
$\mathbb{P}$. If $\langle L|p(x)\rangle_{\lambda}$ denotes the action of the
$\lambda$-linear functional $L$ on the polynomial $p(x)$, then the vector
space operations on $\mathbb{P}^{*}$ are defined by

$\langle L+M|p(x)\rangle_{\lambda}=\langle L|p(x)\rangle_{\lambda}+\langle
M|p(x)\rangle_{\lambda},\quad\langle cL|p(x)\rangle_{\lambda}=c\langle
L|p(x)\rangle_{\lambda},$

where $c$ is a complex constant.

For $f(t)\in\mathcal{F}$, with
$f(t)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}$, we define the
$\lambda$-linear functional on $\mathbb{P}$ by

(5) $\langle f(t)|(x)_{k,\lambda}\rangle_{\lambda}=a_{k},\quad(k\geq 0),$

where

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

By (5), we get

(6) $\langle t^{k}|(x)_{n,\lambda}\rangle_{\lambda}=\langle
1|(t^{k})_{\lambda}(x)_{n,\lambda}\rangle_{\lambda}=n!\delta_{n,k},\quad(n,k\geq
0),$

where $\delta_{n,k}$ is the Kronecker’s symbol.

For $k\geq 0$, the $\lambda$-differential operator $t^{k}$ on $\mathbb{P}$ is
defined by

(7)
$(t^{k})_{\lambda}(x)_{n,\lambda}=\left\\{\begin{array}[]{ccc}(n)_{k}x^{n-k},&\textrm{if
$k\leq n$,}\\\ 0,&\textrm{otherwise.}\end{array}\right.$

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

Let $f(t)=\sum_{k=0}^{\infty}a_{k}\frac{t^{k}}{k!}\in\mathcal{F}$. The order
$o(f(t))$ of $f(t)$ is the smallest integer $k$ which $a_{k}$ does not vanish.

For $f(t),g(t)\in\mathcal{F}$ with $o(f(t))=1,\ o(g(t))=0$, there exists a
unique sequence $s_{n,\lambda}(x)$ of polynomials such that

(8)
$\big{\langle}g(t)(f(t))^{k}\big{|}S_{n,\lambda}(x)\big{\rangle}_{\lambda}=n!\delta_{n,k},\quad(n,k\geq
0).$

The sequence $S_{n,\lambda}(x)$ is called the $\lambda$-Sheffer sequence for
$(g(t),f(t))$, which is denoted by
$S_{n,\lambda}(x)\sim(g(t),f(t))_{\lambda}$.

For any nonzero $\lambda\in\mathbb{R}$, the degenerate exponential functions
are defined as

(9)
$e_{\lambda}^{x}(t)=\sum_{k=0}^{\infty}(x)_{k,\lambda}\frac{t^{k}}{k!},\quad
e_{\lambda}(t)=e_{\lambda}^{1}(t),\quad(\mathrm{see}\ [8,11-13]).$

Let $\log_{\lambda}(t)$ be the compositional inverse function of
$e_{\lambda}(t)$ such that
$\log_{\lambda}\big{(}e_{\lambda}(t)\big{)}=e_{\lambda}\big{(}\log_{\lambda}(t)\big{)}=t$.

Then we have

(10)
$\log_{\lambda}(1+t)=\sum_{n=1}^{\infty}\lambda^{n-1}(1)_{n,1/\lambda}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[6]).$

From (8), we note that $s_{n,\lambda}(x)\sim(g(t),f(t))_{\lambda}$ if and only
if

(11)
$\frac{1}{g(\bar{f}(t))}e_{\lambda}^{x}\big{(}\bar{f}(t)\big{)}=\sum_{n=0}^{\infty}s_{n,\lambda}(x)\frac{t^{n}}{n!},$

for all $x\in\mathbb{C}$, where $\bar{f}(t)$ is the compositional inverse
function of $f(t)$ such that $f(\bar{f}(t))=\bar{f}(f(t))=t$.

For $s_{n,\lambda}(x)\sim(g(t),f(t))_{\lambda}$,
$r_{n,\lambda}(x)\sim(h(t),l(t))_{\lambda}$, we have

(12) $s_{n,\lambda}(x)=\sum_{k=0}^{n}c_{n,k}r_{k,\lambda}(x),$

where

(13)
$c_{n,k}=\frac{1}{k!}\bigg{\langle}\frac{h(\bar{f}(t))}{g(\bar{f}(t))}\big{(}l(\bar{f}(t))\big{)}^{k}\
\bigg{|}\ (x)_{n,\lambda}\bigg{\rangle}_{\lambda},\quad(n\geq k\geq 0).$

Throughout this paper, we assume that $m$ is any fixed positive integer.
In [9], the degenerate Whitney numbers of the second kind are given by

(14)
$e_{\lambda}(t)\frac{1}{k!}\bigg{(}\frac{e_{\lambda}^{m}(t)-1}{m}\bigg{)}^{k}=\sum_{n=k}^{\infty}W_{m,\lambda}(n,k)\frac{t^{n}}{n!},\quad(k\geq
0).$

Recently, Kim-Kim introduced the degenerate Dowling polynomials given by

(15)
$e_{\lambda}(t)e^{x\big{(}\frac{e_{\lambda}^{m}(t)-1}{m}\big{)}}=\sum_{n=0}^{\infty}D_{m,\lambda}(n,x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[9]).$

From (15), we note that

(16) $D_{m,\lambda}(n,x)=\sum_{k=0}^{n}W_{m,\lambda}(n,k)x^{k},\quad(n\geq
0),\quad(\mathrm{see}\ [9]).$

In [7], Kim et al. introduced the partial degenerate Bell polynomials which
are given by

(17)
$e^{x(e_{\lambda}(t)-1)}=\sum_{n=0}^{\infty}\mathrm{Bel}_{n,\lambda}(x)\frac{t^{n}}{n!}.$

Note that $\displaystyle\lim_{\lambda\rightarrow
0}\mathrm{Bel}_{n,\lambda}(x)=\mathrm{Bel}_{n}(x)\displaystyle$. Here
$\mathrm{Bel}_{n}(x)$ are the ordinary Bell polynomials given by

$e^{x(e^{t}-1)}=\sum_{n=0}^{\infty}\mathrm{Bel}_{n}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[2,15]).$

As is well known, the Stirling numbers of the first kind are defined by

(18) $(x)_{n}=\sum_{l=0}^{n}S_{1}(n,l)x^{l},\quad(n\geq 0),\quad(\mathrm{see}\
[2,15]).$

By the inversion of (18), the Stirling numbers of the second kind are defined
by

(19) $x^{n}=\sum_{l=0}^{n}S_{2}(n,l)(x)_{l},\quad(n\geq 0),\quad(\mathrm{see}\
[15,16]).$

In [1], Carlitz introduced the degenerate Bernoulli polynomials defined by

(20)
$\frac{t}{e_{\lambda}(t)-1}e_{\lambda}^{x}(t)=\sum_{n=0}^{\infty}\beta_{n,\lambda}(x)\frac{t^{n}}{n!}.$

When $x=0$, $\beta_{n,\lambda}=\beta_{n,\lambda}(0)$ are called the degenerate
Bernoulli numbers.

