The convolved (u, v)-Lucas first kind p-polynomials are defined using the generating function of the (u, v)-Lucas first kind p-polynomials. The determinantal and permanental representations of the convolved (u, v)-Lucas first kind p-polynomials are used to derive some identities of these polynomials.
Citation: Adikanda Behera, Prasanta Kumar Ray. Determinantal and permanental representations of convolved (u, v)-Lucas first kind p-polynomials[J]. AIMS Mathematics, 2020, 5(3): 1843-1855. doi: 10.3934/math.2020123
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Abstract
The convolved (u, v)-Lucas first kind p-polynomials are defined using the generating function of the (u, v)-Lucas first kind p-polynomials. The determinantal and permanental representations of the convolved (u, v)-Lucas first kind p-polynomials are used to derive some identities of these polynomials.
1.
Introduction
The integer sequence (un)n≥0 is said to be a Lucas sequence of first kind if there exist non zero integers A and B such that un+2=Aun+1+Bun,n≥0 with initials u0=0 and u1=1. Since the last few decades, researchers keep a constant interest in this sequence and have been placed their results to many modern sciences. Lucas sequence of the first kind comprises many sequences, like Fibonacci numbers, Pell numbers, balancing numbers, Jacobsthal numbers etc. that always make a constant attraction to the recent researchers. In one of the communicated papers of the authors, the Lucas first kind p-numbers Lp(j) is defined by the recurrence relation
Lp(j)=aLp(j−1)+bLp(j−p−1),j≥(p+1)
(1.1)
with initials Lp(j)=aj−1, for j=1,2,…,p and Lp(0)=0, where p is taken as non-negative integer and the coefficients a and b are non zero integers. For p=1, (1.1) reduces to the recurrence relation of the Lucas first kind numbers.
A new generalization of the Fibonacci sequence based on its generating function is the convolved Fibonacci numbers F(r)j that have been studied in several manner (for e.g. [3,5,7]) and are defined by
(1−t−t2)−r=∞∑j=0F(r)j+1tj,r∈Z+.
Nowadays it is the most challenging task for the authors to investigate several properties of number and polynomial sequences in a matrix way. Determinantal and permanental representations of numbers, polynomials and functions play a crucial role in many areas of mathematics. Şahin and Ramírez [6] introduced the convolved generalized Lucas polynomials F(r)p,q,j(x) and are defined by
where p(x) and q(x) are polynomials coefficient. They derived several identities using the matrix representations of F(r)p,q,j(x) with real and imaginary entries.
In this article, we generalize Şahin and Ramírez paper by introducing convolved (u,v)-Lucas first kind p-polynomials. Based on determinantal and permanental representations, some similar type identities of [6] are studied for these polynomials using different proof methods.
2.
Convolved (u,v)-Lucas first kind p-polynomials
In this section the (u,v)-Lucas first kind p-polynomials and convolved (u,v)-Lucas first kind p-polynomials are defined. Using some results of convolved (u,v)-Lucas first kind p-polynomials the recurrence relation of these polynomials is also established.
Definition 2.1.Let p be any non negative integer and u(x) and v(x) are polynomials with real coefficients. The (u,v)-Lucas first kind p-polynomials {Lpu,v,j(x)}j∈N are defined by the recurrence relation
Lpu,v,j(x)=u(x)Lpu,v,j−1(x)+v(x)Lpu,v,j−p−1(x)
(2.1)
with initialsLpu,v,0(x)=0 and Lpu,v,j(x)=uj−1(x) for j=1,…,p.
It is noticed that, when we consider u(x)=ax and v(x)=b, equation (2.1) reduced to Lucas first kind p-polynomials {Lp,j(x)} with initial values Lp,0(x)=0 and Lp,j(x)=(ax)j−1 for j=1,2,…,p.
If gpu,v(t) is the generating function of Lpu,v,j+1(x), then it can be easily seen that
gpu,v(t)=∞∑j=0Lpu,v,j+1(x)tj=11−u(x)t−v(x)tp+1.
By virtue of the gnerating function gpu,v(t), the convolved (u,v)-Lucas first kind p-polynomials can be defined as follows.
