A $q$-analogue of $\bar\alpha$-Whitney Numbers

TL;DR

This paper introduces a $q$-analogue of $ar\alpha$-Whitney numbers, exploring their properties and generalizations, including explicit formulas, recurrence relations, and orthogonality, extending classical combinatorial numbers.

Contribution

The paper defines the $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers and their properties, and introduces the $ar{\boldsymbol{\alpha}}$-Whitney-Lah numbers as a new generalization.

Findings

Derived explicit formulas and recurrence relations.

Established orthogonality and inverse relations.

Defined and analyzed the properties of Whitney-Lah numbers.

Abstract

We define the $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers which are reduced to the $\bar{\boldsymbol{\alpha}}$-Whitney numbers when $q\rightarrow1$. Moreover, we obtain several properties of these numbers such as explicit formulas, recurrence relations, generating functions, orthogonality and inverse relations. Finally, we define the $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers as a generalization of the $r$-Whitney-Lah numbers and we introduce their important basic properties.

Equations

$$(x;\bar{\mathbf{\alpha }}|m{)}_n=\sum _{k=0}^n{w}_{m,\bar{\mathbf{\alpha }}}(n,k)x^k,$$

$$x^n=\sum _{k=0}^n{W}_{m,\bar{\mathbf{\alpha }}}(n,k)(x;\bar{\mathbf{\alpha }}|m{)}_k,$$

$$(x;\bar{\mathbf{\alpha }}|m{)}_n=\prod _{j=0}^{n-1}(x-{\alpha }_j-jm)\text{with}(x;\bar{\mathbf{\alpha }}|m{)}_0=1.$$

$$w_{m,\bar{\mathbf{\alpha }}}(n+1,k)=w_{m,\bar{\mathbf{\alpha }}}(n,k-1)-({\alpha }_n+nm)w_{m,\bar{\mathbf{\alpha }}}(n,k),$$

$$W_{m,\bar{\mathbf{\alpha }}}(n+1,k)=W_{m,\bar{\mathbf{\alpha }}}(n,k-1)+({\alpha }_k+km)W_{m,\bar{\mathbf{\alpha }}}(n,k).$$

$$([x{]}_q{)}_n=\prod _{j=0}^{n-1}[x-j{]}_q\text{and}([x{]}_q{)}_o=1.$$

$$([x;\bar{\mathbf{\alpha }}{]}_q{)}_n=\prod _{j=0}^{n-1}[x-{\alpha }_j{]}_q=[x-{\alpha }_0{]}_q[x-{\alpha }_1{]}_q\cdots [x-{\alpha }_{n-1}{]}_q\text{with}([x;\bar{\mathbf{\alpha }}{]}_q{)}_0=1,$$

$$([x;\bar{\mathbf{\alpha }}|m{]}_q{)}_n=\prod _{j=0}^{n-1}[x-{\alpha }_j-jm{]}_q,\text{with}([x;\bar{\mathbf{\alpha }}|m{]}_q{)}_0=1,$$

$$\big{\langle}[x;\bar{\boldsymbol{\alpha}}|m]_{q}\big{\rangle}_{n}=\prod_{j=0}^{n-1}([x]_{q}-[\alpha_{j}+jm]_{q}),\>\text{with}\>\big{\langle}[x;\bar{\boldsymbol{\alpha}}|m]_{q}\big{\rangle}_{0}=1.$$

$$\big{\langle}[x;\bar{\boldsymbol{\alpha}}|m]_{q}\big{\rangle}_{n}=\sum_{k=0}^{n}w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)[x]^{k}_{q},$$

$$([x;\bar{\mathbf{\alpha }}|m{]}_q{)}_n=q^{-{\sum }_{i=0}^{n-1}{\alpha }_i+im}\prod _{j=0}^{n-1}([x{]}_q-[{\alpha }_j+jm{]}_q).$$

$$([x;\bar{\mathbf{\alpha }}|m{]}_q{)}_n=q^{-{\sum }_{i=0}^{n-1}{\alpha }_i+im}\sum _{k=0}^n{w}_{q,m,\bar{\mathbf{\alpha }}}(n,k)[x{]}_q^k.$$

$$w_{q,m,\bar{\mathbf{\alpha }}}(n+1,k)=w_{q,m,\bar{\mathbf{\alpha }}}(n,k-1)-[{\alpha }_n+nm{]}_q{w}_{q,m,\bar{\mathbf{\alpha }}}(n,k),$$