The degenerate Bernoulli polynomials of the second kind are given by

(21)
$\frac{t}{\log_{\lambda}(1+t)}(1+t)^{x}=\sum_{n=0}^{\infty}b_{n,\lambda}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[10]).$

When $x=0$, $b_{n,\lambda}=b_{n,\lambda}(0)$ are called the degenerate
Bernoulli numbers of the second kind.

For any $k\in\mathbb{Z}$, the degenerate polyexponential function is defined
by Kim-Kim as

(22)
$\mathrm{Ei}_{k,\lambda}(x)=\sum_{n=1}^{\infty}\frac{(1)_{n,\lambda}}{(n-1)!n^{k}}x^{n},\quad(\mathrm{see}\
[11]).$

The degenerate poly-Bell polynomials are defined by Kim et al. as

(23)
$\frac{\mathrm{Ei}_{k,\lambda}(\log_{\lambda}(1+t))}{e_{\lambda}(t)-1}e_{\lambda}^{x}(t)=\sum_{n=0}^{\infty}B_{n,\lambda}^{(k)}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[11]).$

When $x=0$, $B_{n,\lambda}^{(k)}=B_{n,\lambda}^{(k)}(0)$ are called the
degenerate poly-Bernoulli numbers.

The degenerate Stirling numbers of the first kind are defined by

(24)
$\frac{1}{k!}\big{(}\log_{\lambda}(1+t)\big{)}^{k}=\sum_{n=k}^{\infty}S_{1,\lambda}(n,k)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[6]).$

Note that $\lim_{\lambda\rightarrow 0}S_{1,\lambda}(n,k)=S_{1}(n,k),\ (n\geq
k\geq 0)$.

By the inversion of (24), the degenerate Stirling numbers of the second kind
are defined by

(25)
$\frac{1}{k!}\big{(}e_{\lambda}(t)-1\big{)}^{k}=\sum_{n=k}^{\infty}S_{2,\lambda}(n,k)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[6]).$

Note that $\displaystyle\lim_{\lambda\rightarrow
0}S_{2,\lambda}(n,k)=S_{2}(n,k)\displaystyle,\ (n\geq k\geq 0)$.

##  3\. Fully degenerate Bell and fully degenerate Dowling polynomials arising
from $\lambda$-umbral calculus

Here we introduce the fully degenerate Bell and the fully degenerate Dowling
polynomials, and investigate some properties and identities relating to them
with the help of $\lambda$-umbral calculus.
First, we define the fully degenerate Bell polynomials which are given by
$\phi_{n,\lambda}(x)\sim(1,\log_{\lambda}(1+t)\big{)}_{\lambda}$. Then, by
(11), we have

(26)
$e_{\lambda}^{x}(e_{\lambda}(t)-1)=\sum_{n=0}^{\infty}\phi_{n,\lambda}(x)\frac{t^{n}}{n!}.$

Note that $\displaystyle\lim_{\lambda\rightarrow
0}\phi_{n,\lambda}(x)=\mathrm{Bel}_{n}(x)\displaystyle$.

Next, we observe that

(27) $\displaystyle\sum_{n=0}^{\infty}\phi_{n,\lambda}(x+y)\frac{t^{n}}{n!}$
$\displaystyle=e_{\lambda}^{x}(e_{\lambda}(t)-1)e_{\lambda}^{y}(e_{\lambda}(t)-1)$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{l=0}^{n}\binom{n}{l}\phi_{l,\lambda}(x)\phi_{n-l,\lambda}(y)\bigg{)}\frac{t^{n}}{n!}.$

From (27), we note that

(28)
$\phi_{n,\lambda}(x+y)=\sum_{l=0}^{n}\binom{n}{l}\phi_{l,\lambda}(x)\phi_{n-l,\lambda}(y),\quad(n\geq
0).$

Further, we see that

(29) $\displaystyle e_{\lambda}^{x}\big{(}e_{\lambda}(t)-1\big{)}$
$\displaystyle=\sum_{k=0}^{\infty}(x)_{k,\lambda}\frac{1}{k!}\big{(}e_{\lambda}(k)-1\big{)}^{k}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}(x)_{k,\lambda}S_{2,\lambda}(n,k)\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by (26) and (29), we obtain the following lemma.

######  Lemma 1.

For $n\geq 0$, we have

$\phi_{n,\lambda}(x)=\sum_{k=0}^{n}S_{2,\lambda}(n,k)(x)_{k,\lambda}.$

Now, we note that

(30) $\displaystyle e_{\lambda}^{x}\big{(}e_{\lambda}(t)-1\big{)}$
$\displaystyle=(1-\lambda+\lambda
e_{\lambda}(t))^{\frac{x}{\lambda}}=(1-\lambda)^{\frac{x}{\lambda}}\bigg{(}1+\frac{\lambda}{1-\lambda}e_{\lambda}(t)\bigg{)}^{\frac{x}{\lambda}}$
$\displaystyle=e_{\lambda}^{x}(-1)\bigg{(}1+\frac{\lambda}{1-\lambda}e_{\lambda}(t)\bigg{)}^{\frac{x}{\lambda}}$
$\displaystyle=e_{\lambda}^{x}(-1)\sum_{k=0}^{\infty}(x)_{k,\lambda}\bigg{(}\frac{1}{1-\lambda}\bigg{)}^{k}\frac{1}{k!}e_{\lambda}^{k}(t)$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{\\{}e_{\lambda}^{x}(-1)\sum_{k=0}^{\infty}\frac{(k)_{n,\lambda}}{k!}\bigg{(}\frac{1}{1-\lambda}\bigg{)}^{k}(x)_{k,\lambda}\bigg{\\}}\frac{t^{n}}{n!}.$

Therefore, by (26) and (30), we obtain the following theorem.

######  Theorem 2 (Dobinski-like formula).

For $n\geq 0$, we have

$\phi_{n,\lambda}(x)=e_{\lambda}^{x}(-1)\sum_{k=0}^{\infty}\frac{(k)_{n,\lambda}}{k!}\bigg{(}\frac{1}{1-\lambda}\bigg{)}^{k}(x)_{k,\lambda}.$

In view of Lemma 1 and (16), we define the fully degenerate Dowling
polynomials by

(31)
$d_{m,\lambda}(n,x)=\sum_{k=0}^{n}W_{m,\lambda}(n,k)(x)_{k,\lambda},\quad(n\geq
0).$

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

From (14) and (31), we note that

(32) $\displaystyle\sum_{n=0}^{\infty}d_{m,\lambda}(n,x)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}W_{m,\lambda}(n,k)(x)_{k,\lambda}\bigg{)}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{(}\sum_{n=k}^{\infty}W_{m,\lambda}(n,k)\frac{t^{n}}{n!}\bigg{)}(x)_{k,\lambda}$
$\displaystyle=\sum_{k=0}^{\infty}e_{\lambda}(t)\frac{1}{k!}\bigg{(}\frac{e_{\lambda}^{m}(t)-1}{m}\bigg{)}^{k}(x)_{k,\lambda}$
$\displaystyle=e_{\lambda}(t)e_{\lambda}^{x}\bigg{(}\frac{e_{\lambda}^{m}(t)-1}{m}\bigg{)}.$

Therefore, by (32), we obtain the generating function of fully degenerate
Dowling polynomials.

