Abstract
Moore–Penrose inverse emerges in statistics, neural networks, machine learning, applied physics, numerical analysis, tensor computations, solving systems of linear equations and in many other disciplines. Especially after the 2000s, the topic of Moore–Penrose inverse has started to attract great attention by researchers and has become a popular subject. In this paper, we investigate the Moore–Penrose inverse of the conditional matrices via convolution product formula. In order to use convolution formula effectively, we derive some useful identities by using some properties of the generalized conditional sequence. Moreover, we express the Moore–Penrose inverse of the conditional matrices in the form of block matrices. Finally, we not only present more general results compared to earlier works, but also provide many novel results using analytical techniques.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The generalized inverse of an integral operator was introduced by Fredholm [1]. The generalized inverse of a matrix was described by Moore who proposed a unique generalized inverse by means of projectors of matrices [2]. Until mid-1950s, there was few research on this subject. Later on, the use of matrix inverses in the solution of the linear equation systems began to increase the interest in this topic. In particular, R. Penrose proposed a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations in 1955 [3]. This work inspired new studies on generalized inverses and began to attract the attention of many researchers. This inverse is called the Moore–Penrose inverse in honor of the works of E. H. Moore and R. Penrose.
Let \(\mathscr {C}^{m \times n}\) be the set of \(m \times n\) complex matrices. For every \(A \in \mathscr {C}^{m \times n}\), the Moore–Penrose inverse of matrix A is the unique \(n \times m\) matrix \(A^{\dagger }\) with the following properties:
where \(A^{*}\) denotes the conjugate transpose of A.
The Moore–Penrose inverse appears in many fields such as applied mathematics, statistics, neural networks, machine learning, applied physics, control system analysis, curve fitting, digital image restoration, numerical analysis, tensor computations and the solution of system of equations (see [4, 5]). Up until now, there have been several studies in different areas related to the Moore–Penrose inverse and its applications (see [1, 6,7,8,9,10,11,12,13,14]). For example, Courrieu developed an algorithm based on a full rank Cholesky factorization which allows for fast computation of Moore–Penrose inverse of the matrices and fast solving of large least square systems, possibly with rank deficient matrices [8]. Baksalary et al. examined some problems with the Moore–Penrose inverse of the sum of two matrices, by combining various facts known in the literature and using some properties of matrix inverses [7]. Sun et al. introduced the Moore–Penrose inverse of tensors with the Einstein product, and the authors found explicit formulas of the Moore–Penrose inverse of some block tensors [13]. Ma et al. examined the perturbation theory for the Moore–Penrose inverse of tensor via Einstein product by using derived representations of some tensor expressions involving the Moore-Penrose inverse [14]. Radičić studied the Moore–Penrose inverse and the group inverse of the k-circulant matrices whose elements are the binomial coefficients [11]. Zhang et al. proposed the Zhang neural networks (ZNN) models for online time-varying full-rank matrix Moore–Penrose inversion [12]. The authors presented the feasibility and effectiveness of ZNN models for online time-varying full-rank matrix Moore–Penrose inversion with the help of computer simulation results and application to inverse kinematic control of redundant robot arms.
In recent years, there has been a huge interest of modern science in the application of the Golden Section and Fibonacci numbers. The Fibonacci numbers, \(\left\{ F_n \right\} _{n=0}^{\infty }\), are the terms of the sequence \(\left\{ 0, 1, 1, 2, 3, 5, \ldots \right\} \) wherein each term is the sum of two consecutive terms, starting with the initial conditions \(F_0 = 0\) and \(F_1 = 1\). As \(n \rightarrow \infty \), the ratio between successive Fibonacci numbers is called as golden ratio, \(\tau = \frac{1 + \sqrt{5}}{2} = 1.618\ldots \), which plays an important role in arts, architecture, engineering, geometry, music, electrostatics, poetry, stock market trading and trigonometry [15]. Up until now, many researchers have studied the applications, generalizations and relations with other disciplines of the Fibonacci and related integer sequences [16,17,18,19,20,21,22,23,24,25,26,27]. For example, Falcón and Plaza proposed the k-Fibonacci numbers, \(\left\{ F_{k,n} \right\} _{n=0}^{\infty }\), by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition [16]. Yazlik and Taskara defined the generalized k-Horadam sequence, \(\left\{ H_{k,n} \right\} _{n=0}^{\infty }\), and they obtained several identities by using determinant [18]. With a different perspective, Edson and Yayenie introduced the notable