$$w_{q,m,\bar{\mathbf{\alpha }}}(n,0)=(-1{)}^n\prod _{i=0}^{n-1}[{\alpha }_i+im{]}_q.$$

$$\begin{split}&\sum_{k=0}^{n+1}w_{q,m,\bar{\boldsymbol{\alpha}}}(n+1,k)[x]^{k}_{q}=\big{\langle}[x;\bar{\boldsymbol{\alpha}}|m]_{q}\big{\rangle}_{n}\>([x]_{q}-[\alpha_{n}+nm]_{q})\\ &=\sum_{k=0}^{n}w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)[x]_{q}^{k}\>([x]_{q}-[\alpha_{n}+nm]_{q})\\ &=\sum_{k=0}^{n}w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)[x]_{q}^{k+1}-[\alpha_{n}+nm]_{q}\sum_{k=0}^{n}w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)[x]_{q}^{k}\\ &=\sum_{k=1}^{n+1}w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k-1)[x]_{q}^{k}-[\alpha_{n}+nm]_{q}\sum_{k=0}^{n}w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)[x]_{q}^{k}.\end{split}$$

$$w_{q,m,\bar{\mathbf{\alpha }}}(n+1,0)=-[{\alpha }_n+nm{]}_q{w}_{q,m,\bar{\mathbf{\alpha }}}(n,0),n=0,1,2,\dots ,$$

$$\mathbf{W}_{1}:=\mathbf{w}_{q,m,\bar{\boldsymbol{\alpha}}}(n):=\big{(}w_{q,m,\bar{\boldsymbol{\alpha}}}(i,j)\big{)}_{0\leq i,j\leq n-1}.$$

$$\left(\begin{array}{cccc}1& 0& 0& 0\\ & & & \\ -[{\alpha }_0{]}_q& 1& 0& 0\\ & & & \\ [{\alpha }_0{]}_q[{\alpha }_1+m{]}_q& -[{\alpha }_0{]}_q-[{\alpha }_1+m{]}_q& 1& 0\\ & & & \\ -[{\alpha }_0{]}_q[{\alpha }_1+m{]}_q[{\alpha }_2+2m{]}_q& [{\alpha }_0{]}_q[{\alpha }_1+m{]}_q+[{\alpha }_0+{\alpha }_1+m{]}_q[{\alpha }_2+2m{]}_q& -[{\alpha }_0{]}_q-[{\alpha }_1+m{]}_q-[{\alpha }_2+2m{]}_q& 1\end{array}\right)$$

$$u(n,k)=u(n-1,k-1)+(a_{n-1}+b_k)u(n-1,k),$$

$$u(n,k)=\sum _{j=0}^k\left(\frac{{\prod }_{i=0}^{n-1}(b_j+a_i)}{{\prod }_{i=0,i\ne j}^k(b_j-b_i)}\right),\forall n,k\in \mathbb{N}.$$

$$u(n,k)=\sum _{j=k}^{n}u(j-1,k-1)\prod _{i=j}^{n-1}(a_i+b_k).$$

$${\sigma }_k(z_1,z_2,\dots ,z_n)=\sum _{1\le j_1<j_2<\cdots <j_k\le n}\prod _{i=1}^k{z}_{j_i},$$

$$\begin{array}{rl}{w}_{q,m,\bar{\mathbf{\alpha }}}(n,k)& =(-1{)}^{n-k}{\sigma }_{n-k}([{\alpha }_0{]}_q,[{\alpha }_1+m{]}_q,\dots ,[{\alpha }_{n-1}+(n-1)m{]}_q)\\ & =(-1{)}^{n-k}\sum _{0\le j_1<\cdots <j_{n-k}\le n-1}\prod _{i=1}^{n-k}[{\alpha }_{j_i}+j_{i}m{]}_q.\end{array}$$

$$w_{q,m,\bar{\mathbf{\alpha }}}(n,k)=(-1{)}^{n-k}\sum _{j=k}^n{w}_{q,m,\bar{\mathbf{\alpha }}}(j-1,k-1)\prod _{i=j}^{n-1}[{\alpha }_i+im{]}_q.$$

$$\sum _{k=0}^n{w}_{q,m,\bar{\mathbf{\alpha }}}(n,k)x^k=\prod _{j=0}^{n-1}(x-[{\alpha }_j+jm{]}_q).$$

$$w_{m,\bar{\mathbf{\alpha }}}(n,k)=(-1{)}^{n-k}\sum _{j=k}^n{w}_{m,\bar{\mathbf{\alpha }}}(j-1,k-1)\prod _{i=j}^{n-1}({\alpha }_i+im).$$

