In this paper, we generalize the \(k\)-Jacobsthal sequences and call them the generalized \((k,t)\)-Jacobsthal \(p\)-sequences. Also, we obtain combinatorial identities. Then, the generalized\((k,t)\)-Jacobsthal \(p\)-matrix is used to factorize the Pascal matrix. Finally, using the Riordan method, we obtain two factorizations of the Pascal matrix involving the generalized \((k,t)\)-Jacobsthal \(p\)-sequences.
In Mathematics, sequences such as Fibonacci, Pell, Mersenne, etc., play an important role (see [1, 8, 15, 16, 21]). A Jacobsthal number is one of many sequences studied in mathematics and other fields. The Jacobsthal number \(J_n\) is defined as \[\begin{aligned} \nonumber J_n=J_{n-1}+2J_{n-2},~ n\geq 2, \end{aligned}\] with initial conditions \(J_0=0\) and \(J_1=1\) [11]. The Jacobsthal numbers have been generalized in several ways [4, 5, 6].
In 2008, the Jacobsthal Lucas \(E\)-matrix and \(R\)-matrix were given which are similar to the Fibonacci \(Q\)-matrix [12]. In [2], Gaussian Jacobsthal sequences were introduced and the corresponding generating functions were given. In 2016, upper and lower bounds were obtained on matrices whose elements are \(k\)-Jacobsthal sequences [19]. A definition of the adjacency-Jacobsthal sequence can be found in [7].
In [14], obtained factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices. In 2011, introduced new factorizations of the Pascal matrix via Fibonomial coefficients called the Fibo-Pascal matrix, involving the \(k\)-Fibonacci matrix and \(k\)-Pell matrix [18]. In [9], the \(k\)-Fibonacci matrix and the Pascal matrix were studied. In 2022, gave a factorization of the Pascal matrix involving the \(t\)-extension of the \(p\)-Fibonacci matrix [10]. In [13], the \((d,k)\)-Fibonacci polynomial was introduced and a factorization of the Pascal matrix based on these sequences was presented.
Our motivation here is a generalization of the \(k\)-Jacobsthal sequence, which we used to factorize Pascal matrices, which can be used later in other fields.
Here, we introduce the generalized \(k\)-Jacobsthal sequence and obtain combinatorial identities. Also, we give a Factorization of the Pascal matrix involving these sequences.
The remainder of this paper is organized as follows. The Jacobsthal numbers are generalized in Section 2 and new sequences are obtained. In Section 3, we give three Factorization of the Pascal matrix involving the generalized \((k,t)\)-Jacobsthal \(p\)-sequences.
These are some definitions and concepts that will be useful during this process:
According to Jacobsthal’s sequence, one of these generalizations can be stated as follows:
Definition 2.1. For \(n\geq 0\) and \(k \geq 2\), the generalized Jacobsthal sequence \(\lbrace J(k,n)\rbrace\) is defined as \[\begin{aligned} J(k,n) = kJ(k, n-1) + 2J(k, n- 2) \mbox{ for } n \geq 2, \end{aligned}\] with initial conditions \(J(k, 0) = 0\) and \(J(k, 1) = 1\) [20].
For example, if \(k=3\), we have \[\begin{aligned} J(3,n) =3 J(3, n-1)+ 2J(3, n- 2)~\mbox{ for } ~n \geq 2, \end{aligned}\] and thus \(\lbrace J(2,n)\rbrace_0^\infty=\lbrace 0, 1, 3,11, \ldots\rbrace\).
Definition 2.2. The \(n\times n\) lower triangular Pascal matrix, denoted by \(P_n=[p_{ij}]\), is defined as follows [3]: \[p_{ij}=\left \{ \begin{array}{ll} \displaystyle{i-1\choose j-1},&\text {if}~ i\geq j,\\ \quad 0, & \text{otherwise}. \end{array}\right.\]
The Riordan group was introduced in [17] as follows.