generalization of the Fibonacci sequence, biperiodic Fibonacci sequence, which is generated by the recurrence relation \(q_n=a q_{n-1}+q_{n-2}\) (when n is even) or \(q_n=b q_{n-1}+q_{n-2}\) (when n is odd), where a and b are nonzero real numbers [19]. Moreover, the authors derived the extended Binet’s formula, generating function and several identities of \(\left\{ q_n\right\} _{n=0}^{\infty }\). In a similar way, Bilgici presented the biperiodic Lucas numbers which is generated by the recurrence relation \(l_n = b l_{n-1} + l_{n-2}\) (when n is even) or \(l_n = a q_{n-1} + q_{n-2}\) (when n is odd), where a and b are nonzero real numbers [21]. Moreover, he gave generating functions, the Binet formulas and some special identities of \(\left\{ l_n \right\} _{n=0}^{\infty }\). Yazlık et al. illustrated a new generalization of the Fibonacci and Lucas p-numbers, biperiodic Fibonacci and Lucas p-numbers [20]. The authors built up the tree diagrams for the biperiodic Fibonacci and Lucas p-sequences, and they derived the recurrence relations of these sequences by using these diagrams. Moreover they obtained the Binet formulas of the biperiodic Fibonacci and Lucas p-sequences by using Vandermonde matrices. Edson et al. defined a further generalization of the Fibonacci sequence, k-periodic Fibonacci sequence, which is defined using a non-linear recurrence relation that depends on k real parameters, and is an extension of the biperiodic Fibonacci sequence [24]. By analogy to the studies [19, 21, 23] Tan and Leung proposed the generalized biperiodic Horadam sequence and investigated some congruence properties of the generalized Horadam sequence [22]. Throughout this paper, we call the generalized biperiodic Horadam sequence as generalized conditional sequence.
Up to the present, several researchers have studied the Moore–Penrose inverse of some matrices whose elements are the classical special number sequences (see [10, 28,29,30]). For example, Miladinovic and Stanimirovic studied the pseudoinverse of the generalized singular Fibonacci matrix and they derived some combinatorial identities by using generalized singular Fibonacci matrices [10]. Shen and He proposed the Moore–Penrose inverse of the matrix whose nonzero entries are the classical Horadam numbers [28]. As a generalization of the studies [10, 28], Shen et al. studied the Moore–Penrose inverse of the strictly lower triangular Toeplitz matrix and they derived a convolution formula containing the Horadam numbers. Moreover, the authors derived various combinatorial identities by using convolution formula [29].
In this paper, drawing inspiration from the previous works on the Moore–Penrose inverse, we focus on the following topics in order to obtain novel results:
-
1.
Constructing a singular matrix with generalized conditional sequences,
-
2.
Establishing a new convolution formula with the help of generalized conditional sequences,
-
3.
Proving auxiliary identities in order to use the convolution formula effectively,
-
4.
Obtaining the Moore–Penrose inverse of a singular matrix explicitly by using convolution formula,
-
5.
Expressing the Moore–Penrose inverse of a singular matrix in the form of block matrices,
-
6.
Providing more general results for different values of the initial conditions and the coefficients of the generalized conditional sequence.
2 Preliminaries and main results
In this section, we introduce some definitions and preliminary facts which are used in this paper.
Definition 1
[22] For any arbitrary numbers s and t and nonzero real numbers a, b and c, the generalized conditional sequence is defined by the recurrence relation
It’s not difficult to see from the following table that the generalized conditional sequence can be reduced infinite special number sequences for the special cases of a, b, c, s and t.
a | b | c | s | t | Generalized conditional sequence |
a | b | c | 0 | 1 | Generalized conditional Fibonacci sequence |
a | b | c | 2 | b | Generalized conditional Lucas sequence |
a | b | 1 | 0 | 1 | Biperiodic Fibonacci sequence |
a | b | 1 | 2 | a | Biperiodic Lucas sequence |
a | b | 2 | 0 | 1 | Biperiodic Jacobsthal sequence |
a | b | 2 | 2 | a | Biperiodic Jacobsthal–Lucas sequence |
k | k | 1 | 0 | 1 | k-Fibonacci sequence |
k | k | 1 | 2 | k | k-Lucas sequence |
k | k | 2 | 0 | 1 | k-Jacobsthal sequence |
k | k | 2 | 2 | k | k-Jacobsthal–Lucas sequence |
2 | 2 | k | 0 | 1 | k-Pell sequence |
2 | 2 | k | 2 | 2 | k-Pell–Lucas sequence |
2 | 2 | 1 | 0 | 1 | Pell sequence |
2 | 2 | 1 | 2 | 1 | Pell–Lucas sequence |
1 | 1 | 1 | 0 | 1 | Fibonacci sequence |
1 | 1 | 1 | 2 | 1 | Lucas sequence |
1 | 1 | 2 | 0 | 1 | Jacobsthal sequence |
1 | 1 | 2 | 2 | 1 | Jacobsthal–Lucas sequence |
\(\vdots \) | \(\vdots \) | \(\vdots \) | \(\vdots \) | \(\vdots \) | \(\vdots \) |
The French mathematician Jacques–Marie Binet found an explicit formula of the Fibonacci sequence in 1843 and it was called as Binet’s formula after this discovery. The next definition explains generalized Binet formula of the sequence \(\left\{ W_{n}^{(s,t)} \right\} _{n=0}^{\infty }\).