$$[x]^{n}_{q}=\sum_{k=0}^{n}W_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)\big{\langle}[x;\bar{\boldsymbol{\alpha}}|m]_{q}\big{\rangle}_{k},$$

$$[x{]}_q^n=\sum _{k=0}^n{q}^{{\sum }_{i=0}^{k-1}{\alpha }_i+im}{W}_{q,m,\bar{\mathbf{\alpha }}}(n,k)([x;\bar{\mathbf{\alpha }}|m{]}_q{)}_k.$$

$$W_{q,m,\bar{\mathbf{\alpha }}}(n+1,k)=W_{q,m,\bar{\mathbf{\alpha }}}(n,k-1)+[{\alpha }_k+km{]}_q{W}_{q,m,\bar{\mathbf{\alpha }}}(n,k),$$

$$W_{q,m,\bar{\mathbf{\alpha }}}(n,0)=[{\alpha }_0{]}_q^n.$$

Full text

Keywords: $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers, $\bar{\boldsymbol{\alpha}}$-Whitney numbers, $r$-Whitney numbers, $q$-Stirling numbers; $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers.

2010 Mathematics Subject Classification: Primary 05A19 ; Secondary 05A15, 05E05.

1 Introduction

El-Desouky et al. [5] introduced the $\bar{\boldsymbol{\alpha}}$-Whitney numbers of both kinds as a new family of numbers generalizing many types of numbers such as $r$-Whitney numbers, Whitney numbers, $r$-Stirling numbers, Jacobi-Stirling numbers and Legendre-Stirling numbers.

The $\bar{\boldsymbol{\alpha}}$-Whitney numbers of the first kind $w_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ and second kind $W_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ are defined by

[TABLE]

and

[TABLE]

where $\bar{\boldsymbol{\alpha}}=(\alpha_{0},\alpha_{1},\ldots,\alpha_{n-1})$, and

[TABLE]

The $\bar{\boldsymbol{\alpha}}$-Whitney numbers of the first and second kind satisfying recurrence relations of the form:

[TABLE]

[TABLE]

Note that the $\bar{\boldsymbol{\alpha}}$-Whitney numbers coincide with the $r$-Whitney numbers and Whitney numbers by setting $\bar{\boldsymbol{\alpha}}=(r,r,\ldots,r)$ and $\bar{\boldsymbol{\alpha}}=(1,1,\ldots,1)$, respectively. Many properties of the $\bar{\boldsymbol{\alpha}}$-Whitney numbers, $r$-Whitney numbers and Whitney numbers can be found in [5, 4, 1, 8, 10, 11, 12].

The organization of this article is as follows. In the next two sections, we define the $q$-analogue of the $\bar{\boldsymbol{\alpha}}$-Whitney numbers of the first and second kind denoted by $w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)$ and $W_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)$, respectively, and obtain their recurrence relations, explicit formulas and generating functions. In the third section, we obtain the orthogonality property of the both kinds of the $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers which yields to the inverse relations. Moreover we give some important special cases. In the fourth section, we define the $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers and deduce its recurrence relation, explicit formula and matrix representation.

Let $0<q<1$, $x$ a real number, $[x]_{q}=1+q+\cdots+q^{x-1}$ and $[0]_{q}=0$. The $q$-factorial of $x$ is defined by $[x]_{q}!=[x]_{q}[x-1]_{q}\cdots[2]_{q}[1]_{q}$, and the $q$-falling factorial of order $n$ is defined by

[TABLE]

Moreover, the following definitions and notation are introduced.

[TABLE]

[TABLE]

and

[TABLE]

2 The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the first kind

Definition 1**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the first kind $w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)$ are defined by

[TABLE]

where $w_{q,m,\bar{\boldsymbol{\alpha}}}(0,0)=1$ and $w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)=0$ for $k>n$ or $k<0$.

Since for the q-numbers we have $[x-y]_{q}=q^{-y}([x]_{q}-[y]_{q})$. Then

[TABLE]

Thus Eq. (2.1) in Definition 1 can be written in the equivalent form

[TABLE]

In particular, note that $w_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)$ is reduced to the $w_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ when $q\rightarrow 1$.

Theorem 1**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the first kind satisfy the recurrence relation

[TABLE]

where $n\geq k\geq 1$, and

[TABLE]

Proof.

