# Construction of a generalization of the Leibnitz numbers and their
properties

YILMAZ SIMSEK  Department of Mathematics, Faculty of Science University of
Akdeniz TR-07058 Antalya-TURKEY<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to give a novel generalization of the Leibnitz
numbers derived from application of the Beta function to the modification for
the Bernstein basis functions. We also give some properties of the Leibnitz
numbers with the aid of their generating functions derived from the Volkenborn
integral on the set of $p$-adic integers. We also give some novel identities
and relations involving the Leibnitz numbers, the Daehee numbers, the Changhee
numbers, inverse binomial coefficients, and combinatorial sums. Finally, by
coding computation formula for the generalization of the Leibnitz numbers in
Mathematica 12.0 with their implementation, we compute few values of these
numbers with their tables. Finally, by using the applications of Volkenborn
integral to Mahler coefficients, we derive some novel formulas involving the
Leibnitz numbers.

2010 Mathematics Subject Classification. 05A15, 11B65, 11B83, 11S80, 26C05,
33B15.

Keywords and phrases. Leibnitz numbers, Daehee numbers, Stirling numbers,
Special numbers and polynomials, Generating functions, Beta function,
Bernstein basis functions, Volkenborn integral, $p$-adic integers.

The present paper was supported by the Scientific Research Project
Administration of Akdeniz University.

##  1\. Introduction

###  1.1. About Professor Lee Chae Jang:

I met Professor Lee Chae Jang at the international conference organized by the
Jangjeon Mathematical Society in Mysore, India, in 2003. Professor Taekyun
Kim, Professor Seog-Hoon Rim, and Professor Lee Chae Jang are founders of the
Jangjeon Mathematics Society. This society has been not only published two
international mathematical journals including the Advanced Studies in
Contemporary Mathematics (ASCM) and the Proceedings of the Jangjeon
Mathematical Society (PJMS), but also organized International Conference of
The Jangjeon Mathematical Society. Until I met Professor Jang, he has always
contributed to mathematics in a disciplined manner and has always been an
example for young scientists, without losing his passion for mathematics. I
had a lot of interactions and views with Professor Jang, both culturally and
from a mathematical point of view.

For Professor Jang’s great contribution to mathematics and his guidance for
young scientists;

I dedicate this article to his retirement with my best wishes.

I wish the rest of his life to be happy, fruitful, success and healthy with
all his family and friends.

It is known by many researchers, who work on the subject of special numbers
and their applications in recent years, that this subject has became among the
leading topics of mathematics and especially analytic number theory. The
Leibnitz numbers, known as the famous German mathematician Gottfried Wilhelm
Leibnitz (1646 - 1716), are considered in this paper. These numbers, which
have rarely been addressed until now, are studied by using the techniques of
generating functions and their Volkenborn integral representation in this
paper. These numbers are also closely related to the Leibniz Harmoic Triangle
numbers. The denominators of some of these numbers are also directly related
to the pronic numbers. Within the scope of this study, it has been proved that
these numbers are also related to Daehee numbers, Changhee numbers and the
combinatorial numbers we found.

###  1.2. Definitions and Notations

Let $\mathbb{N}$ and $\mathbb{C}$ denote the set of natural numbers and the
set of complex numbers, respectively, and also
$\mathbb{N}_{0}=\mathbb{N}\cup\left\\{0\right\\}$.

Let $x_{(n)}=x\left(x-1\right)\dots\left(x-n+1\right)$ with $x_{(0)}=1$ and
$n\in\mathbb{N}$.

The Leibnitz numbers, $\boldsymbol{l}\left(n,k\right)$, are defined by

(1) $\boldsymbol{l}\left(n,k\right)=\frac{1}{\left(n+1\right)\binom{n}{k}}$

whose generating function is given as follows:

(2)
$\mathcal{G}_{l}\left(t,u\right):=\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}u^{n}=\frac{\log\left(1-u\right)+\log\left(1-ut\right)}{\left(1-u\right)\left(1-tu\right)-1},$

where $|u|<1$; $k=0,1,2,\ldots,n$ and $n\in\mathbb{N}_{0}$ (cf. [2, Exercise
16, p. 127]).