Definition 2
[22] The Binet formula of the generalized conditional sequence is
where \(D=\frac{W_{1}^{(s,t)} - \frac{\beta }{a} W_{0}^{(s,t)}}{\alpha -\beta }\), \(E=\frac{W_{1}^{(s,t)}-\frac{\alpha }{a} W_{0}^{(s,t)}}{\alpha -\beta }\), \(\alpha \) and \(\beta \) are the zeros of the polynomial \(x^{2}-a b x-a b c\), that is, \(\alpha =\frac{a b+\sqrt{a^{2} b^{2}+4 a b c}}{2}\) and \(\beta =\frac{a b-\sqrt{a^{2} b^{2}+4 a b c}}{2}\). Moreover, \(\xi (n)=n-2\left\lfloor \frac{n}{2}\right\rfloor \) is the parity function, i.e., \(\xi (n)=0\) when n is even and \(\xi (n)=1\) when n is odd. Let assume \(\Delta =a^{2} b^{2}+4 a b c \ne 0\). Also we have \(\alpha +\beta =a b, \alpha -\beta =\sqrt{a^{2} b^{2}+4 a b c}\) and \(\alpha \beta =-a b c\).
The following definition explains a lower triangular and strictly lower triangular matrix \(\mathscr {W}_n^{(r, k)}\), whose nonzero elements are the generalized conditional sequence satisfying \(W_{k+1}^{(s,t)} \ne 0\). The non-positive integer r indicates the number of the zero diagonals including the main diagonal and below.
Definition 3
For any integers r and k satisfying \(r < 0\) and \(k \geqslant 0\), and the generalized conditional sequence \(\left\{ W_{n}^{(s,t)}\right\} _{n \in N}\) with \(W_{k+1}^{(s,t)} \ne 0\), the \(n \times n\) matrix \(\mathscr {W}_n^{(r, k)}=\left[ \omega _{i, j}^{(r, k)}\right] \) is defined by
Due to the the specific structure of the matrix \(\mathscr {W}_n^{(r, k)}\), we can express it by the following block matrix form
where \(\mathscr {O}_{p \times q}\) denotes the \(p \times q\) zero matrix.
Example 1
The \(5 \times 5\) generalized conditional matrix for \((r,k) = (-2,2)\) is equal to
The underlying idea in convolution is to combine a kernel list with consecutive sublists of a list of data. Next definition explains the convolution formula.
Definition 4
[31] For any two arrays \(x=\left\{ x_1, x_2, \dots , x_n \right\} \) and \(y=\left\{ y_1, y_2, \dots , y_n \right\} \), the convolution of x and y is defined by
The next theorem describes a convolution formula which involves the generalized conditional sequences with corresponding powers of \(\left( \frac{-c \left( \frac{b}{a}\right) ^{\frac{\xi (m-1)}{2} } W_{m}^{(s,t)} }{\left( \frac{b}{a}\right) ^{\frac{\xi (m)}{2} } W_{m+1}^{(s,t)} } \right) \), where \(m\geqslant 0\) and \(W_{m+1}^{(s, t)} \ne 0\).