######  Theorem 3.

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

Note that $\displaystyle\lim_{\lambda\rightarrow
0}d_{m,\lambda}(n,x)=D_{m}(n,x)$. Here $D_{m}(n,x)$ are the ordinary Dowling
polynomials given by

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

Let us assume that

$\mathbb{P}_{n}=\\{p(x)\in\mathbb{C}[x]\ |\ \deg p(x)\leq n\\},\quad(n\geq
0).$

Then $\mathbb{P}_{n}$ is an $(n+1)$-dimensional vector space over
$\mathbb{C}$.

For $p(x)\in\mathbb{P}_{n}$, we let

(33) $p(x)=\sum_{k=0}^{n}C_{k}\phi_{k,\lambda}(x).$

Then, by (8), we get

(34)
$\displaystyle\big{\langle}(\log_{\lambda}(1+t))^{k}\big{|}p(x)\rangle_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}C_{l}\big{\langle}(\log_{\lambda}(1+t))^{k}\big{|}\phi_{l,\lambda}(x)\big{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}C_{l}k!\delta_{k,l}=k!C_{k}.$

Thus, by (34), we get

(35)
$C_{k}=\frac{1}{k!}\big{\langle}(\log_{\lambda}(1+t))^{k}\big{|}p(x)\big{\rangle}_{\lambda}.$

Therefore, by (33) and (35), we obtain the following theorem.

######  Theorem 4.

For $p(x)\in\mathbb{P}_{n}$, we have

$p(x)=\sum_{k=0}^{n}C_{k}\phi_{k,\lambda}(x),$

where

$C_{k}=\frac{1}{k!}\big{\langle}(\log_{\lambda}(1+t))^{k}\big{|}p(x)\big{\rangle}_{\lambda}.$

Let us take

$p(x)=\beta_{n,\lambda}(x)=\sum_{l=0}^{n}\binom{n}{l}\beta_{n-l,\lambda}(x)_{l,\lambda}\in\mathbb{P}_{n}.$

Then, by Theorem 4, we get

(36) $\beta_{n,\lambda}(x)=\sum_{k=0}^{n}C_{k}\phi_{k,\lambda}(x),$

where

(37) $\displaystyle C_{k}$
$\displaystyle=\frac{1}{k!}\big{\langle}(\log_{\lambda}(1+t))^{k}\big{|}\beta_{n,\lambda}(x)\big{\rangle}_{\lambda}$
$\displaystyle=\frac{1}{k!}\sum_{l=0}^{n}\binom{n}{l}\beta_{n-l,\lambda}\big{\langle}(\log_{\lambda}(1+t))^{k}\big{|}(x)_{l,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\binom{n}{l}\beta_{n-l,\lambda}\sum_{j=k}^{\infty}S_{1,\lambda}(j,k)\frac{1}{j!}\langle
t^{j}|(x)_{l,\lambda}\rangle_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\binom{n}{l}\beta_{n-l,\lambda}S_{1,\lambda}(l,k).$

Therefore, by (36) and (37), we obtain the following theorem.

######  Theorem 5.

For $n\geq 0$, we have

$\beta_{n,\lambda}(x)=\sum_{k=0}^{n}\bigg{\\{}\sum_{l=0}^{n}\binom{n}{l}\beta_{n-l,\lambda}S_{1,\lambda}(l,k)\bigg{\\}}\phi_{k,\lambda}(x).$

Let us take $p(x)=(x)_{n,\lambda}\in\mathbb{P}_{n}$. Then, by Theorem 4, we
get

(38) $(x)_{n,\lambda}=\sum_{k=0}^{n}C_{k}\phi_{k,\lambda}(x),$

where

(39) $\displaystyle C_{k}$
$\displaystyle=\frac{1}{k!}\big{\langle}(\log_{\lambda}(1+t))^{k}\big{|}(x)_{n,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\sum_{m=k}^{\infty}S_{1,\lambda}(m,k)\frac{1}{m!}\langle
t^{m}|(x)_{n,\lambda}\rangle_{\lambda}$ $\displaystyle=S_{1,\lambda}(n,k).$

Therefore, by (38) and (39), we obtain the following theorem.

######  Theorem 6.

For $n\geq 0$, we have

$(x)_{n,\lambda}=\sum_{k=0}^{n}S_{1,\lambda}(n,k)\phi_{k,\lambda}(x).$

We note that

$\displaystyle
B_{n,\lambda}^{(k)}(x)\sim\bigg{(}\frac{e_{\lambda}(t)-1}{\mathrm{Ei}_{k,\lambda}(\log_{\lambda}(1+t))},t\bigg{)}_{\lambda},$
$\displaystyle\phi_{n,\lambda}(x)\sim(1,\log_{\lambda}(1+t))_{\lambda}.$

We assume that

(40) $B_{n,\lambda}^{(k)}(x)=\sum_{k=0}^{n}C_{n,k}\phi_{k,\lambda}(x).$

Then, by (12), we get

(41) $\displaystyle C_{n,k}$
$\displaystyle=\frac{1}{k!}\bigg{\langle}\frac{\mathrm{Ei}_{k,\lambda}(\log_{\lambda}(1+t)}{e_{\lambda}(t)-1}\big{(}\log_{\lambda}(1+t)\big{)}^{k}\bigg{|}(x)_{n,\lambda}\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=k}^{n}\frac{S_{1,\lambda}(l,k)}{l!}\bigg{\langle}\frac{\mathrm{Ei}_{k,\lambda}(\log_{\lambda}(1+t)}{e_{\lambda}(t)-1}t^{l}\bigg{|}(x)_{n,\lambda}\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=k}^{n}S_{1,\lambda}(l,k)\binom{n}{l}\sum_{j=0}^{\infty}B_{j,\lambda}^{(k)}\frac{1}{j!}\langle
t^{j}|(x)_{n-l,\lambda}\rangle_{\lambda}$
$\displaystyle=\sum_{l=k}^{n}\binom{n}{l}S_{1,\lambda}(l,k)B_{n-l,\lambda}^{(k)}.$

Therefore, by (40) and (41), we obtain the following theorem.

######  Theorem 7.

For $n\geq 0$, we have

$B_{n,\lambda}^{(k)}(x)=\sum_{k=0}^{n}\bigg{\\{}\sum_{l=k}^{n}\binom{n}{l}S_{1,\lambda}(l,k)B_{n-l,\lambda}^{(k)}\bigg{\\}}\phi_{k,\lambda}(x).$

From (21), we have

(42)
$b_{n,\lambda}(x)\sim\bigg{(}\frac{t}{e_{\lambda}(t)-1},e_{\lambda}(t)-1\bigg{)}_{\lambda}.$