As seen from the equation (2), the function $\mathcal{G}_{l}\left(t,u\right)$
is the generating function for the polynomials:

$L_{n}(t):=\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}$

whose coefficients are the Leibnitz numbers and also whose degree is $n$. That
is, the ordinary generating function for the polynomials $L_{n}(t)$ is given
as follows:

(3) $\mathcal{G}_{l}\left(t,u\right)=\sum_{n=0}^{\infty}L_{n}(t)u^{n}.$

Observe that

$\displaystyle L_{n}(1)=\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right),$

and

$\displaystyle L_{n}(0)=\boldsymbol{l}\left(n,0\right)=\frac{1}{n+1}.$

The Leibnitz numbers, $\boldsymbol{l}\left(n,k\right)$, are also given by
following finite combinatorial sum:

(4)
$\boldsymbol{l}\left(n,k\right)=\sum\limits_{v=0}^{k}\left(-1\right)^{k-v}\frac{1}{n-v+1}\binom{k}{v},$

where $k=0,1,2,\ldots,n$ and $n\in\mathbb{N}_{0}$ (cf. [2, Exercise 16, p.
127]).

With the initial condition

$\boldsymbol{l}\left(n,0\right)=\frac{1}{n+1},$

the Leibnitz numbers satisfy the following recurrence relation:

$\boldsymbol{l}\left(n,k\right)=\frac{k}{n+1}\boldsymbol{l}\left(n-1,k-1\right),$

where $k=1,2,\ldots,n$ and $n\in\mathbb{N}$ (cf. [2, Exercise 16, p. 127]).

In [17], whose content could not be reached but whose existence is known, Zhao
and Wuyungaowa claimed in its abstract that they gave a series of identities
involving Leibniz numbers, Stirling numbers, harmonic numbers, and arctan
numbers by making use of generating functions. They also claimed that give the
asymptotic expansion of certain sums related to Leibniz numbers by the
Laplace’s method. On the other hand, there is no data to comment on whether
the Leibniz numbers mentioned there and the Leibnitz numbers discussed in this
study point to the same concept or the relationship of the results. Due to the
expression Leibniz numbers in the abstract of the relevant study, we cite it
here.

The Daehee numbers, $D_{n}$, are defined by

(5)
$\mathcal{G}_{D}\left(u\right):=\frac{\log\left(1+u\right)}{u}=\sum\limits_{n=0}^{\infty}D_{n}\frac{u^{n}}{n!}$

where

(6) $D_{n}=(-1)^{n}\frac{n!}{n+1}$

(cf. [12]).

The Changhee numbers, $Ch_{n}$, are defined by

(7)
$\mathcal{G}_{Ch}\left(u\right):=\frac{2}{2+u}=\sum\limits_{n=0}^{\infty}Ch_{n}\frac{u^{n}}{n!}$

where

(8) $Ch_{n}=(-1)^{n}\frac{n!}{2^{n}}$

(cf. [13]).

The numbers, $Y_{n}\left(\lambda\right)$, are defined by

(9) $\mathcal{G}_{Y}\left(u,\lambda\right):=\frac{2}{\lambda\left(1+\lambda
u\right)-1}=\sum\limits_{n=0}^{\infty}Y_{n}\left(\lambda\right)\frac{u^{n}}{n!}$

where

(10)
$Y_{n}\left(\lambda\right)=\left(-1\right)^{n}\frac{2n!}{\lambda-1}\left(\frac{\lambda^{2}}{\lambda-1}\right)^{n}$

(cf. [33], [37]).

The relation between Changhee numbers and the numbers
$Y_{n}\left(\lambda\right)$ is given as follows:

(11) $Ch_{n}=\left(-1\right)^{n+1}Y_{n}\left(-1\right)$

(cf. [33], [37]).

Next, we summarize the content of this paper as follows:

In Section 2, we give combinatorial identities and relations related to the
Leibnitz numbers, Daehee numbers and Changhee numbers. We also give some
computational formulas for these numbers.

In Section 3, we give further remarks and observations on the Leibnitz
numbers. Moreover, by using finite sums derived from application of the
integral to the modification for the Bernstein basis functions, we introduce a
generalization of the Leibnitz numbers.

In Section 4, we give Mathematica implementation of the generalized Leibnitz
numbers and by this implementation, we compute few values of these numbers
with their tables.

In Section 5, by applying Volkenborn integral on the set of $p$-adic integers,
we derive some novel formulas involving the Leibnitz numbers.

##  2\. Combinatorial identities and relations involving the Leibnitz numbers,
Daehee numbers and Changhee numbers

By using functional equations of the generating functions for the the Leibnitz
numbers, the Daehee numbers and special series, we find many formulas,
identities and relations involvilg the Changhee numbers and combinatorial
numbers and sums.