Definition 2.3. Let \(R^{'}=[r_{ij}]_{i,j\geq 0}\) be an infinite matrix with complex entries. Let \(C_i(t)=\sum\limits_{n\geq 0}^\infty r_{n,i}t^n\) be the generating function of the ith column of \(R^{'}\). We call \(R^{'}\) a Riordan matrix if \(c_i(t)=g(t)[f(t)]^i\), where \[g(t)=1+g_1t+g_2t^2+g_3t^3+\cdots,~~~~~f(t)=t+f_2t^2+f_3t^3+\cdots.\]
In this case we write \(R=(g(t),f(t))\) and denote by \(R\) the set of Riordan matrices. Then the set \(R\) is a group under matrix multiplication \(*\), with the following properties:
(i) \((g(t),f(t))*(h(t),l(t))=(g(t)h(f(t)),l(f(t))),\)
(ii) \(I=(1,t)\) is the identity element,
(iii) the inverse of \(R\) is given by \(R^{-1}=(\dfrac{1}{g(\bar{f}(t)},\bar{f} (t))\), where \(\bar{f}(t)\) is the compositional inverse of \(f(t)\), that is,\(f(\bar{f}(t))=\bar{f}(f(t))=t.\)
In this section, we define the generalized \((k,t)\)-Jacobsthal \(p\)-sequences and some results are given which will be used later.
Definition 3.1. For integers \(k\geq 1\), \(p \geq 1\) and \(t\geq 2\), the generalized \((k,t)\)-Jacobsthal \(p\)-sequences denoted \(\lbrace J_n^p(k,t)\rbrace\) are defined as \[\begin{aligned} J_n^p(k,t)=kJ_{n-1}^p(k,t)+2J_{n-p-1}^p(k,t)+\cdots+J_{n-p- t}^p(k,t),~ n\geq t+p+1, \end{aligned} \tag{1}\] where \(J_0^p(k,t)=J_1^p(k,t)=\dots=J_{t+p-2}^p(k,t)=0\) and \(J_{t+p-1}^p(k,t)=1\).
Example 3.2. Let \(p=1\) and \(k=3\).
(i) If \(t=2\), according to Definition 3.1, we have \[\begin{aligned} J_n^1(3,2)=3J_{n-1}^1(3,2)+2J_{n-2}^1(3,2)+J_{n-3}^1(3,2),~ n\geq 4. \end{aligned} \tag{2}\]
Therefore, \(\lbrace J_n^1(3,2)\rbrace_0^\infty=\lbrace 0,0, 1, 3,11,40,145,526, \cdots \rbrace\).
(ii) If \(t=3\) , according to Definition 3.1, we have \[\begin{aligned} J_n^1(3,3)=3J_{n-1}^1(3,3)+2J_{n-2}^1(3,3)+J_{n-3}^1(3,3)+J_{n-4}^1(3,3),~ n\geq 5. \end{aligned} \tag{3}\]
So, \(\lbrace J_n^1(3,3)\rbrace_0^\infty=\lbrace 0,0, 0,1, 3,11,40,146,532, \cdots \rbrace\).
(iii) If \(t=4\), according to Definition 3.1, we have \[\begin{aligned} J_n^1(3,4)=3J_{n-1}^1(3,4)+2J_{n-2}^1(3,4)+J_{n-3}^1(3,4)+J_{n-4}^1(3,4)+J_{n-5}^1(3,4),~ n\geq 6. \end{aligned} \tag{4}\]
So, \(\lbrace J_n^1(3,4)\rbrace_0^\infty=\lbrace 0,0, 0,0,1, 3,11,40,146,532, 1939,\cdots \rbrace\).