Since \big{\langle}[x;\bar{\boldsymbol{\alpha}}|m]_{q}\big{\rangle}_{n+1}=\big{\langle}[x;\bar{\boldsymbol{\alpha}}|m]_{q}\big{\rangle}_{n}\>([x]_{q}-[\alpha_{n}+nm]_{q}).

Using Eq. (2.1), we get

[TABLE]

Equating the coefficients of $[x]_{q}^{k}$ on both sides yields (2.2).

For $k=0$, we find

[TABLE]

successive application gives (2.3). ∎

Definition 2**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney matrix of the first kind is the $n\times n$ lower triangular matrix defined by

[TABLE]

For example when $n=4$ the matrix $\mathbf{w}_{q,m,\bar{\boldsymbol{\alpha}}}(n)$ is given by

[TABLE]

In particular, we note that $\mathbf{w}_{q,m,\bar{\boldsymbol{\alpha}}}(n)$ is reduced to the $\bar{\boldsymbol{\alpha}}$-Whiyney matrix of the first kind [5] when $q\rightarrow 1$. In addition at $q\rightarrow 1$ and $\bar{\boldsymbol{\alpha}}=(r,r,\ldots,r)$ the $\mathbf{w}_{q,m,\bar{\boldsymbol{\alpha}}}(n)$ is reduced to the $r$-Whitney matrix of the first kind [12].

Mansour et al. [9] derived a closed formula for all sequences satisfying a certain recurrence relation as follows:

Theorem 2**.**

[9, Theorem  1.1]. Suppose $(a_{i})_{i\geq 0}$ and $(b_{i})_{i\geq 0}$ are sequences of numbers with $b_{i}\neq b_{j}$ when $i\neq j$ and

[TABLE]

with boundary conditions $u(n,0)=\prod_{i=0}^{n-1}(a_{i}+b_{0})$ and $u(0,k)=\delta_{0k}$, where $\delta_{jk}$ is the Kronecker delta function, then

[TABLE]

Remark 1**.**

[9, p. 25]**

The recurrence for $u(n,k)$ is given by

[TABLE] 2. 2.

*In the case when $b_{i}=0$ for all $i$, then $u(n,k)$ is the *$(n-k)$th elementary symmetric function of $a_{0},\;a_{1},\;\ldots,\;a_{n-1}$. The elementary symmetric function $\sigma_{k}$ is defined by

[TABLE]

where $\sigma_{0}=1$ and $\sigma_{k}=0$ when $n<k$ or $k<0$.

Theorem 3**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the first kind are given by

[TABLE]

and the following recurrence relation holds:

[TABLE]

Proof.

Taking $a_{i}=-[\alpha_{i}+im]_{q}$ and $b_{i}=0$, one can use Remark 1 to obtain Eq. (2.7) and Eq. (2.8). ∎

From Eq. (2.7), we can obtain the generating function

[TABLE]

As $q\rightarrow 1$, Eq. (2.8) reduces to a new recurrence relation for the $\bar{\boldsymbol{\alpha}}$-Whitney numbers of the first kind given by

[TABLE]

3 The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the second kind

Definition 3**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the second kind $W_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ are defined by

[TABLE]

where $W_{q,m,\bar{\boldsymbol{\alpha}}}(0,0)=1$ and $W_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)=0$ for $k>n$ or $k<0$.

We notice that Eq. (3.1) in Definition 3 can be written in the equivalent form

[TABLE]

Theorem 4**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the second kind satisfy the recurrence relation

[TABLE]

where $n\geq k\geq 1$, for $k=0$ we have

[TABLE]

Proof.

Since we have $[x]_{q}^{n+1}=[x]_{q}^{n}\>([x]_{q}-[\alpha_{k}+km]_{q}+[\alpha_{k}+km]_{q}).$ Using (3.1), we get

[TABLE]

Equating the coefficients of \big{\langle}[x;\bar{\boldsymbol{\alpha}}|m]_{q}\big{\rangle}_{k} on both sides, we obtain Eq. (3.2).