By using (2), we get

(12)
$\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}u^{n}=\frac{1}{t+1}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{1}{n-k+1}\left(\frac{t}{t+1}\right)^{k}\left(1+t^{n-k}\right)u^{n}.$

Comparing the coefficients $u^{n}$ on both-sides of the above equation, we get
the following theorem:

######  Theorem 2.1.

(13)
$\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}=\frac{1}{t+1}\sum_{k=0}^{n}\frac{1}{n-k+1}\left(\frac{t}{t+1}\right)^{k}\left(1+t^{n-k}\right).$

Substituting $t=1$ into (13), we get the finite summation of the Leibnitz
numbers as in the following corollary:

######  Corollary 2.2.

(14)
$\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)=\sum_{k=0}^{n}\frac{1}{\left(n-k+1\right)2^{k}}.$

######  Theorem 2.3.

(15)
$\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}=\frac{1}{1+t}\sum\limits_{j=0}^{n}\left(-1\right)^{j}\frac{D_{j}}{j!}\left(\frac{t}{1+t}\right)^{n-j}\left(1+t^{j+1}\right).$

###### Proof.

By using (2) and (5), we get the following functional equation of generating
functions:

(16)
$\mathcal{G}_{l}\left(t,u\right)=\frac{\mathcal{G}_{D}\left(-u\right)+t\mathcal{G}_{D}\left(-ut\right)}{1+t-ut}$

which yields

$\displaystyle\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}u^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{1+t}\sum\limits_{n=0}^{\infty}u^{n}\frac{t^{n}}{\left(1+t\right)^{n}}\left(\sum\limits_{n=0}^{\infty}\frac{\left(-1\right)^{n}D_{n}}{n!}u^{n}+t\sum\limits_{n=0}^{\infty}\frac{\left(-1\right)^{n}D_{n}u^{n}}{n!}t^{n}\right).$

Hence, we get

$\displaystyle\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}u^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{1+t}\sum\limits_{n=0}^{\infty}\sum\limits_{j=0}^{n}\left(-1\right)^{j}\frac{D_{j}}{j!}\left(\frac{t}{1+t}\right)^{n-j}u^{n}$
$\displaystyle+\frac{t}{1+t}\sum\limits_{n=0}^{\infty}\sum\limits_{j=0}^{n}\left(-1\right)^{j}\frac{D_{j}}{j!}t^{j}\left(\frac{t}{1+t}\right)^{n-j}u^{n}.$

Comparing the coefficients of $u^{n}$ on both sides of the above equation, we
arrive at the desired result. ∎

Combining (15) with (6), we have the following corollary:

######  Corollary 2.4.

(17)
$\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}=\frac{1}{1+t}\sum\limits_{j=0}^{n}\frac{1}{j+1}\left(\frac{t}{1+t}\right)^{n-j}\left(1+t^{j+1}\right).$

Observe that the equation (17) is equivalent to the equation (13).

In the next, by combining (17) with (1), we also have the following corollary:

######  Corollary 2.5.

(18)
$\sum\limits_{k=0}^{n}\frac{t^{k}}{\left(n+1\right)\binom{n}{k}}=\frac{1}{1+t}\sum\limits_{j=0}^{n}\frac{1+t^{j+1}}{j+1}\left(\frac{t}{1+t}\right)^{n-j}.$

Substituting $t=1$ into (18) yields the following result:

######  Corollary 2.6.

(19)
$\sum\limits_{k=0}^{n}\frac{1}{\left(n+1\right)\binom{n}{k}}=\sum\limits_{j=0}^{n}\frac{2^{j-n}}{j+1}.$

Observe that the combination of (1) with equation (19) is equivalent to
equation (14).

Combining (19) with (8), we have the following result:

######  Theorem 2.7.

(20)
$\sum\limits_{k=0}^{n}\frac{1}{\left(n+1\right)\binom{n}{k}}=\sum\limits_{j=0}^{n}\frac{\left(-1\right)^{n-j}\left(n-j\right)!}{\left(j+1\right)Ch_{n-j}}.$

Substituting (11) into (20), we arrive at the following corollary:

######  Corollary 2.8.

$\sum\limits_{k=0}^{n}\frac{1}{\left(n+1\right)\binom{n}{k}}=-\sum\limits_{j=0}^{n}\frac{\left(n-j\right)!}{\left(j+1\right)Y_{n-j}\left(-1\right)}.$

Substituting $t=1$ into (2) and combining the final eqaution with (5), we get
the following functional equation:

######  Theorem 2.9.

Let $n\in\mathbb{N}$. Then we have

(21)
$\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)-\frac{1}{2}\sum\limits_{k=0}^{n-1}\boldsymbol{l}\left(n-1,k\right)=\frac{(-1)^{n}}{n!}D_{n}.$

###### Proof.

We set the following functional equation involving generating functions for
the Leibnitz numbers and the Daehee numbers:

(22)
$\mathcal{G}_{D}\left(-u\right)=\left(1-\frac{u}{2}\right)\mathcal{G}_{l}\left(1,u\right).$