Lemma 3.3. Let \(g_{J_n^p(k,t)}\) be the generating function of the generalized \((k,t)\)-Jacobsthal \(p\)-numbers, then \[\label{eq3.5} g_{J_n^p(k,t)}=\dfrac{x^{t+p-1}}{1-kx-2x^{p+1}-x^{p+2}-\cdots-x^{t+p}}. \tag{5}\]
Proof. We have \[\begin{aligned} g_{J_n^p(k,t)} &=\sum\limits_{n=1}^\infty J_n^p(k,t)x^n\\ &=J_1^p(k,t) x+J_2^p(k,t) x^2+\cdots+J_{t+p-1}^p(k,t) x^{t+p-1}+\sum\limits_{n=t+p}^\infty J_n^p(k,t) x^n\\ &=x^{t+p-1}+\sum\limits_{n=t+p}^\infty k{J_{n-1}^p(k,t)+ 2J_{n-p-1}^p(k,t)+J_{n-p-2}^p(k,t)+\cdots+J_{n-t}^p(k,t) }x^n\\ &=x^{t+p-1}+ \sum\limits_{n=t+p+1}^\infty (kJ_{n-1}^p(k,t) x^n +2\sum\limits_{n=t+p+1}^\infty J_{n-p-1}^p(k,t) x^n+\sum\limits_{n=t+p+1}^\infty J_{n-p-2}^p(k,t) x^n\\ &~~~+\cdots+ \sum\limits_{n=t+p+1}^\infty J_{n-t-p}^p(k,t)) x^n\\ &=x^{t+p-1}+k x \sum\limits_{n=1}^\infty J_{n}^p(k,t) x^n+ 2x^{p+1}\sum\limits_{n=1}^\infty J_{n}^p(k,t) x^n+\cdots+ x^{t+p} \sum\limits_{n=1}^\infty J_{n}^p(k,t) x^n\\ &=x^{t+p-1}+ xkg_{J_{n}^p(k,t)}+2x^{p+1}g_{J_{n}^p(k,t)} +\cdots+x^{t+p}g_{J_{n}^p(k,t)}. \end{aligned}\]
Thus, \[g_{J_{n}^p(k,t)}=\dfrac{x^{t+p-1}}{1-kx-2x^{p+1}-x^{ p+2}-\cdots-x^{t+p}}.\] ◻
Lemma 3.4. The generating function of the generalized \((k,t)\)-Jacobsthal \(p\)-numbers has the following exponential representation \[g_{J_n^p(k,t)}=x^{t+p-1}\exp \sum\limits_{i=1}^\infty\dfrac{x^i}{i}(k+2x^{p-2}+x^{p-3}+\cdots+x^{t+p-1})^i,\] where \(t\geq 2\).
Proof. From Eq. (5), we have \[\ln\dfrac{g_{J_n^p(k,t)}}{x^{t+p-1}}=-\ln(1-kx-2x^{p+1}-x^{p+2}-\cdots-x^{t+p}).\] \[\begin{aligned} &-\ln(1-kx-2x^{p+1}-x^{p+2}-\cdots-x^{t+p}) =-[-x(k+2x^{p}+\cdots+x^{t+p-1})\\ &-\dfrac{1}{2}x^2(k+2x^{p}+\cdots+x^{t+p-1})^2-\cdots-\dfrac{1}{n}x^n(k+2x^{p}+\cdots+x^{t+p-1})^n-\cdots], \end{aligned}\] which gives the result. ◻
In this section, we obtain the inverse of the generalized \((k,1)\)-Jacobsthal \(1\)-matrix. Also, we give a factorization of generalized \((k,1)\)-Jacobsthal \(1\)-matrix and get some results from it. First, we define the generalized \((k,t)\)-Jacobsthal \(p\)-matrix.