Thus, by (12) and (13), we get

(43) $\phi_{n,\lambda}(x)=\sum_{k=0}^{n}C_{n,k}b_{k,\lambda}(x),$

where, for $k=0$,

(44) $\displaystyle C_{n,0}$
$\displaystyle=\bigg{\langle}\frac{e_{\lambda}(t)-1}{e_{\lambda}(e_{\lambda}(t)-1)-1}\bigg{|}(x)_{n,\lambda}\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\frac{\beta_{l,\lambda}}{l!}\big{\langle}(e_{\lambda}(t)-1)^{l}\big{|}(x)_{n,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\beta_{l,\lambda}S_{2,\lambda}(n,l);$

for $k\geq 1$,

(45) $\displaystyle C_{n,k}$
$\displaystyle=\frac{1}{k!}\bigg{\langle}\frac{e_{\lambda}(t)-1}{e_{\lambda}(e_{\lambda}(t)-1)-1}\big{(}e_{\lambda}(e_{\lambda}(t)-1)-1\big{)}^{k}\bigg{|}(x)_{n,\lambda}\bigg{\rangle}_{\lambda}$
$\displaystyle=\frac{1}{k!}\big{\langle}(e_{\lambda}(t)-1)\big{(}e_{\lambda}(e_{\lambda}(t)-1)-1\big{)}^{k-1}\big{|}(x)_{n,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\frac{1}{k!}\big{\langle}e_{\lambda}(t)\big{(}e_{\lambda}(e_{\lambda}(t)-1)-1\big{)}^{k-1}\big{|}(x)_{n,\lambda}\big{\rangle}_{\lambda}-\frac{1}{k!}\big{\langle}\big{(}e_{\lambda}(e_{\lambda}(t)-1)-1\big{)}^{k-1}\big{|}(x)_{n,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{n-1}\binom{n}{j}(1)_{n-j,\lambda}\sum_{l=0}^{k-1}\binom{k-1}{l}(-1)^{k-1-l}\big{\langle}e_{\lambda}^{l}(e_{\lambda}(t)-1)\big{|}(x)_{j,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{n-1}\sum_{l=0}^{k-1}\binom{n}{j}\binom{k-1}{l}(1)_{n-j,\lambda}(-1)^{k-1-l}\sum_{i=0}^{\infty}\phi_{i,\lambda}(l)\frac{1}{i!}\big{\langle}t^{i}|(x)_{j,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\frac{1}{k!}\sum_{j=0}^{n-1}\sum_{l=0}^{k-1}\binom{n}{j}\binom{k-1}{l}(1)_{n-j,\lambda}(-1)^{k-1-l}\phi_{j,\lambda}(l).$

Therefore, by (43), (44) and (45), we obtain the following theorem.

######  Theorem 8.

For $n\geq 0$, we have

$\phi_{n,\lambda}(x)=\sum_{l=0}^{n}\beta_{l,\lambda}S_{2,\lambda}(n,l)+\sum_{k=1}^{n}\bigg{\\{}\frac{1}{k!}\sum_{j=0}^{n-1}\sum_{l=0}^{k-1}\binom{n}{j}\binom{k-1}{l}(1)_{n-j,\lambda}(-1)^{k-1-l}\phi_{j,\lambda}(l)\bigg{\\}}b_{k,\lambda}(x).$

From Theorem 3, we note that

(46) $\displaystyle d_{m,\lambda}(n,x)$
$\displaystyle\sim\bigg{(}(mt+1)^{-\frac{1}{m}},\
\frac{1}{\lambda}\big{(}(mt+1)^{\frac{\lambda}{m}}-1\big{)}\bigg{)}_{\lambda}$
$\displaystyle\sim\bigg{(}(mt+1)^{-\frac{1}{m}},\
\frac{1}{m}\log_{\frac{\lambda}{m}}(mt+1)\bigg{)}_{\lambda}.$

For $p(x)\in\mathbb{P}_{n}$, let us assume that

(47) $p(x)=\sum_{k=0}^{n}C_{k}d_{m,\lambda}(k,x).$

Then we note that

(48)
$\displaystyle\bigg{\langle}(mt+1)^{-\frac{1}{m}}\bigg{(}\frac{1}{m}\log_{\frac{\lambda}{m}}(1+mt)\bigg{)}^{k}\bigg{|}p(x)\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}C_{l}\bigg{\langle}(mt+1)^{-\frac{1}{m}}\bigg{(}\frac{1}{m}\log_{\frac{\lambda}{m}}(1+mt)\bigg{)}^{k}\bigg{|}d_{m,\lambda}(l,x)\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}C_{l}l!\delta_{l,k}=C_{k}k!.$

By (48), we get

(49)
$C_{k}=\frac{1}{k!}\bigg{\langle}(mt+1)^{-\frac{1}{m}}\bigg{(}\frac{1}{m}\log_{\frac{\lambda}{m}}(1+mt)\bigg{)}^{k}\bigg{|}p(x)\bigg{\rangle}_{\lambda}.$

Therefore, by (47) and (49), we obtain the following theorem.

######  Theorem 9.

For $p(x)\in\mathbb{P}_{n}$, we have

$p(x)=\sum_{k=0}^{n}C_{k}d_{m,\lambda}(k,x),$

where

$C_{k}=\frac{1}{k!}\bigg{\langle}(mt+1)^{-\frac{1}{m}}\bigg{(}\frac{1}{m}\log_{\frac{\lambda}{m}}(1+mt)\bigg{)}^{k}\bigg{|}p(x)\bigg{\rangle}_{\lambda}.$

Let us take $p(x)=\beta_{n,\lambda}(x)$. Then we have

(50) $\beta_{n,\lambda}(x)=\sum_{k=0}^{n}C_{k}d_{m,\lambda}(k,x),$

where

(51) $\displaystyle C_{k}$
$\displaystyle=\frac{1}{k!}\bigg{\langle}(mt+1)^{-\frac{1}{m}}\bigg{(}\frac{1}{m}\log_{\frac{\lambda}{m}}(1+mt)\bigg{)}^{k}\bigg{|}\beta_{n,\lambda}(x)\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\binom{n}{l}\beta_{n-l,\lambda}\frac{1}{k!}\bigg{\langle}(mt+1)^{-\frac{1}{m}}\bigg{(}\frac{1}{m}\log_{\frac{\lambda}{m}}(1+mt)\bigg{)}^{k}\bigg{|}(x)_{l,\lambda}\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\binom{n}{l}\beta_{n-l,\lambda}\sum_{j=k}^{l}S_{1,\frac{\lambda}{m}}(j,k)m^{j-k}\frac{1}{j!}\langle(mt+1)^{-\frac{1}{m}}t^{j}|(x)_{l,\lambda}\rangle_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\binom{n}{l}\beta_{n-l,\lambda}\sum_{j=k}^{l}S_{1,\frac{\lambda}{m}}(j,k)\binom{l}{j}m^{j-k}\langle(mt+1)^{-\frac{1}{m}}|(x)_{l-j,\lambda}\rangle_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\sum_{j=k}^{l}\binom{n}{l}\beta_{n-l,\lambda}S_{1,\frac{\lambda}{m}}(j,k)m^{j-k}\binom{l}{j}\sum_{i=0}^{\infty}(-1)^{i}\bigg{(}\frac{1}{m}\bigg{)}^{i}\frac{1}{i!}\big{\langle}(\log(mt+1)\big{)}^{i}|(x)_{l-j,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=0}^{n}\sum_{j=k}^{l}\sum_{i=0}^{l-j}\binom{n}{l}\binom{l}{j}\beta_{n-l,\lambda}S_{1,\frac{\lambda}{m}}(j,k)m^{j-k}(-1)^{i}\bigg{(}\frac{1}{m}\bigg{)}^{i}S_{1}(l-j,i)m^{l-j}$
$\displaystyle=\sum_{l=0}^{n}\sum_{j=k}^{l}\sum_{i=0}^{l-j}\binom{n}{l}\binom{l}{j}\beta_{n-l,\lambda}S_{1,\frac{\lambda}{m}}(j,k)m^{l-k-i}(-1)^{i}S_{1}(l-j,i).$

Therefore, by (50) and (51), we obtain the following theorem.