When $k=0$, we get $W_{q,m,\bar{\boldsymbol{\alpha}}}(n+1,0)=[\alpha_{0}]_{q}\,W_{q,m,\bar{\boldsymbol{\alpha}}}(n,0),\>n=0,1,2,\ldots$ . Thus $W_{q,m,\bar{\boldsymbol{\alpha}}}(n,0)=[\alpha_{0}]_{q}^{n}\,W_{q,m,\bar{\boldsymbol{\alpha}}}(0,0)=[\alpha_{0}]_{q}^{n}$. ∎

Definition 4**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney matrix of the second kind is the $n\times n$ lower triangular matrix defined by

[TABLE]

For example when $n=4$, the matrix $\mathbf{W}_{q,m,\bar{\boldsymbol{\alpha}}}(n)$ is given by

[TABLE]

When $q\rightarrow 1$ the matrix $\mathbf{W}_{q,m,\bar{\boldsymbol{\alpha}}}(n)$ is reduced to the $\bar{\boldsymbol{\alpha}}$-Whiyney matrix of the second kind [5], also at $q\rightarrow 1$ and $\bar{\boldsymbol{\alpha}}=(r,r,\ldots,r)$ the $\mathbf{W}_{q,m,\bar{\boldsymbol{\alpha}}}(n)$ is reduced to the $r$-Whitney matrix of the second kind [4, 12].

Theorem 5**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the second kind $W_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)$ have the explicit formula

[TABLE]

and satisfy the recurrence relation

[TABLE]

Proof.

Taking $a_{i}=0$ and $b_{i}=[\alpha_{i}+im]_{q}$ in (2.5) and (2.6), yield (3.4) and (3.5), respectively. ∎

As $q\rightarrow 1$, the recurrence relation (3.5) reduces to a new recurrence relation for the $\bar{\boldsymbol{\alpha}}$-Whitney numbers of the second kind given by

[TABLE]

Using (3.4) we obtain the exponential generating function of the $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the second kind $W_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)$

[TABLE]

Theorem 6**.**

The generating function of $W_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)$ is given by

[TABLE]

where $k=1,2,3,\ldots$, and

[TABLE]

Proof.

Equation (3.8) can easily obtained from the definition of generating function

[TABLE]

From (3.2), we get

[TABLE]

Thus we obtain the recurrence relation for the generating function $Y_{k,q}(t)$

[TABLE]

Hence

[TABLE]

Applying successively this recurrence, we get Eq. (3.7). ∎

The previous theorem shows that the numbers $W_{q,m,\bar{\boldsymbol{\alpha}}}(n,k)$ are the complete symmetric function of the numbers $[\alpha_{0}]_{q},\>[\alpha_{1}+m]_{q},\>\ldots,\>[\alpha_{k}+km]_{q}$ of order $n-k$.

We obtain from Eq. (3.7)

[TABLE]

Expanding the right hand side and comparing the coefficients of $t^{n-k}$ yields

[TABLE]

4 Orthogonality and Inverse Relations

The orthogonality and the inverse relations for the $\bar{\boldsymbol{\alpha}}$-Whitney numbers of both kinds were obtained in [5]. In this section, we establish analogous properties for the $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of both kinds.

Theorem 7**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the first and second kind satisfy the following orthogonality relations:

[TABLE]

and

[TABLE]

Proof.

Using (2.1) and (3.1) give

[TABLE]

Comparing the coefficients of $[x]^{j}_{q}$ gives

[TABLE]

The second relation can be proved similarly. ∎

The orthogonality properties give the following identities

[TABLE]

The following theorem can easily be deduced from Theorem 7.

Theorem 8**.**

The $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers of the first and second kind satisfy the following inverse relations

[TABLE]

Proof.

If the condition

[TABLE]

holds, then

[TABLE]

By Theorem 7, we get

[TABLE]

The converse can be shown similarly. ∎

Special cases:

Setting $m=1$ and $\bar{\boldsymbol{\alpha}}=(r,r,\ldots,r):=\mathbf{r}$, then (2.1) and (3.1), respectively, give

[TABLE]

where $s_{q}(n,k,r)$ and $S_{q}(n,k,r)$ are the non-central q-Stirling numbers of the first and second kind, respectively, see [3]. 2. 2.

Setting $m=1$ and $\bar{\boldsymbol{\alpha}}=(0,0,\ldots,0):=\mathbf{0}$, hence (2.1) and (3.1), respectively, give

[TABLE]

where $s_{q}(n,k)$ and $S_{q}(n,k)$ are the q-Stirling numbers of the first and second kind, respectively, see [2, 7]. 3. 3.

Setting $m=1$ and $\alpha_{j}+j=\beta_{j}$, for $j=0,1,\ldots,n-1$, then (2.1) and (3.1), respectively, give

[TABLE]

where $s_{q,\bar{\boldsymbol{\beta}}}(n,k)$ and $S_{q,\bar{\boldsymbol{\beta}}}(n,k)$ are the generalized q-Stirling numbers of the first and second kind (q-Comtet numbers), respectively, see [6].