Definition 2.2.The convolved (u,v)-Lucas first kind p-polynomials {L(p,r)u,v,j(x)}j∈N for p≥1 are defined by
Using (2.3), we yield convolved (u,v)-Lucas first kind p-polynomials L(p,r)u,v,j(x) for j=0,1,2,3,4,5, and 6 with different (p,r) values, which are listed in both Table 1 and Table 2.
Table 1.
Convolved (u,v)-Lucas first kind p-polynomials.
Now we are in a position to find the recurrence relation of the convolved (u,v)-Lucas first kind p-polynomials.
Theorem 2.4.The recurrence relation of the convolved (u,v)-Lucas first kind p-polynomials {L(p,r)u,v,j(x)}j∈N obey the second order recurrence relation
3.
Determinantal representations of convolved (u,v)-Lucas first kind p-polynomials
In this section we consider various Hessenberg matrices with some adjustable real or imaginary entries. Based upon these matrices we establish some results involving determinantal representations of convolved (u,v)-Lucas first kind p-polynomials.
The following result is useful while proving the subsequent theorems.
Lemma 3.1.[1] Let Aj=(ail)j×j with 1≤i,l≤j be the lower Hessenberg matrix for all j≥1 and define det(A0)=1. Then, det(A1)=a11 and for j≥2
Proof. Using induction on j, the result is clearly holds for j=1 by (2.4). Assume that the result is true for all positive integers less than or equal to j−1, i.e. det(F(p,r)u,v,j)=L(p,r)u,v,j+1(x). By virtue of Lemma 3.1 and the relation (2.4), we have
4.
Permanental representations of convolved (u,v)-Lucas first kind p-polynomials
In this section we consider various Hessenberg matrices and upon these matrices we establish some results involving permanental representations of convolved (u,v)-Lucas first kind p-polynomials. Moreover, we consider some non-singular matrices and establish the first column of inverse of these matrices is written in convolved (u,v)-Lucas first kind p-polynomials.
The following result is useful while proving the subsequent theorems.
Lemma 4.1.[4] Let Aj=(ail)j×j with 1≤i,l≤j be the lower Hessenberg matrix for all j≥1, and define per(A0)=1. Then per(A1)=a11, and for j≥2,
Proof. By the induction on j the result is true for j=1. Let us consider the result is true for all positive integers less than or equal to j−1, i.e. per(G(p,r)u,v,j)=L(p,r)u,v,j+1(x). Then by using Lemma 4.1, we have
At the end of this section, we present two important results concerning convolved (u,v)-Lucas first kind p-polynomials. We omit the proofs of these results because they are similar to the methods which are adopted in Theorem 9 of [6].
Theorem 4.6.Let ˜F(p,r)u,v,j+1 be the (j+1)−by−(j+1) non singular matrix given by
˜F(p,r)u,v,j+1=[100…00⋮F(p,r)u,v,j001],
where F(p,r)u,v,j is the Hessenberg matrix of order j defined in Theorem 3.2. Then the first column of (˜F(p,r)u,v,j+1)−1 is
The authors declare there are no conflicts of interest in this paper.
References
[1]
N. D. Cahill, J. R. D'Errico, D. A. Narayan, Fibonacci determinants, College Math. J., 33 (2002), 221-225. doi: 10.1080/07468342.2002.11921945
[2]
Y. H. Chen, C. Y. Yu, A new algorithm for computing the inverse and the determinant of aHessenberg matrix, Appl. Math. Comput., 218 (2011), 4433-4436.
[3]
P. Moree, Convoluted convolved Fibonacci numbers, J. Integer Seq., 7 (2004), 1-14.
[4]
A. A. Öcal, N. Tuglu, E. Altinişik, On the representation of k-generalized Fibonacci and Lucasnumbers, Appl. Math. Comput., 170 (2005), 584-596.
[5]
J. L. Ramírez, Some properties of convolved k-Fibonacci numbers, ISRN Combinatorics., 2013 (2013), 1-6.
[6]
A. Şahin, J. L. Ramírez, Determinantal and permanental representations of convolved Lucaspolynomials, Appl. Math. Comput., 281 (2016), 314-322.
[7]
X. Ye, Z. Zhang, A common generalization of convolved generalized Fibonacci and Lucaspolynomials and its applications, Appl. Math. Comput., 306 (2017), 31-37.