######  Theorem 10.

For $n\geq 0$, we have

$\beta_{n,\lambda}(x)=\sum_{k=0}^{n}\bigg{\\{}\sum_{l=0}^{n}\sum_{j=k}^{l}\sum_{i=0}^{l-j}\binom{n}{l}\binom{l}{j}\beta_{n-l,\lambda}S_{1,\frac{\lambda}{m}}(j,k)m^{l-k-i}(-1)^{i}S_{1}(l-j,i)\bigg{\\}}d_{m,\lambda}(k,x).$

Let us take $p(x)=(x)_{n,\lambda}\in\mathbb{P}_{n}$. Then, by Theorem 9, we
get

(52) $(x)_{n,\lambda}=\sum_{k=0}^{n}C_{k}d_{m,\lambda}(k,x),$

where

(53) $\displaystyle C_{k}$
$\displaystyle=\frac{1}{k!}\bigg{\langle}(mt+1)^{-\frac{1}{m}}\bigg{(}\frac{1}{m}\log_{\frac{\lambda}{m}}(1+mt)\bigg{)}^{k}\bigg{|}(x)_{n,\lambda}\bigg{\rangle}_{\lambda}$
$\displaystyle=\frac{1}{m^{k}}\sum_{l=k}^{n}S_{1,\frac{\lambda}{m}}(l,k)\frac{m^{l}}{l!}\big{\langle}(mt+1)^{-\frac{1}{m}}t^{l}\big{|}(x)_{n,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=k}^{n}S_{1,\frac{\lambda}{m}}(l,k)m^{l-k}\binom{n}{l}\big{\langle}e^{-\frac{1}{m}\log(mt+1)}\big{|}(x)_{n-l,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=k}^{n}S_{1,\frac{\lambda}{m}}(l,k)m^{l-k}\binom{n}{l}\sum_{i=0}^{n-l}\bigg{(}-\frac{1}{m}\bigg{)}^{i}\frac{1}{i!}\big{\langle}\big{(}\log(mt+1)\big{)}^{i}\big{|}(x)_{n-l,\lambda}\big{\rangle}_{\lambda}$
$\displaystyle=\sum_{l=k}^{n}S_{1,\frac{\lambda}{m}}(l,k)m^{l-k}\binom{n}{l}\sum_{i=0}^{n-l}(-1)^{i}\bigg{(}\frac{1}{m}\bigg{)}^{i}\sum_{j=i}^{\infty}S_{1}(j,i)m^{j}\frac{1}{j!}\langle
t^{j}|(x)_{n-l,\lambda}\rangle_{\lambda}$
$\displaystyle=\sum_{l=k}^{n}S_{1,\frac{\lambda}{m}}(l,k)m^{l-k}\binom{n}{l}\sum_{i=0}^{n-l}(-1)^{i}m^{-i}S_{1}(n-l,i)m^{n-l}$
$\displaystyle=\sum_{l=k}^{n}\sum_{i=0}^{n-l}\binom{n}{l}S_{1,\frac{\lambda}{m}}(l,k)S_{1}(n-l,i)m^{n-k-i}(-1)^{i}.$

Therefore, by (52) and (53),we obtain the following theorem.

######  Theorem 11.

For $n\geq 0$, we have

$(x)_{n,\lambda}=\sum_{k=0}^{n}\bigg{\\{}\sum_{l=k}^{n}\sum_{i=0}^{n-l}\binom{n}{l}S_{1,\frac{\lambda}{m}}(l,k)S_{1}(n-l,i)m^{n-k-i}(-1)^{i}\bigg{\\}}d_{m,\lambda}(k,x).$

Recalling that $d_{m,\lambda}(n,x)\sim\big{(}(mt+1)^{-\frac{1}{m}},\
\frac{1}{m}\log_{\frac{\lambda}{m}}(mt+1)\big{)}_{\lambda}$,
$\phi_{n,\lambda}(x)\sim(1,\log_{\lambda}(1+t))_{\lambda}$,
let us assume that

(54) $d_{m,\lambda}(n,x)=\sum_{k=0}^{n}C_{n,k}\phi_{k,\lambda}(x).$

Then, by (13), we get

(55) $\displaystyle C_{n,k}$
$\displaystyle=\frac{1}{k!}\bigg{\langle}e_{\lambda}(t)\bigg{(}\log_{\lambda}\bigg{(}1+\frac{1}{m}(e_{\lambda}^{m}(t)-1)\bigg{)}\bigg{)}^{k}\bigg{|}(x)_{n,\lambda}\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{j=k}^{n}S_{1,\lambda}(j,k)\frac{1}{j!}\bigg{\langle}e_{\lambda}(t)\bigg{(}\frac{1}{m}\big{(}e_{\lambda}^{m}(t)-1\big{)}\bigg{)}^{j}\bigg{|}(x)_{n,\lambda}\bigg{\rangle}_{\lambda}$
$\displaystyle=\sum_{j=k}^{n}S_{1,\lambda}(j,k)\sum_{l=j}^{\infty}W_{m,\lambda}(l,j)\frac{1}{l!}\langle
t^{l}|(x)_{n,\lambda}\rangle_{\lambda}$
$\displaystyle=\sum_{j=k}^{n}S_{1,\lambda}(j,k)W_{m,\lambda}(n,j).$

Therefore, by (54) and (55), we obtain the following theorem.

######  Theorem 12.

For $n\geq 0$, we have

$d_{m,\lambda}(n,x)=\sum_{k=0}^{n}\bigg{\\{}\sum_{j=k}^{n}S_{1,\lambda}(j,k)W_{m,\lambda}(n,j)\bigg{\\}}\phi_{k,\lambda}(x).$

Now, we observe that

(56) $\displaystyle
e_{\lambda}^{x}\bigg{(}\frac{1}{m}\big{(}e_{\lambda}^{m}(t)-1\big{)}\bigg{)}=\sum_{k=0}^{\infty}(x)_{k,\lambda}\bigg{(}\frac{1}{m}\bigg{)}^{k}\frac{1}{k!}\big{(}e_{\frac{\lambda}{m}}(mt)-1\big{)}^{k}$
$\displaystyle=\sum_{k=0}^{\infty}(x)_{k,\lambda}m^{-k}\sum_{n=k}^{\infty}S_{2,\frac{\lambda}{m}}(n,k)m^{n}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}\bigg{(}\frac{x}{m}\bigg{)}_{k,\frac{\lambda}{m}}S_{2,\frac{\lambda}{m}}(n,k)\bigg{)}m^{n}\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\phi_{n,\frac{\lambda}{m}}\bigg{(}\frac{x}{m}\bigg{)}m^{n}\frac{t^{n}}{n!}.$

From (11) and (56), we note that

(57)
$m^{n}\phi_{n,\frac{\lambda}{m}}\bigg{(}\frac{x}{m}\bigg{)}\sim\bigg{(}1,\frac{(mt+1)^{\frac{\lambda}{m}}-1}{\lambda}\bigg{)}_{\lambda}\sim\bigg{(}1,\frac{1}{m}\log_{\frac{\lambda}{m}}(mt+1)\bigg{)}_{\lambda}.$

Let us assume that

(58)
$m^{n}\phi_{n,\frac{\lambda}{m}}\bigg{(}\frac{x}{m}\bigg{)}=\sum_{k=0}^{n}C_{n,k}d_{m,\lambda}(k,x).$