Definition 4.1. The \(n\times n\) generalized \((k,t)\)-Jacobsthal \(p\)-matrix \(p\geq 1\), denoted by \(M^{p,t}_{(n,k)}=[m^{p,t}_{(k,ij)}]\), is defined as follows: \[m^{p,t}_{(k,ij)}= J_{i-j+1}^p(k,t).\]
For example, suppose that \(n=7,~k=2,~t=1\) and \(p=1\), we have \[\begin{aligned} M^{1,1}_{(7,2)}&=\begin{bmatrix} J_{1}^1(2,1)&J_{0}^1(2,1)&J_{-1}^1(2,1)&J_{-2}^1( 2,1)&J_{-3}^1(2,1)&J_{-4}^1(2,1)&J_{-5}^1(2,1)\\ J_{2}^1(2,1)&J_{1}^1(2,1)&J_{0}^1(2,1)&J_{-1}^1( 2,1)&J_{-2}^1(2,1)&J_{-3}^1(2,1)&J_{-4}^1(2,1)\\ J_{3}^1(2,1)& J_{2}^1(2,1)&J_{1}^1(2,1)&J_{0}^1(2,1)&J_{-1}^1( 2,1)&J_{-2}^1(2,1)&J_{-3}^1(2,1)\\ J_{4}^1(2,1)& J_{3}^1(2,1)& J_{2}^1(2,1)&J_{1}^1(2,1)&J_{0}^1(2,1)&J_{-1}^1( 2,1)&J_{-2}^1(2,1)\\ J_{5}^1(2,1)& J_{4}^1(2,1)& J_{3}^1(2,1)& J_{2}^1(2,1)&J_{1}^1(2,1)&J_{0}^1(2,1)&J_{-1}^1( 2,1)\\ J_{6}^1(2,1)& J_{5}^1(2,1)& J_{4}^1(2,1)& J_{3}^1(2,1)& J_{2}^1(2,1)&J_{1}^1(2,1)&J_{0}^1(2,1)\\ J_{7}^1(2,1)& J_{6}^1(2,1)& J_{5}^1(2,1)& J_{4}^1(2,1)& J_{3}^1(2,1)& J_{2}^1(2,1)&J_{1}^1(2,1) \end{bmatrix}\\ &=\begin{bmatrix} 1&0&0&0&0&0&0\\ 2&1&0&0&0&0&0\\ 6&2&1&0&0&0&0\\ 16&6&2&1&0&0&0\\ 44&16&6&2&1&0&0\\ 120&44&16&6&2&1&0\\ 328&120&44&16&6&2&1 \end{bmatrix}. \end{aligned}\]
Remark 4.2. Using Definition 2.1, for \(n<0\), \(J_{n}^P(k,t)=0\). Set \(M_{(n,k)}:=M^{1,1}_{(n,k)}\) and \(J_n(k):=J_{n}^1(k,1)\).
Theorem 4.3. For the inverse of the generalized \((k,1)\)-Jacobsthal \(1\)-matrix, denoted by \((M_{(n,k)})^{-1}=[m^{'}_{(ij,k)}]\), we have \[m^{'}_{ij}(k)=\left \{ \begin{array}{ll} 1,&\text{if}~i=j,\\ -k,&\text{if}~j=i-1,\\ -2, &\text{if}~ j=i-2,\\ 0, &\text{otherwise.} \end{array}\right.\]
Proof. To find the inverse of inverse of the generalized \((k,1)\)-Jacobsthal \(1\)-matrix, we define the \(n\times n\) matrix \(F_{(k,n)}=[f_{ij}^k]\) as follows: \[F_{(k,n)}=\begin{bmatrix} 1&0&0&\ldots &0&0\\ J_2(k) &1&0&\ldots &0&0\\ \vdots &\vdots &\vdots&\ddots & \vdots & \vdots\\ J_{n-1}(k)&0&0&\ldots &1&0\\ J_n(k)&0&0&\ldots &0&1\\ \end{bmatrix}.\\ \]
Clearly, \(F_{(k,n)}\) is invertible and \[(F_{(k,n)})^{-1}=\begin{bmatrix} 1&0&0&\ldots &0&0\\ -J_2(k) &1&0&\ldots &0&0\\ \vdots &\vdots &\vdots&\ddots & \vdots & \vdots\\ -J_{n-1}(k)&0&0&\ldots &1&0\\ -J_n(k)&0&0&\ldots &0&1\\ \end{bmatrix}.\\ \]