By using the above equation, we obtain

$\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)u^{n}-\frac{1}{2}\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)u^{n+1}=\sum\limits_{n=0}^{\infty}(-1)^{n}D_{n}\frac{u^{n}}{n!}.$

Therefore

$1+\sum\limits_{n=1}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)u^{n}-\frac{1}{2}\sum\limits_{n=1}^{\infty}\sum\limits_{k=0}^{n-1}\boldsymbol{l}\left(n-1,k\right)u^{n}=D_{0}+\sum\limits_{n=0}^{\infty}(-1)^{n}D_{n}\frac{u^{n}}{n!}.$

Comparing the coefficients of $u^{n}$ on both sides of the above equation
yields the desired result. ∎

Combining (21) with (6), we obtain the following result:

######  Theorem 2.10.

Let $n\in\mathbb{N}$. Then we have

$\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)-\frac{1}{2}\sum\limits_{k=0}^{n-1}\boldsymbol{l}\left(n-1,k\right)=\frac{1}{n+1}.$

######  Remark 1.

By using (22), assuming that $\left|u\right|<1$, we obtain

$\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)u^{n}=\sum\limits_{n=0}^{\infty}\frac{u^{n}}{n+1}\sum\limits_{n=0}^{\infty}\frac{u^{n}}{2^{n}}.$

Therefore

$\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)u^{n}=\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\frac{2^{k-n}}{k+1}u^{n}.$

Comparing the coefficients of $u^{n}$ on both sides of the above equation
yields the Eq.-(19).

##  3\. Generalized Leibnitz numbers

Here, we give further remarks and observations on the Leibnitz numbers and
their relations with finite sums derived from application of the integral to
the modification for the Bernstein basis functions. By using the following
well-known beta functions:

$B\left(\alpha,\beta\right)=\int\limits_{0}^{1}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt=B\left(\beta,\alpha\right)$

where $\operatorname{Re}\left(\alpha\right)>0$ and
$\operatorname{Re}\left(\beta\right)>0$ (cf. [18], [36]), one has the
following novel integral formula:

(23)
$\int\limits_{a}^{b}\left(x-a\right)^{\alpha-1}\left(b-x\right)^{\beta-1}dx=\left(b-a\right)^{\alpha+\beta-1}B\left(\alpha,\beta\right)$

(cf. [36, p.10, Eq. (69)]).

Let $x\in\left[a,b\right]$ and $k=0,1,2,\ldots,n$. By combining (23) with the
modification for the Bernstein basis functions is given as follows:

$B_{k}^{n}\left(x;a,b\right)=\binom{n}{k}\left(\frac{x-a}{b-a}\right)^{k}\left(\frac{b-x}{b-a}\right)^{n-k}\text{;\
\ \ }a<b$

where (cf. [5], [28]; see references therein), we have

$\int\limits_{a}^{b}B_{k}^{n}\left(x;a,b\right)dx=\binom{n}{k}\sum\limits_{j=0}^{k}\sum\limits_{l=0}^{n-k}\left(-1\right)^{n-j-l}\binom{k}{j}\binom{n-k}{l}\frac{a^{k-j}b^{n+j-k+1}-a^{n-l+1}b^{l}}{n+j-k-l+1}$

Thus, by the above integral representation, we also get

$\displaystyle\sum\limits_{j=0}^{k}\sum\limits_{v=0}^{n-k}\left(-1\right)^{n-j-v}\binom{k}{j}\binom{n-k}{v}\frac{a^{k-j}b^{n+j-k+1}-a^{n-v+1}b^{v}}{n+j-k-v+1}$
$\displaystyle=$ $\displaystyle\left(b-a\right)B\left(k+1,n-k+1\right),$

(cf. [28, Eq. (28)]).

By using (3), it is time to introduce generalized Leibnitz numbers by the
following definition:

######  Definition 3.1.

Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let
$k=0,1,2,\ldots,n$ and $n\in\mathbb{N}_{0}$. Then, the generalized Leibnitz
numbers $\mathcal{L}\left(n,k;a,b\right)$ are defined by

(25)
$\mathcal{L}\left(n,k;a,b\right)=\sum\limits_{j=0}^{k}\sum\limits_{v=0}^{n-k}\left(-1\right)^{n-j-v}\binom{k}{j}\binom{n-k}{v}\frac{a^{k-j}b^{n+j-k+1}-a^{n-v+1}b^{v}}{n+j-k-v+1}.$

Note that in (25) we assume that $0^{0}=1$.