Then, by (13), (46) and (57), we get

(59) $\displaystyle
C_{n,k}=\frac{1}{k!}\bigg{\langle}e_{\lambda}^{-1}(t)t^{k}\bigg{|}(x)_{n,\lambda}\bigg{\rangle}_{\lambda}=\binom{n}{k}\bigg{\langle}e_{\lambda}^{-1}(t)\bigg{|}(x)_{n-k,\lambda}\bigg{\rangle}_{\lambda}=\binom{n}{k}(-1)_{n-k,\lambda}.$

Thus, by (58) and (59), we get

$\phi_{n,\frac{\lambda}{m}}\bigg{(}\frac{x}{m}\bigg{)}=m^{-n}\sum_{k=0}^{n}\binom{n}{k}(-1)_{n-k,\lambda}d_{m,\lambda}(k,x).$

##  4\. Conclusion

In this paper, we introduced the fully degenerate Bell polynomials
$\phi_{n,\lambda}(x)$, which are degenerate versions of the ordinary Bell
polynomials $\mathrm{Bel}_{n}(x).$ Here $\mathrm{Bel}_{n}(x)$ is the natural
extension of the Bell number $\mathrm{Bel}_{n}$, which counts the number of
partitions of a set with $n$ elements into disjoint nonempty subsets. Further,
we introduced the fully degenerate Dowling polynomials $d_{m,\lambda}(n,x)$
which are degenerate versions of the ordinary Dowling polynomials
$D_{m}(n,x)$. Here $D_{m}(n,x)$ is the natural extension of the Whitney
numbers of the second kind $W_{m}(n,k)$, which is equal to the number of
coloured partitions of $\left\\{1,\dots,n+1\right\\}$ into $k+1$ nonempty
subsets satisfying the conditions stated in Introduction with $r=1$.

We derived some identities and properties for those degenerate polynomials by
using the recently introduced $\lambda$-umbral calculus. Especially, we
derived formulas expressing any polynomial in terms of the fully degenerate
Bell and the fully degenerate Dowling polynomials and applied them to the
degenerate Bernoulli polynomials and degenerate falling factorial polynomials.

It is one of our future projects to continue to work on various degenerate
versions of many special numbers and polynomials.

## References

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# Correlation of Fréchet Audio Distance With Human Perception of Environmental
Audio Is Embedding Dependant

1st Modan Tailleur  École Centrale Nantes
CNRS, LS2N, UMR 6004
F-44000 Nantes, France
<EMAIL_ADDRESS>1st Junwon Lee  Gaudio Lab, Inc. / KAIST
Seoul / Daejeon, South Korea
<EMAIL_ADDRESS>2nd Mathieu Lagrange  École Centrale Nantes
CNRS, LS2N, UMR 6004
F-44000 Nantes, France
<EMAIL_ADDRESS>3rd Keunwoo Choi  Gaudio Lab, Inc.
Seoul, South Korea
<EMAIL_ADDRESS>4th Laurie M. Heller  Department of Psychology
Carnegie Mellon University
Pittsburgh, PA, U.S.
<EMAIL_ADDRESS>5th Keisuke Imoto  Doshisha University
Kyoto, Japan
<EMAIL_ADDRESS>6th Yuki Okamoto Ritsumeikan University
Kusatsu, Japan
<EMAIL_ADDRESS>
###### Abstract

This paper explores whether considering alternative domain-specific embeddings
to calculate the Fréchet Audio Distance (FAD) metric can help the FAD to
correlate better with perceptual ratings of environmental sounds. We used
embeddings from VGGish, PANNs, MS-CLAP, L-CLAP, and MERT, which are tailored
for either music or environmental sound evaluation. The FAD scores were
calculated for sounds from the DCASE 2023 Task 7 dataset. Using perceptual
data from the same task, we find that PANNs-WGM-LogMel produces the best
correlation between FAD scores and perceptual ratings of both audio quality
and perceived fit with a Spearman correlation higher than 0.5. We also find
that music-specific embeddings resulted in significantly lower results.
Interestingly, VGGish, the embedding used for the original Fréchet
calculation, yielded a correlation below 0.1. These results underscore the
critical importance of the choice of embedding for the FAD metric design.

###### Index Terms:

Environmental Sound Synthesis, Objective Audio Quality, Neural Audio
Embeddings, Evaluation Metrics

##  I Introduction

Generative audio synthesis has become a popular research topic in which deep
neural nets are typically driven by textual prompts[1, 2, 3, 4, 5]. Those
systems must be evaluated on high-level perceptual features such as audio
quality and alignment with categories for meaningful comparisons. However,
evaluating synthetic audio through perceptual evaluation remains a cumbersome
process, despite its validity.

To address this challenge, various metrics have been developed for use in
prototyping and large-scale quality assessment [6, 2, 7]. Among these metrics,
the Fréchet Audio Distance (FAD) [8] is widely used. FAD compares the
distribution of a reference set with that of synthetic audio using VGGish
embeddings [9].

Recent work [10] demonstrates that considering alternative embeddings that
have been trained on music data is a relevant approach to better assess the
quality of music generation systems.

Similarly, in this paper, we investigate whether changing the embeddings can
lead to an increased correlation of the FAD with human perceptual ratings of
both audio quality and perceived fit to categories of environmental sounds
(i.e., general audio excluding speech and music). Given the fact that VGGish
embeddings have been trained on environmental audio, we expected that VGGish
embeddings would perform well, but our findings show the opposite. VGGish
embeddings report low correlations, as do the embeddings trained on music
data. Fortunately, we find that more recent embeddings specifically trained on
environmental audio are quite effective, suggesting that the choice of the
embedding is a crucial part of FAD metric design.

The rest of the paper is organized as follows. Section II delves into related
works, followed by the presentation of the selected embeddings in Section III.
Section IV presents the experiments and Section V the correlation results
between the FAD scores when considering several state-of-the-art neural audio
embeddings and perceptual evaluations using the DCASE Task 7 2023 dataset[11].
The code corresponding to this study is made publicly available.111Code
repository: https://github.com/mathieulagrange/dcaseFadEmbedding

##  II Related work

The Fréchet Audio Distance (FAD) [8] has been proposed as an adaptation of the
Fréchet Inception Distance (FID) [12] for audio quality assessment. FID and
FAD compare the distribution of two datasets in a given embedding space.
VGGish[13] was originally proposed as the feature extractor for FAD. To
evaluate a synthesis model, a set of desired audio serves as the reference.
Firstly, two multivariate Gaussian distributions, which have the same means
and covariances as the embedding sets, are considered. Then, the Fréchet
distance between the two distributions $r$ and $t$ is calculated as follows:

$FAD(r,t)=\left\|\mu_{r}-\mu_{t}\right\|_{2}+\text{tr}\left(\Sigma_{r}+\Sigma_{t}-2\sqrt{\Sigma_{r}\Sigma_{t}}\right)$
(1)

where $\mu_{x}$ and $\Sigma_{x}$ are respectively the mean and covariance
matrix of a given distribution $x$. The FAD calculation compares the two
datasets in terms of fit to domain with the comparison of means, but also in
terms of diversity by including a form of covariance comparison in the
equation. A low FAD score thus indicates that the two datasets contain similar
sound sources, and a similar diversity. If the reference dataset can be
considered of high audio quality it is generally assumed that a low FAD
distance implies that the evaluated dataset is also of good audio quality.