Throughout this paper, for the sake of simplicity, we will use the following notation
Theorem 1
Let m, r be two integers with \(m\geqslant 0\) and \(r \geqslant 2\). If \(c\ne 0 \), \(W_{m+1}^{(s,t)} \ne 0\) and \(\alpha , \beta \ne \frac{-c \sqrt{ab} \left( \frac{b}{a}\right) ^{\frac{\xi (m-1)}{2} } W_{m}^{(s,t)} }{\left( \frac{b}{a}\right) ^{\frac{\xi (m)}{2} } W_{m+1}^{(s,t)} }\), then we have
Proof
In a clear way, Eq. (7) hold for \(W_{m}^{(s,t)} = 0\). Hence, we take into account the case \(W_{m}^{(s,t)} \ne 0\). By virtue of the Binet formula (3) and doing simple transformations, we obtain
By using identities \(\alpha \beta = -abc\) and \( \alpha + \beta = ab \), we obtain denumerator of Con(r, m) as
After some algebraic operations, the numerator of Con(r, m) can be transformed into the following form:
Taking into account the Binet formula (3), \(Num\left( Con(r,m)\right) \) and \(Denum\left( Con(r,m)\right) \), we get
By simplifying the above equation, we obtain
Therefore, the proof is completed. \(\square \)
Lemma 1
Let r be an arbitrary positive integer and \(W_{k+1}^{(s,t)} \ne 0\). If \(\alpha = \frac{-c \sqrt{ab} \left( \frac{b}{a}\right) ^{\frac{\xi (k-1)}{2} } W_{k}^{(s,t)} }{\left( \frac{b}{a}\right) ^{\frac{\xi (k)}{2} } W_{k+1}^{(s,t)} }\) or \(\beta = \frac{-c \sqrt{ab} \left( \frac{b}{a}\right) ^{\frac{\xi (k-1)}{2} } W_{k}^{(s,t)} }{\left( \frac{b}{a}\right) ^{\frac{\xi (k)}{2} } W_{k+1}^{(s,t)} }\), then we get
Proof
Clearly, equality (8) is valid for \(c=0\). So, we consider the case \(c\ne 0\). If \(\alpha = \frac{-c \sqrt{ab} \left( \frac{b}{a}\right) ^{\frac{\xi (k-1)}{2} } W_{k}^{(s,t)} }{\left( \frac{b}{a}\right) ^{\frac{\xi (k)}{2} } W_{k+1}^{(s,t)} }\), then we get
After some algebraic operations, we have
Thus, we obtain \(\left( \frac{b}{a}\right) ^{\frac{\xi (k-1)}{2} } W_{k}^{(s,t)} \left( \frac{b}{a}\right) ^{\frac{\xi (k+1)}{2} } W_{k+2}^{(s,t)} = \left( \left( \frac{b}{a}\right) ^{\frac{\xi (k)}{2} } W_{k+1}^{(s,t)} \right) ^2\). On the other hand, by virtue of the Binet formula (3), we have
Furthermore, with the help of (9), we obtain
In a similar way, we can verify that the equality (8) is valid for \(\beta = \frac{-c \sqrt{ab} \left( \frac{b}{a}\right) ^{\frac{\xi (k-1)}{2} } W_{k}^{(s,t)} }{\left( \frac{b}{a}\right) ^{\frac{\xi (k)}{2} } W_{k+1}^{(s,t)} }\). Hence, the proof is completed. \(\square \)
Lemma 2
Let \(\left\{ W_{n}^{(s,t)} \right\} _{n=0}^{\infty }\) be the generalized conditional sequence satisfying \(W_{k+1}^{(s,t)} \ne 0\). Then the inverse of the matrix \(\mathscr {W}_{n}^{(k)}\), \(\Phi _{n} = \left[ \phi _{ij} \right] _{n \times n}\), is defined by
where k is an arbitrary integer satisfying \(0 \leqslant k < n\).
Proof
Let us denote the matrices \(\mathscr {W}_{n}^{(k)} = \left[ \omega _{ij} \right] _{n \times n}\) and \(\mathscr {X}_{n} = \left[ x_{ij} \right] _{n \times n} = \mathscr {W}_{n}^{(k)} \Phi _{n} \). Due to the structure of the matrices, we can observe that \(x_{i,j} = 0\) for \(i < j\).
For \(i = j\), we obtain
\(\square \)
For \(i = j+1\), we obtain
For the last case, \(i > j\), we obtain
By taking \(r=i-j\), we can obtain \(\omega _{i, j} = \left( \frac{b}{a}\right) ^{\frac{\xi (k+r)}{2} } W_{k+r+1}^{(s,t)} \) and \(\omega _{i, j+1} = \left( \frac{b}{a}\right) ^{\frac{\xi (k+r-1)}{2} } W_{k+r}^{(s,t)} \). Therefore, we get
By virtue of Theorem 1, we obtain
Therefore, we prove that \(\mathscr {X}_{n}\) is the \(n \times n\) identity matrix. In a similar way, we can verify that \(\Phi _{n} \mathscr {W}_{n}^{(k)} = \mathscr {X}_{n}\). So, the proof is completed.