######  Remark 2.

Setting $a=0$ and $b=1$ in (25) and using (1) yields

$\frac{1}{\left(n+1\right)\binom{n}{k}}=\sum\limits_{v=0}^{n-k}\left(-1\right)^{n-k-v}\binom{n-k}{v}\frac{1}{n-v+1},$

(cf. [23, 25, 27], [28, Eq. (29)]), which is equivalent to (4).

Observe that further identities and new number families may be discovered by
using the methods in paper [25, 23, 27, 28], of the generalized Leibnitz
numbers, which seem to be closely related to the modification of the Bernstein
basis functions.

Open Problems:

  * •

How can we constuct generating functions for the (generalized) Leibnitz
numbers?

  * •

Is there any zeta-type functions, on the set of complex numbers, which
interpolates the (generalized) Leibnitz numbers?

##  4\. Mathematica implementation of the numbers
$\mathcal{L}\left(n,k;a,b\right)$

In this section, in order to give some applications of the computation
formulas given in the previous sections, we present Mathematica implementation
(see: Implementation 1) for the numbers $\mathcal{L}\left(n,k;a,b\right)$ by
coding (25) in Mathematica 12.0.

Implementation 1: The following Mathematica code returns the values of the
generalized Leibnitz numbers $\mathcal{L}\left(n,k;a,b\right)$.

⬇

1Unprotect[Power];

2Power[0,0]=1;

3Protect[Power];

4GLeibnitzNum[n_,k_,a_,b_]:=Sum[Sum[((-1)^(n-j-v))*Binomial[k,j]*Binomial[n-k,v]*((a^(k-j))*(b^(n+j-k+1))-(a^(n-v+1))(b^v))/(n+j-k-v+1),
{v,0,n-k}], {j,0,k}]

Then, by (1), we compute few values of the numbers
$\mathcal{L}\left(n,k;a,b\right)$, and give their tables as follows:

Table 1. For $k=1$ and $n\in\\{0,1,2,3,4\\}$, few values of the generalized
Leibnitz numbers $\mathcal{L}\left(n,k;a,b\right)$ . Table 2. For $k=2$ and
$n\in\\{0,1,2,3,4\\}$, few values of the generalized Leibnitz numbers
$\mathcal{L}\left(n,k;a,b\right)$ . Table 3. For $n\in\\{0,1,2,3\\}$ and
$k\in\\{0,1,2\\}$, few values of the generalized Leibnitz numbers
$\mathcal{L}\left(n,k;a,b\right)$.

Substituting $a=0$ and $b=1$, the numbers $\mathcal{L}\left(n,k;a,b\right)$
are reduced to the classical Leibnitz numbers
$\boldsymbol{l}\left(n,k\right)$.

Table 4. For $n\in\\{0,1,2,\dots,8\\}$ and $k\in\\{0,1,2,\dots,8\\}$, few
values of the generalized Leibnitz numbers $\mathcal{L}\left(n,k;0,1\right)$,
namely $\boldsymbol{l}\left(n,k\right)$.

##  5\. The Volkenborn integral representation of the Leibnitz numbers on the
set of $p$-adic integers

In this section, by applying Volkenborn integral on the set of $p$-adic
integers not only to the Mahler coefficients, but also to uniformly
differential function on the set of $p$-adic integers, we obtain some novel
formulas involving the Leibnitz numbers.

Here, we follow notations of the following references: [8, 11, 21, 34]; and
the references cited therein.

Some notations and definitions for $p$-adic integrals are given as follows:

Let $m\in\mathbb{N}$. Let $ord_{p}(m)$ denote the greatest integer $k$
($k\in\mathbb{N}_{0})$ such that $p^{k}$ divides $m$ in $\mathbb{Z}$. If
$m=0$, then $ord_{p}(m)=\infty$.

$\left|.\right|_{p}$ is a norm on $\mathbb{Q}$. This norm is given by

$\left|x\right|_{p}=\left\\{\begin{array}[]{cc}p^{-ord_{p}(x)}&\text{if }x\neq
0,\\\ 0&\text{if }x=0.\end{array}\right.$

Let $\mathbb{Z}_{p}$ be a set of $p$-adic integers which is given by

$\mathbb{Z}_{p}=\left\\{x\in\mathbb{Q}_{p}:\left|x\right|_{p}\leq 1\right\\}.$

Let $f$ define on $\mathbb{Z}_{p}$. The function $f$ is called a uniformly
differential function at a point$\ a\in\mathbb{Z}_{p}$ if $f$ satisfies the
following conditions:

If the difference quotients
$\Phi_{f}:\mathbb{Z}_{p}\times\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p}$ such
that

$\Phi_{f}(x,y)=\frac{f(x)-f(y)}{x-y}$

have a limit $f^{\prime}(z)$ as $(x,y)\rightarrow(0,0)$ ($x$ and $y$ remaining
distinct). A set of uniformly differential functions is briefly indicated by
$f\in UD(\mathbb{Z}_{p})$ or $f\in
C^{1}(\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p}\mathbb{)}$.