Whether it be for audio or image evaluation, using Fréchet distance has a few
drawbacks. Trying to match an embedding space can lead to models of very
different quality having similar Fréchet distance scores, as one could be
fitting the embedding space more accurately without improving the perceptual
quality[14]. The majority of embeddings used are trained representations,
implying a potentially strong dependency on the dataset and task used for
training. Consequently, for accurate evaluation using Fréchet Distance, the
dataset under assessment must exhibit similarity to the training set of the
embedding. This emphasizes the potentially significant influence of the
embedding choice on the Fréchet Distance calculation process.

For image generation evaluation, Kynkäänniemi, Tuomas, et. al.[14] showed that
simply matching the top-N classifications histogram between the reference and
generated set improved the FID score without further improvement in generative
model. This indicates the high dependency of FID on ImageNet[15], which is
used to train the Inception embedding. The high dependency on the choice of
embedding may be much worse in audio domain. ImageNet dataset has carefully
tailored 1k classes to cover the wide range of in-the-wild images. Conversely,
VGGish was trained to classify only 3k labels, that are not even necessarily
relevant to sound. This may limit the generalizability of audio embeddings,
depending on the training data size and task.

Gui et al. [10] explored the limitations of FAD in music generation, as
VGGish-based FAD struggles to accurately predict the perceptual features of
generated musical audio.The authors investigated different embeddings and
found that VGGish yields notably poor FAD scores compared to alternative
representations. Although they have shown that embeddings such as CLAP [6, 16]
are more suitable for music generation, their effectiveness in improving the
FAD metric for Environmental audio generation remains to be evaluated.

##  III Embeddings

Experimental details on all embeddings examined are presented in Table I. Our
objective with this selection is to investigate whether domain-specific
embeddings significantly influence the relevance of the Fréchet Audio Distance
(FAD) metric. We consider VGGish [17] as our baseline, as it has a proven
record of use for FAD calculation. Next, we consider MERT [18] as a recent
embedding primarily trained on music data, as well as a CLAP model trained
specifically on voice and music. Given that those latter embeddings are not
trained on environmental audio, we hypothesize that they should perform
poorly. Other CLAP models and PANNs[19] models, on the other end, are trained
using environmental audio. As CLAP models are trained using partly PANN
architectures, or at least have considered PANNs in their framework in their
evaluation protocol, they are expected to outperform PANNs.

###  III-A VGGish

VGGish [17] is an audio classifier trained on a subset of a large audio
dataset extracted from YouTube videos called YouTube-100M, which contains
350,000h of audio data with video-level class labels. YouTube-100M covers a
wide range of audio content, spanning everyday sounds, sound effects, and
music, captured in diverse real-world scenarios. It’s worth noting that within
this dataset the video-level classes may not necessarily be directly related
to the audio content, as a source can be present in a video without generating
any sound. VGGish uses log-mel spectrograms with 64 frequency bins and 10-ms
hops as input, and it has about 70M parameters. Following the methodology
outlined by Gui et al. [10], we employ the VGGish model to process 1-second
audio segments, with 50% overlap.

TABLE I: Description of Embedding Models. The size of the Receptive Field (RF) is the maximal duration of audio considered by the model to compute the embedding. | Model | Audio | Embedding | RF
---|---|---|---|---
| Size | Rate | Size / Rate | Size
VGGish [17] | 72M | 16 kHz | 128 / 1 Hz | 1 s
MERT [18] | 72M | 24 kHz | 768 / 76 Hz | 5 s
MS-CLAP [20] | 158M | 44 kHz | 1024 / 1 Hz | 7 s
L-CLAP [16] | 158M | 48 kHz | 512 / 1 Hz | 10 s
PANNs-CNN14-16k [19] | 80M | 16 kHz | 2048 / .1 Hz | 10 s
PANNs-CNN14-32k [19] | 80M | 32 kHz | 2048 / .1 Hz | 10 s
PANNs-WGM-Logmel [19] | 80M | 32 kHz | 2048 / .1 Hz | 10 s

###  III-B MERT

The Music undERstanding model with large-scale self-supervised Training (MERT)
[18] generates embeddings learned with teacher-student methods, using a
combination of teachers including an acoustic teacher based on Residual Vector
Quantization - Variational AutoEncoder (RVQ-VAE) and a musical teacher based
on the Constant-Q Transform (CQT). The student model is a BERT-style
transformer encoder. The MERT models are specialized in music, being trained
on an in-house private music dataset comprising 160k hours of audio data.
Among the several available models, we chose the one with 95M parameters
(MERT-95M).

###  III-C PANNs

PANNs [19] are classifiers trained on AudioSet[21]. The dataset originated
from YouTube videos and is around 5,000h long, with 527 different general
sound classes. Unlike YouTube-100M, AudioSet employs human labeling at the
audio level. The majority of these models leverage log-Mel spectrograms
featuring 64 mel bins and 10-ms hops as input. Released in various sizes and
trained at different sample rates, these models offer versatility in
application. Among them, the CNN14 model emerges as the most commonly utilized
variant, with 16kHz (PANN-CNN14-16kHz) or 32kHz (PANN-CNN14-32kHz) sampling
rates. However, the model that leads to the best accuracy, according to their
paper, is the Wavegram-Logmel-CNN (PANN-WGM-LogMel). This model uses the audio
waveform as input, which is transformed into a learned spectro-temporal
representation along with a Mel-spectrogram. Subsequently, both
representations are fed into the rest of the network. PANN-CNN14 and PANN-WGM-
LogMel models contain 80M parameters.

###  III-D MS-CLAP

The embedding Contrastive Language-Audio Pretraining (CLAP) [6] is trained to
learn multimodal representations, using both an audio and a text encoder.
Symmetric cross-entropy loss was exploited for language and audio cross-modal
contrastive learning. It uses PANN-CNN14-32kHz for audio encoding and BERT for
text encoding. It is trained on 128,000 audio/caption of FSD50k, ClothoV2,
AudioCaps and MACS, representing about 250h of audio. For this study, we use
the CLAP model released in September 2023 called “sep 23”.

###  III-E L-CLAP

LAION has also trained a CLAP architecture [16] similar to the one used for
MS-CLAP. They trained their best model using HTS-AT [22] as the audio encoder
and RoBERTa [23] as the text encoder. They proved that training their model on
LAION-Audio-630k and AudioSet with keyword-to-caption augmentation
significantly improves the performances of their CLAP model. LAION-Audio-630k
comprises a diverse range of 4,000 hours of audio recordings depicting human
activities, natural sounds, and audio effects, sourced from eight publicly
available websites. In total, their model for environmental audio called
“630k-audioset-best” (L-CLAP-audio) is reported to be trained on about 10,000h
of audio. They also released models specialized in music, which are trained on
music and speech from their data collection. We chose the model “music
audioset epoch 15 esc 90.14” (L-CLAP-mus) for comparison.

##  IV Experiments

###  IV-A Data

In this study, we leverage the DCASE 2023 Challenge Task 7 dataset[11],
encompassing 700 sound excerpts of 7 different categories: dog bark, footstep,
gunshot, keyboard, moving motor vehicle, rain, and sneeze/cough. Each sound
excerpt is a mono 16-bit 22,050 Hz 4-second audio sourced from three distinct
datasets: UrbanSound8K[24], FSD50K[25], and BBC Sound Effects222https://sound-
effects.bbcrewind.co.uk/. To ensure high relevance, diversity, and clarity,
the challenge organizers manually selected and validated the excerpts.