5 The $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers

The signless Lah numbers $L(n,k)=\frac{n!}{k!}\binom{n-1}{k-1}$ were first studied by Lah [13] and they expressed in terms of the signless stirling numbers $s(n,k)$ of the first kind, and the stirling numbers $S(n,k)$ of the second kind

[TABLE]

Choen and Jung [4] defined the $r$-Whitney-Lah numbers $L_{m,r}(n,k)$ by

[TABLE]

Analogously, we define the $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers $L_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ as follows:

[TABLE]

where $L_{m,\bar{\boldsymbol{\alpha}}}(0,0)=1$ and $L_{m,\bar{\boldsymbol{\alpha}}}(n,k)=0$ for $n<k$ or $k<0$.

Theorem 9**.**

The $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers $L_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ may be obtained from

[TABLE]

Proof.

Replacing $x$ by $-x$ in Eq. (1.1), we get

[TABLE]

Hence

[TABLE]

∎

Theorem 10**.**

The $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers satisfy the recurrence relation

[TABLE]

where $n\geq k\geq 1$, for $k=0$ we have

[TABLE]

Proof.

We can write

[TABLE]

Using (5.2), we get

[TABLE]

Equating the coefficients of $(x;\bar{\boldsymbol{\alpha}}|m)_{k}$ on both sides, we obtain (5.4).

For $k=0$, we find

[TABLE]

Consequently, we get

[TABLE]

∎

Special cases:

The $L_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ is reduced to $L(n,k)$ when $m=1$ and $\bar{\boldsymbol{\alpha}}=(0,0,\ldots,0)$. 2. 2.

The $L_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ is reduced to $L_{m,r}(n,k)$ when $\bar{\boldsymbol{\alpha}}=(r,r,\ldots,r)$. 3. 3.

The $L_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ is reduced to the $r$-Lah numbers $L_{r}(n+r,k+r)$ when $m=1$ and $\bar{\boldsymbol{\alpha}}=(r,r,\ldots,r)$, see [14].

Defining the $\bar{\boldsymbol{\alpha}}$-Whitney-Lah matrix as

[TABLE]

For example when $n=4$ the matrix $\mathbf{L}_{m,\bar{\boldsymbol{\alpha}}}(n)$ is given by

[TABLE]

In particular, when $\bar{\boldsymbol{\alpha}}=(r,r,\ldots,r)$ the $\mathbf{L}_{m,\bar{\boldsymbol{\alpha}}}(n)$ is reduced to the $r$-Whitney-Lah matrix [12].

Theorem 11**.**

The $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers $L_{m,\bar{\boldsymbol{\alpha}}}(n,k)$ have the explicit formula

[TABLE]

and the recurrence relation

[TABLE]

Proof.

The proof follows by setting $a_{i}=b_{i}=\alpha_{i}+im$ in (2.5) and (2.6). ∎

In particular, by setting $\bar{\boldsymbol{\alpha}}=(r,r,\ldots,r)$ we obtain the explicit formula and recurrence relation for $L_{m,r}(n,k)$ as follows:

Corollary 1**.**

The $r$-Whitney-Lah numbers $L_{m,r}(n,k)$ satisfy the following:

[TABLE]

[TABLE]

Choen and Jung [4] showed that

[TABLE]

Thus from (5.8) and (5.10) we obtain the following combinatorial identity

[TABLE]

5.1 Matrix representations

Let $\mathbf{w},\mathbf{W}$ and $\mathbf{L}$ denote infinite lower triangular matrices whose $(n,k)$-th entries are $w_{m,\bar{\boldsymbol{\alpha}}}(n,k)$, $W_{m,\bar{\boldsymbol{\alpha}}}(n,k)$, and $L_{m,\bar{\boldsymbol{\alpha}}}(n,k)$, respectively. Furthermore, let $\mathbf{D}$ be the infinite diagonal matrix whose $(n,k)$-th entry is $D(n,k)=(-1)^{n}\delta{nk}$, hence $\mathbf{D}^{-1}=\mathbf{D}$, and $\mathbf{D}\mathbf{D}^{-1}=\mathbf{I}$. Equation (5.1) can be written in the matrix form

[TABLE]

El-Desouky et al. [5] showed that $\mathbf{w}^{-1}=\mathbf{W}$, $\mathbf{W}^{-1}=\mathbf{w}$. Thus

[TABLE]