The well-known Volkenborn integral (bosonic $p$-integral) is given by

$\int\limits_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{1}\left(x\right)=\underset{N\rightarrow\infty}{\lim}\frac{1}{p^{N}}\sum_{x=0}^{p^{N}-1}f\left(x\right),$

where denotes the Haar distribution, which is defined by

$\mu_{1}\left(x\right)=\mu_{1}\left(x+p^{N}\mathbb{Z}_{p}\right)=\frac{1}{p^{N}}$

(cf. [6]-[13], [21]; see also the references cited in each of these earlier
works).

In order to achieve the results of this section, we let

(26) $f\left(x\right)=\sum\limits_{n=0}^{\infty}a_{n}\binom{x}{n}\in
C^{1}(\mathbb{Z}_{p}\rightarrow\mathbb{K)},$

where $\binom{x}{n}=\frac{x_{(n)}}{n!}$ denotes the Mahler coefficients.
Applying the Volkenborn integral to the function $f\left(x\right)$ in terms of
the Mahler coefficients $\binom{x}{n}$, we have the following well-known
formula:

(27)
$\int\limits_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{1}\left(x\right)=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{n+1}a_{n},$

(cf. [21, p. 168-Proposition 55.3]).

In order to give generating function (2), we apply the Volkenborn integral to
the following uniformly differential function on $\mathbb{Z}_{p}$:

$f\left(x,u;t\right)=\left(1-u\right)^{x}\left(1-tu\right)^{x},$

where $u$, $x\in\mathbb{Z}_{p}$.

Substituting the above function into the following well-known integral
equation:

$\int\limits_{\mathbb{Z}_{p}}f(x+1)d\mu_{1}\left(x\right)=\int\limits_{\mathbb{Z}_{p}}f(x)d\mu_{1}\left(x\right)+f^{\prime}(k)$

(cf. [21]), we get

(28)
$\int\limits_{\mathbb{Z}_{p}}\left(1-u\right)^{x}\left(1-tu\right)^{x}d\mu_{1}\left(x\right)=\frac{\log\left[\left(1-u\right)\left(1-tu\right)\right]}{\left(1-u\right)\left(1-tu\right)-1}.$

Consequently, the function on the right of equation (28) gives the generating
function given in equation (2) for the Leibnitz numbers.

Combining (28) with the binomial series

$\sum\limits_{n=0}^{\infty}x_{(n)}\frac{u^{n}}{n!}=\left(1+u\right)^{x},$

and using (27), we obtain

$\sum\limits_{n=0}^{\infty}(-1)^{n}\frac{u^{n}}{n!}\sum\limits_{k=0}^{n}\binom{n}{k}t^{n-k}\int\limits_{\mathbb{Z}_{p}}x_{(k)}x_{(n-k)}d\mu_{1}\left(x\right)=\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}u^{n}.$

Comparing the coefficients of $u^{n}$ on both sides of the above equation
yields the following theorem:

######  Theorem 5.1.

Let $n\in\mathbb{N}_{0}$. Then we have

(29)
$\frac{(-1)^{n}}{n!}\sum\limits_{k=0}^{n}\binom{n}{k}t^{n-k}\int\limits_{\mathbb{Z}_{p}}x_{(k)}x_{(n-k)}d\mu_{1}\left(x\right)=\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}.$

Combining (29) with the following formula:

$\int\limits_{\mathbb{Z}_{p}}x_{(n)}x_{(m)}d\mu_{1}\left(x\right)=\sum_{k=0}^{m}(-1)^{m+n-k}\binom{n}{k}\binom{m}{k}\frac{k!(n+m-k)!}{n+m-k+1}$

where $m,n\in\mathbb{N}_{0}$ (cf. [35]), we arrive at the following theorem:

######  Theorem 5.2.

Let $n\in\mathbb{N}_{0}$. Then we have

(30)
$\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)t^{k}=\frac{1}{n!}\sum_{k=0}^{n}\sum_{j=0}^{n-k}\left(-1\right)^{j}\binom{n}{k}\binom{n-k}{j}\binom{k}{j}\frac{j!\left(n-j\right)!}{n-j+1}t^{n-k}.$

Substituting $t=1$ into (30), we get the following corollary:

######  Corollary 5.3.

(31)
$\sum\limits_{k=0}^{n}\boldsymbol{l}\left(n,k\right)=\frac{1}{n!}\sum_{k=0}^{n}\sum_{j=0}^{n-k}\left(-1\right)^{j}\binom{n}{k}\binom{n-k}{j}\binom{k}{j}\frac{j!\left(n-j\right)!}{n-j+1}.$

## References

  * [1] S. N. Bernstein, Démonstration du théoreme de Weierstrass fondée sur la calcul des probabilités, Communications of the Kharkov Mathematical Society 13 (1912), 1–2.
  * [2] C. A. Charalambides, Enumerative Combinatorics, Chapman & Hall /CRC, Boca Raton, London New York, 2002.
  * [3] J. Choi, Note on Apostol-Daehee polynomials and numbers, Far East J. Math. Sci. 101 (8) (2017), 1845–1857.
  * [4] L. Comtet, Advanced Combinatorics, D. Reidel Publication Company, Dordrecht-Holland, Boston-U.S.A., 1974.
  * [5] R. Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, R. Academic Press, SanDiego, 2002.
  * [6] L. C. Jang, W. J. Kim and Y. Simsek, A study on the $p$-adic integral representation on associated with Bernstein and Bernoulli polynomials. Adv. Differ. Equ. 2010 Article ID 163217 (2010), https://doi.org/10.1155/2010/163217
  * [7] T. Kim, On a $q$-analogue of the $p$-adic $\log$ gamma functions and related integrals, J. Number Theory 76 (1999), 320–329.
  * [8] T. Kim, $q$-Volkenborn integration, Russ. J. Math. Phys. 19 (2002), 288–299.
  * [9] T. Kim, L. C. Jang, D. W. Park and C. Adiga , $q$-Volkenborn integration II, Notes Number Theory Discrete Math. 9 (4) (2003), 83–89.
  * [10] T. Kim, A note on $q$-Volkenborn integration, Proc. Jangjeon Math. Soc. 8 (1) (2005), 13–17.
  * [11] T. Kim, $q$-Euler numbers and polynomials associated with $p$-adic $q$-integral and basic $q$-zeta function, Trend Math. Information Center Math. Sciences 9 (2006), 7–12.
  * [12] D.S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. (Ruse), 7 (120) (2013), 5969–5976.
  * [13] D.S. Kim, T. Kim and J. Seo, A note on Changhee numbers and polynomials, Adv. Stud. Theor. Phys. 7 (2013), 993–1003.
  * [14] I. Kucukoglu, B. Simsek and Y. Simsek, An approach to negative hypergeometric distribution by generating function for special numbers and polynomials, Turk. J. Math. 43 (2019), 2337–2353.
  * [15] I. Kucukoglu and Y. Simsek, On a family of special numbers and polynomials associated with Apostol-type numbers and poynomials and combinatorial numbers, Appl. Anal. Discrete Math., 13 (2019), 478–494.
  * [16] I. Kucukoglu and Y. Simsek, Identities and relations on the $q$-Apostol type Frobenius-Euler numbers and polynomials, J. Korean Math. Soc. 56 (1) (2019), 265–284.
  * [17] F.-Z. Zhao and Wuyungaowa, Some Identities for Leibniz Numbers, Ars Combinatoria 97 (2010).
  * [18] E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
  * [19] S.-H. Rim, T. Kim and S.S. Pyo, Identities between harmonic, hyperharmonic and Daehee numbers, J. Inequal Appl. 2018 (168) (2018), 1–12 .
  * [20] S. Roman, The Umbral Calculus, Dover Publications, New York, 2005.
  * [21] W. H. Schikhof, Ultrametric Calculus: An Introduction to $p$-adic Analysis, Cambridge Studies in Advanced Mathematics 4, Cambridge University Press, Cambridge, 1984.
  * [22] Y. Simsek, Construction a new generating function of Bernstein type polynomials, Appl. Math. Comput. 218 (3) (2011), 1072–1076.
  * [23] Y. Simsek Unification of the Bernstein-type polynomials and their applications, Bound. Value Probl. 2013 (56) (2013), DOI: 10.1186/1687-2770-2013-56
  * [24] Y. Simsek, Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions, Fixed Point Theory Appl. 2013 (80) (2013), 1–13.
  * [25] Y. Simsek, Generating functions for the Bernstein type polynomials: a new approach to deriving identities and applications for these polynomials, Hacet. J. Math. Stat. 43 (1) (2014), 1–14.
  * [26] Y. Simsek, A new class of polynomials associated with Bernstein and beta polynomials, Math. Methods Appl. Sci. 37 (2014), 676–685.
  * [27] Y. Simsek, Beta-type polynomials and their generating functions, Appl. Math. Comput. 254 (2015), 172–182.
  * [28] Y. Simsek, Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers, Math. Meth. Appl. Sci. 38 (14) (2015), 3007–3021.
  * [29] Y. Simsek, Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 26 (3) (2016), 555–566.
  * [30] Y. Simsek, Identities on the Changhee numbers and Apostol-type Daehee polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 27 (2) (2017), 199–212.
  * [31] Y. Simsek, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Appl. Anal. Discrete Math. 12 (2018), 1–35, https://doi.org/10.2298/AADM1801001S.
  * [32] Y. Simsek, Combinatorial identities and sums for special numbers and polynomials, Filomat 32 (20) (2018), 6869–6877.
  * [33] Y. Simsek Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and $p$-adic $q$-integrals, Turk. J. Math. 42 (2018), 557–577.
  * [34] Y. Simsek, Explicit formulas for $p$-adic integrals: Approach to $p$-adic distributions and some families of special numbers and polynomials, Montes Taurus J. Pure Appl. Math. 1 (1) (2019), 1–76.
  * [35] Y. Simsek, Formulas for $p$-adic $q$-integrals including falling-rising factorials, combinatorial sums and special numbers, https://arxiv.org/pdf/1702.06999.pdf
  * [36] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Acedemic Publishers, Dordrecht, Boston and London, 2001.
  * [37] H. M. Srivastava, I. Kucukoglu and Y. Simsek, Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials, J. Number Theory 181 (2017), 117–146.
  * [38] E. W. Weisstein, Leibniz Harmonic Triangle, From MathWorld-A Wolfram Web Resource, https://mathworld.wolfram.com/LeibnizHarmonicTriangle.html.
  * [39] E. W. Weisstein, Pronic Number, From MathWorld-A Wolfram Web Resource, https://mathworld.wolfram.com/PronicNumber.html.