Supplementary to this dataset are the audio generated using the baseline and 8
submitted algorithms, with each contributing an additional set of 700 sound
excerpts synthesized by their respective systems. The duration of the whole
dataset (recorded and synthesized) is about 8h. The 8 systems from the
participants were top-ranked in terms of FAD score. Twenty sounds from each
system underwent perceptual evaluation for both audio quality and category fit
by 91 raters (47 hrs). Each category within each system has been evaluated,
resulting in a total of 63 evaluations for each criterion. The correlations
shown in section V are thus based on those 20 evaluated audios per category
and per system.

###  IV-B Uncertainty estimation

To assess the uncertainty associated with each Spearman correlation
calculation, we introduce Gaussian noise with a standard deviation of 1 to the
perceptual evaluation scores. Subsequently, we repeat this process 100 times
to generate 100 different noisy sets, each containing 63 perceptual
evaluations for both category fit and audio quality.

By computing the mean and standard deviation of the Spearman correlation
coefficients across these 100 noisy sets, we obtain estimates of the
variability and uncertainty inherent in the correlation calculations.

##  V Results

Given the FAD definition given in Eq. 1, lower FAD scores indicate a high
similarity between datasets. Consequently, a reliable FAD metric should
demonstrate an inverse correlation with high perceptual quality when computed
between a reference set and various generated sets. To simplify the
presentation of results, we use the inverse of the FAD ($\text{FAD}^{-1}$) so
that a high quality $\text{FAD}^{-1}$ metric should achieve a high positive
correlation with perceptual attributes that are higher when better.

Figure 1: Spearman correlation coefficient ($n=63$) between $\text{FAD}^{-1}$.
and perceptual evaluation of audio quality and category fit for different
embeddings. Error bars display standard deviation.

###  V-A Overall Correlation

As shown in Figure 1, the PANNs-WGM-LogMel FAD and the MS-CLAP FAD demonstrate
significantly high correlations with both category fit and audio quality, with
PANN-WGM-LogMel being significantly higher than MS-CLAP FAD. In contrast, both
the MERT-95M FAD and the VGGish FAD demonstrate very weak correlation with
perceptual evaluation. Additionally, the L-CLAP models perform less
effectively compared to MS-CLAP and PANN-WGM-LogMel, with L-CLAP-audio showing
a better correlation score than L-CLAP-mus.

###  V-B Per-category Correlation

In Figure 2, we present the per-category correlation results. Interestingly,
substantial variability in correlation is observed across different
categories. PANNs-WGM-LogMel displays greater stability across categories,
while performing better than the other embeddings on half of the categories.
VGGish demonstrates good performance in categories such as sneeze/cough and
gunshot, but performs extremely poorly in every other category. Similarly,
CLAP exhibits low correlations in categories such as moving motor vehicle,
sneeze/cough, and gunshot, while demonstrating better performance in other
categories, competing or beating PANNs-WGM-LogMel in some of them. These
results should be treated cautiously though as the Spearman correlation
coefficients are calculated with only 9 data points per category.

###  V-C Influence of Dimensionality

As shown in Table I, each embedding model varies in size. Considering the
findings presented in Figure 1, we want to investigate if the dimensionality
of the embedding may bias the performance of the FAD metric. To refute the
hypothesis that a higher dimensional embedding may gain an unfair advantage,
we conduct a dimensionality reduction of every embedding to match that of
VGGish ($n$=128). The dimensionality reduction is performed by projecting the
set of embeddings on the 128 Eigenvectors of the highest Eigenvalues using
Principal Component Analysis (PCA).

It is observed that diminishing the size of the embeddings had minimal impact
on the correlations with $\text{FAD}^{-1}$, decreasing them by approximately
0.01 for each embedding. This tends to show us that the size of the embedding
does not significantly influence the performance of the FAD metric.

###  V-D FAD-based Category Mapping

Little is known about how the FAD may lead to a similarity graph that is
linked to the perceptual attributes of sounds. To examine this, we grouped the
7 sound categories of the 700 audios of the evaluation set of DCASE Task 7
2023 dataset into 3 meta-categories: Impact (Footsteps, Gunshot, Keyboard),
Living (Dog Bark, Sneeze/Cough), and Texture (Moving Motor Vehicle, Rain). A
satisfactory embedding should result in the FAD grouping similar categories
together while separating highly dissimilar categories. Given that the 7 sound
classes themselves are highly distinct, they should also be quite separated
from each other in the projection.

Figure 3 shows a 2D mapping of the similarities of every pair of categories
from the DCASE Task 7 2023 dataset, calculated with Multidimensional Scaling
(MDS) with FAD using three different embeddings as input. We find that the
three embeddings successfully group similar categories together, as evidenced
by the Voronoi separation diagram. Additionally, the split between the
different groups is somewhat more pronounced for MS-CLAP and PANN-WGM-LogMel
than for VGGish. Furthermore, for MS-CLAP and PANN-WGM-LogMel, all 7 sound
sources are further apart from each other, which indicates that these two
embeddings are the best at separating the different categories.

Figure 2: Spearman correlation coefficient ($n=9$) between $\text{FAD}^{-1}$.
and perceptual evaluation of audio quality and category fit for VGGish, PANNs
CNN14 Wavegram Logmel and CLAP. Figure 3: 2D Projection of inter-category FAD
similarity matrix using Multi-dimensional scaling (MDS) on DCASE Task 7 2023
dataset.

##  VI Conclusion

In this paper, we explored the use of alternative embeddings to assess audio
quality and alignment with categories in environmental audio, aiming to
improve the validity of the Fréchet Audio Distance (FAD) metric. We compared
several embeddings, including VGGish, PANNs, MS-CLAP, L-CLAP, and MERT, using
the DCASE Task 7 2023 dataset.

We find that there is a strong dependency of the embedding in the FAD metric,
which appears to be closely linked to the domain of the embedding training
dataset. In fact, music-trained embeddings, such as MERT-95M and CLAP Laion
Music, perform less effectively than those trained on environmental audio.
Furthermore, while VGGish was trained to classify 3k video-level labels, these
labels might not necessarily be relevant to sound, potentially limiting its
generalizability for tasks like DCASE TASK 7 2023 systems evaluation. This
observation also aligns with findings reported by Gui et al. [10].

Interestingly, among the PANN models, only PANN-WGM-LogMel consistently
outperforms MS-CLAP. One explanation could be that only the PANN-CNN14-32kHz
model was evaluated and/or incorporated by MS-CLAP and L-CLAP. However,
according to Kong et al. [19], PANN-WGM-LogMel is the best performing model
for classification on Audioset. Thus, a CLAP model trained with PANN-WGM-
LogMel as an audio classifier could be a relevant improvement.

It is important to note that the performance of the embeddings varies
significantly depending upon the category. This indicates that domain
knowledge is important for selecting an appropriate embedding for the FAD
calculation.

The study suggests that specialized embeddings tailored to specific tasks are,
as of today, crucial for the relevance of the FAD metric. Further
investigation is recommended, for example, by considering a more diverse
number of categories for which perceptual ratings are available.

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