Example 2
For \(n=5\) and \(k=0\), we get
If we take \(a=b=A\), \(c=B\), \(s=a\) and \(t=b\) in Lemma 2, we obtain the inverse of the matrix \(\mathscr {U}_{n}^{(k)}\) whose elements are the classical Horadam numbers.
Corollary 1
[29] Let \(\left\{ U_n^{(a, b)}\right\} _{n \in N}\) be the Horadam sequence satisfying \(U_{k+1}^{(a, b)} \ne 0\). Then the inverse of the matrix \(\mathscr {U}_n^{(k)}\) is the matrix \(\mathscr {R}_n=\left[ r_{i, j}\right] _{n \times n}\) defined by
where k is an arbitrary integer satisfying \(0 \leqslant k<n\).
Theorem 2
For any integers r and k satisfying \(r < 0\) and \(k \geqslant 0\), and the generalized conditional sequence \(\left\{ W_{n}^{(s,t)}\right\} _{n \in N}\) with \(W_{k+1}^{(s,t)} \ne 0\), the Moore–Penrose inverse of the matrix \(\mathscr {W}_n^{(r, k)}\) is given by the following block matrix form
Proof
As the blocks are null or invertible, by virtue of the Lemma 2, the proof is obvious so we omit it. \(\square \)
Example 3
The Moore–Penrose inverse of the \(5 \times 5\) generalized conditional matrix for \((r,k) = (-2,2)\) is equal to
If we take \(a=b=A\), \(c=B\), \(s=a\) and \(t=b\) in Theorem 2, we obtain the Moore–Penrose inverse of the matrix \(\mathscr {U}_{n}^{(s,k)}\) whose elements are the classical Horadam numbers.
Corollary 2
[29] Let \(s<0, k \geqslant 0\) be arbitrary integers, and \(\left\{ U_n^{(a, b)}\right\} _{n \in N}\) be the Horadam sequence satisfying \(U_{k+1}^{(a, b)} \ne 0\). Then the Moore–Penrose inverse of the matrix \(\mathscr {U}_n^{(s, k)}\) is the \(n \times n\) block matrix \(\mathscr {Q}_n\) given by
where \(\mathscr {R}_{n+s}=\left[ r_{i, j}\right] \) is an \((n+s) \times (n+s)\) matrix given by
3 Conclusion and discussions
The topic of generalized inverses has become one of the most important and the most interesting research fields of applied and computational mathematics in recent years. One of the most important inversion method is the Moore–Penrose inverse, which has been actively studied by researchers for years [4, 5, 7, 8, 10, 11, 13, 14, 28, 29, 31]. Although there are many methods for calculating the Moore–Penrose inverse, it is commonly used for Singular Value Decomposition (SVD) when performing computations. Despite this method is robust, it cannot compute the results faster when the matrix size is large. In this paper, we obtain the Moore–Penrose inverse of a singular matrix whose elements are the generalized conditional sequence by using convolution formula. Since the results are obtained with analytical methods, it reduces the computational costs compared to other methods. Moreover, we give some important identities in order to find the Moore–Penrose inverse of the matrix \(\mathscr {W}_n^{(r, k)}\). For special values of a, b, c, s and t, our results can be reduced into the works [29, 31]. Thus we provide more general results compared to the previous studies. To sum up, the results we have presented have eliminated the difficulties in computation of the Moore–Penrose inverse of the singular matrices.