Hence, \[M_{(n,k)}=F_{(k,n)}\times (I_1\oplus F_{(k,n-1)})\times (I_2\oplus F_{(k,n-2)} )\times \cdots\times(I_{n-2}\oplus F_{(2,2)} ),\] where \(I_j\) is an identity matrix. Since \((I_t\oplus F_{(k,n-t)} )^{-1}=I_t\oplus (F_{{(k,n-t)}})^{-1}\), we have \[(M_{(n,k)})^{-1}=(I_{n-2}\oplus( F_{(k,2)} )^{-1})\times \cdots \times (I_1\oplus (F_{(k,n-1)})^{-1} )\times(F_{(k,n)})^{-1}.\]
Therefore, \[m^{'}_{ij}(k)=\left \{ \begin{array}{ll} 1,&\text{if}~i=j,\\ -k,&\text{if}~j=i-1,\\ -2, &\text{if}~ j=i-2,\\ 0, &\text{otherwise.} \end{array}\right.\] ◻
Example 4.4. For \(n=4\), we have \[\begin{aligned} F_{(k,4)}=&\begin{bmatrix} 1&0&0&0\\ J_2(k) &1&0&0\\ J_{3}(k)&0&1&0\\ J_4(k)&0&0&1\\ \end{bmatrix}=\begin{bmatrix} 1&0&0&0\\ k &1&0&0\\ k^2+2&0&1&0\\ k^3+4k&0&0&1\\ \end{bmatrix}.\\ M_{(k,4)}=&\begin{bmatrix} 1&0&0&0\\ J_2(k) &1&0&0\\ J_{3}(k)&J_2(k)&1&0\\ J_4(k)&J_3(k)&J_2(k)&1\\ \end{bmatrix}=\begin{bmatrix} 1&0&0&0\\ k &1&0&0\\ k^2+2&k&1&0\\ k^3+4k&k^2+2&k&1\\ \end{bmatrix}.\\ I_1\oplus F_{(k,3)}=&\begin{bmatrix} 1&0&0&0\\ 0 &1&0&0\\ 0&k&1&0\\ 0&k^2+2&k&1\\ \end{bmatrix}.\\ I_2\oplus F_{(2,k)}=&\begin{bmatrix} 1&0&0&0\\ 0 &1&0&0\\ 0&0&1&0\\ 0&0&k&1\\ \end{bmatrix}.\\ M_{(4,k)}=&F_{(4,k)}\times (I_1\oplus F_{(3,k)})\times (I_2\oplus F_{(2,k)} ). \end{aligned}\]
Therefore, for \(k\geq 1\), \[(M_{(4,k)})^{-1}=\begin{bmatrix} 1&0&0&0\\ -k&1&0&0\\ 0&-k&1&0\\ -2&0&-k&1\\ \end{bmatrix}.\]
Here, we give a factorization of the the generalized \((k,1)\)-Jacobsthal \(1\)-matrix. First, we introduce the matrix \(V^k_{n}\).
Definition 4.5. Entries of the \(n\times n\) matrix \(V^k_{n}=[v^k_{ij}]\) are defined as following: \[\label{Eq4} v^k_{ij}=\displaystyle{i-1\choose j-1}-k\displaystyle{i-2\choose j-1}-2\displaystyle{i-3\choose j-1}. \tag{6}\]
For \(i , j\geq 2\), using relation (6), we can write \[v^k_{ij}= v^k_{i-1j-1}+ v^k_{i-1j},\] where \(v^k_{11}=1\), \(v^k_{1j}=0\), \(j\geq 2\).
For \(k=1\) and \(n=4\), we have \[V^1_4=\begin{bmatrix} 1&0&0&0\\ 1-k&1&0&0\\ -1-k&2-k&1&0\\ -1-k&1-2k&3-k&1\\ \end{bmatrix}.\]
By the above information, we prove the following theorem.