V\yon\ar[d, equal]\ar[r, bimr-biml, "g"]&
\cdots\ar[r, bimr-biml, "g"]&|[alias=V]|
V\yon\ar[rr, bimr-biml, "T"]&&
\0\ar[d, equal]\\
V\yon\ar[rrrr, bimr-biml, "T"', "" name=T]&&&&
\0
\ar[from=V, to=V|-T, Rightarrow, "\alpha\circ\cdots\circ\alpha"]
\ar[from=V', to=VV, equal]
\end{tikzcd}
\]

Explain why (<ref>) models an initialization, i.e. a way to choose a starting color in each vertex.

Recall that the $(V\yon,0)$-bicomodule $T$ assigns to each vertex $v\in V$ some set $T_v$ of “colors”, whereas the $(V\yon,0)$-bicomodule $V$ assigns each vertex a singleton set. A map $V\to T$ between them chooses one color for each vertex, which we're calling the “starting color”.

§ SUMMARY AND FURTHER READING

In this chapter we gave more of the theory of $\tri$-comonoids, sometimes called polynomial comonads. We began by defining an adjunction

\[
\adj{\smcat^\sharp}{U}{\cofree{-}}{\poly}
\]

where $\cofree{p}$ is the cofree comonoid $\cofree{p}$ associated to any polynomial $p$, and we gave intuition for it in terms of $p$-trees. A $p$-tree is a (possibly infinite) tree for which each vertex is labeled by a position $i\in p(\1)$ and its outgoing arrows are each labeled by an element of $p[i]$. The category corresponding to $\cofree{p}$ has $p$-trees as its objects; the morphisms emanating from such a $p$-tree are the finite rooted paths up the tree, and the codomain of such a path is the tree rooted at its endpoint.

We then briefly discussed some other properties of $\smcat^\sharp$ including a formal proof of the fact that it has all limits and colimits. Colimits in $\smcat^\sharp$ are created by, i.e. fit nicely with, colimits in $\poly$, but limits are quite strange; we did not really discuss them here, but for example the product in $\smcat^\sharp$ of the walking arrow $\bullet\to\bullet$ with itself has infinitely-many objects!

We then moved on to left-, right-, and bicomodules between polynomial comonoids. In particular, we showed that left $\cat{C}$-comodules can be identified with functors $\cat{C}\to\poly$, and that $(\cat{C},\cat{D})$-bicomodules correspond to parametric right adjoint functors (prafunctors) $\smset^\cat{D}\to\smset^\cat{C}$. This idea was due to Richard Garner; it is currently unpublished, but can be found in video form here: <https://www.youtube.com/watch?v=tW6HYnqn6eI>. What we call prafunctors are sometimes called familial functors between (co-)presheaf categories; see <cit.>, <cit.>, and <cit.> for more on this notion.

§ EXERCISE SOLUTIONS