References
Fredholm, I.: Sur une classe d’équations fonctionnelles. Acta Math. 27, 365–390 (1903)
Moore, E.H.: On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26, 394–395 (1920)
Penrose, R.: A generalized inverse for matrices. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 51, pp. 406–413. Cambridge University Press (1955)
Ben-Israel, A., Greville, T.N.: Generalized Inverses: Theory and Applications. Springer, Berlin (2003)
Wang, G., Wei, Y., Qiao, S., Lin, P., Chen, Y.: Generalized Inverses: Theory and Computations. Springer, Berlin (2018)
Baksalary, O.M., Trenkler, G.: The Moore–Penrose inverse: a hundred years on a frontline of physics research. Eur. Phys. J. H 46(1), 1–10 (2021)
Maria Baksalary, O., Sivakumar, K., Trenkler, G.: On the Moore–Penrose inverse of a sum of matrices. Linear and Multilinear Algebra 71, 1–17 (2022)
Courrieu, P.: Fast computation of Moore–Penrose inverse matrices. Neural Inf. Process. Lett. Rev. 8(2) (2005)
Greville, T.: Some applications of the pseudoinverse of a matrix. SIAM Rev. 2(1), 15–22 (1960)
Miladinovic, M., Stanimirovic, P.: Singular case of generalized Fibonacci and Lucas matrices. J. Korean Math. Soc. 48(1), 33–48 (2011)
Radičić, B.: The inverse and the Moore–Penrose inverse of a \(k\)-circulant matrix with binomial coefficients. Bull. Belg. Math. Soc. Simon Stevin 27(1), 29–42 (2020)
Zhang, Y., Yang, Y., Tan, N., Cai, B.: Zhang neural network solving for time-varying full-rank matrix Moore–Penrose inverse. Computing 92(2), 97–121 (2011)
Sun, L., Zheng, B., Bu, C., Wei, Y.: Moore–Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64(4), 686–698 (2016)
Ma, H., Li, N., Stanimirović, P.S., Katsikis, V.N.: Perturbation theory for Moore–Penrose inverse of tensor via Einstein product. Comput. Appl. Math. 38(3), 1–24 (2019)
Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, Hoboken (2001)
Falcon, S., Plaza, Á.: The \(k\)-Fibonacci sequence and the Pascal \(2\)-triangle. Chaos Solitons Fractals 33(1), 38–49 (2007)
Catarino, P., Campos, H.: Incomplete \(k\)-Pell, \(k\)-Pell–Lucas and modified \(k\)-Pell numbers. Hacet. J. Math. Stat. 46(3), 361–372 (2017)
Yazlik, Y., Taskara, N.: A note on generalized \(k\)-Horadam sequence. Comput. Math. Appl. 63(1), 36–41 (2012)
Edson, M., Yayenie, O.: A new generalization of Fibonacci sequence and extended Binet’s formula. Integers 9(6), 639–654 (2009)
Yazlik, Y., Köme, C., Madhusudanan, V.: A new generalization of Fibonacci and Lucas \(p\)-numbers. J. Comput. Anal. Appl. 25(4), 657–669 (2018)
Bilgici, G.: Two generalizations of Lucas sequence. Appl. Math. Comput. 245, 526–538 (2014)
Tan, E., Leung, H.-H.: A note on congruence properties of the generalized bi-periodic Horadam sequence. Hacet. J. Math. Stat. 49, 1–10 (2019)
Yayenie, O.: A note on generalized Fibonacci sequences. Appl. Math. Comput. 217(12), 5603–5611 (2011)
Edson, M., Lewis, S., Yayenie, O.: The \(k\)-periodic Fibonacci sequence and an extended Binet’s formula. J. Integers 11(6), 739–751 (2011)
Marques, D., Trojovskỳ, P.: On characteristic polynomial of higher order generalized Jacobsthal numbers. Adv. Differ. Equ. 2019(1), 392 (2019)
Ait-Amrane, N.R., Belbachir, H.: Bi-periodic \(r\)-Fibonacci sequence and bi-periodic \(r\)-Lucas sequence of type \(s\). Hacet. J. Math. Stat. 51(3), 1–20 (2022)
Tan, E., Leung, H.-H.: Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences. Adv. Differ. Equ. 2020(1), 1–11 (2020)
Shen, S.-Q., He, J.-J.: Moore–Penrose inverse of generalized Fibonacci matrix and its applications. Int. J. Comput. Math. 93(10), 1756–1770 (2016)
Shen, S., Liu, W., Feng, L.: Inverse and Moore–Penrose inverse of Toeplitz matrices with classical Horadam numbers. Oper. Matrices 11(4), 929–939 (2017)
Köme, C.: Moore–Penrose inverse of the singular conditional matrices and its applications. Indian J. Pure Appl. Math. (2022). https://doi.org/10.1007/s13226-022-00352-4
Stanimirović, P., Miladinović, M.: Inversion of the generalized Fibonacci matrix by convolution. Int. J. Comput. Math. 88(7), 1519–1532 (2011)
Funding
Open access funding provided by the Scientific and Technological Research Council of Türkiye (TÜBİTAK).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Köme, C., Yazlik, Y. Inverse and Moore–Penrose inverse of conditional matrices via convolution. J. Appl. Math. Comput. 70, 417–433 (2024). https://doi.org/10.1007/s12190-023-01974-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-023-01974-5