Theorem 4.6. For the Pascal matrix \(P_n\), we have \(P_n=M_{(n,k)}V_n^k\).
Proof. The matrix \(M_{(n,k)}\) is invertible. If we get \((M_{(n,k)})^{-1}P_n=V_n^k\), then Theorem is proved. Let \((M_{(n,k)})^{-1}P_n=A_n\) where \(A_n=(a_{i,j})_{1\leq i,j \leq n}\), i.e., \[a_{i,j}=\sum\limits_{u=j}^im^{'}_{iu}(k)v^k_{uj}.\]
Since \((M_{(n,k)})^{-1}\) and \(P_n\) are lower triangular matrices, by the definition of \((M_{(n,k)})^{-1}\), we have \[\begin{aligned} a_{i,j}&=\sum\limits_{u=j}^i m^{'}_{iu}(k)\displaystyle{u-1\choose j-1}\\ &=m^{'}_{ii-2}(k)\displaystyle{i-3\choose j-1}+m^{'}_{ii-1}(k)\displaystyle{i-2\choose j-1} +m^{'}_{ii}(k)\displaystyle{i-1\choose j-1}\\ &=-2\displaystyle{i-3\choose j-1}-k\displaystyle{i-2\choose j-1}+\displaystyle{i-1\choose j-1}=(v_{ij}^k)_{1\leq i,j\leq n }. \end{aligned}\] ◻
Corollary 4.7. For \(t,u\in \mathbb {N}\), \[\displaystyle{t-1\choose u-1}=P_{tu}=\sum\limits_{j=u}^tm_{tj}(k)v_{ju}^k=m_{t1}(k)v_{1u}^k+m_{t2}(k)v_{2u}^k+\cdots+m_{t t-1}(k)v_{t-1u}^k+m_{tt}(k)v_{tu}^k.\]
For \(u=1\), we have \[P_{t1}=\sum\limits_{j=u}^tm_{tj}(k)v_{ju}^k=m_{t1}(k)v_{11}^k+m_{t2}(k)v_{21}^k+\cdots+m_{t t-1}(k)v_{t-11}^k+m_{tt}(k)v_{t1}^k.\]
Proof. By Theorem 4.6, we have \(P_n=M_{(n,k)}V_n^k\). Then \[P_n=v_{11}+J_{t-1}(k)v_{21}+\cdots+ J_{2}(k)v_{t-11}+J_{1}(k)v_{t1}.\]
Let \(u=1\). Since \[v_{i1}=\left \{ \begin{array}{ll} 1, &\text{if}~i=1,\\ 0, &\text{if}~i\leq 3,\\ -1-k, &\text{if}~i\geq 4, \end{array} \right. \tag{7}\] we have the result. ◻
Here, we define an infinite Generalized \((k,1)\)-Jacobsthal \(1\)-matrix.
Definition 4.8. For \(k\geq 1\),the Generalized \((k,1)\)-Jacobsthal \(p\)-matrix, denoted by \(O(x)=[J_n(k)]\), is defined as follows: \[F(x)=\begin{bmatrix} 1&0&0&0&0&0&\cdots\\ k&1&0&0&0&0&\cdots\\ k^2+2&k&1&0&0&0&\cdots\\ k^3+4k&k^2+2&k&1&0&0&\cdots\\ k^4+6k^2+4& k^3+4k&k^2+2&k&1&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}=(g_{O(x)}(t),f_{O(x)}(t)).\]
The matrix \(O(x)\) is an element of the set of Riordan matrices. Since the first column of \(O(x)\) is \[(1, k, k^2+2, k^3+4k,k^4+6k^2+4,\cdots )^T.\]
Then it is obvious that \(g_{O(x)}(t)=\sum\limits_{n=0}^\infty J_n(k)t^n=\dfrac{t}{1-kt-2t^2}.\) In the matrix \(O(x)\) each entry has a rule the upper two rows, that is, \[J_n(k)=\left \{ \begin{array}{ll} 0, & n<1,\\ 1, & n=1,\\ kJ_{n-1}(k)+2J_{n-2}(k), & n>1. \end{array}\right.\]
Then \(J_{O(x)}(t)=t\), that is \[O(x)=(g_{O(x)}(t),f_{O(x)}(t))=(\dfrac{1}{1-kt-2t^2}, t),\] hence \(O(x)\) is \(R\). For these factorization, we need to define \(V_n^k(x)=(v_{ij}^k)\), as follows: \[v_{ij}^k(x) =\displaystyle{i-1\choose j-1}-k\displaystyle{i-2\choose j-1}-2\displaystyle{i-3\choose j-1}, \tag{8}\] we have the infinite matrix \(V_n^k(x)\) as follows: \[V_n^k(x)=\begin{bmatrix} 1&0&0&0&0&\cdots\\ 1-k&1&0&0&0&\cdots\\ -1-k&2-k&1&0&0&\cdots\\ -1-k&1-2k&3-k&1&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}. \tag{9}\]
Theorem 4.9. Let \(V_n^k(x)\) be the infinite matrix as (4) and \(O(x)\) be the infinite generalized \((k,1)-\)Jacobsthal \(1-\) matrix . Then \(P(x)=O(x)*V_n^k(x)\), where \(P\) is the Pascal matrix.
Proof. From the definitions of the infinite Pascal matrix and the infinite the generalized \((k,1)-\)Jacobsthal \(1-\) matrix we have the following Riordan representing \[P(x)=(\dfrac{1}{1-t},\dfrac{t}{1-t}), ~~O(x)=(\dfrac{1}{1-kt-2t^2}, t).\]
Now we can find the Riordan representation of infinite matrix \[V_n^k(x)=(g_{V_n^k(x)}(t), f_{V_n^k(x)}(t)),\] as follows: \[V_n^k(x)=\begin{bmatrix} 1&0&0&0&0&\cdots\\ 1-k&1&0&0&0&\cdots\\ -1-k&2-k&1&0&0&\cdots\\ -1-k&1-2k&3-k&1&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}.\]
From the first column of the matrix \(V_n^k(x)\) we obtain \(V_n^k(x)=O(x)^{-1}*P(x)\) and \[O(x)^{-1}=(g_{O(x)}(t),f_{O(x)}(t))^{-1}=(1-kt-2t^{ 2}, t),\] we have \[V_n^k(x)=(\dfrac{1-kt-2t^{2}}{1-t},\dfrac{t}{1-t}),\] which completes the proof. ◻
Now we define the \(n \times n\) matrix \(B(x)=(b_{ij}(x))\) as follows \[b_{ij}(x)=\displaystyle{i-1\choose j-1}-k\displaystyle{i-1\choose j}-2\displaystyle{i-1\choose j+1},\] we have the infinite matrix \(B(x)\) as follows \[\label{eq4.5} B(x)=\begin{bmatrix} 1&0&0&0&\cdots\\ 1-k&1&0&0&\cdots\\ -2k-1&1-k&1&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots \end{bmatrix}. \tag{10}\]
Lemma 4.10. Let \(B(x)\) be the matrix in (10). Then we have \(P(x)=B(x)*F(x)\).
Proof. The proof is similar to that of Theorem 4.3. ◻
Our approach was to introduce generalized \((k,t)\)-Jacobsthal \(p\)-sequences. The combinatorial identities we obtained were also obtained. Then, using generalized \((k,t)\)-Jacobsthal \(p\)-matrices, we factorized the Pascal matrix. Finally, by using the Riordan method, we got two factorizations of the Pascal matrix involving generalized \((k,t)\)-Jacobsthal \(p\)-numbers.
The authors declare that there are no conflicts of interest regarding the publication of this paper.