SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v16i12.2213Research ArticleSome Identities on Sums of Finite Products of the Pell,Fibonacci, and Chebyshev PolynomialsKishoreJugaljkish11111@gmail.com1VermaVipin2Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional UniversityPhagwara , Punjab, 144411IndiaSVKM's Narsee Monjee Institute of Management Studies (NMIMS) UniversityMumbai, Maharashtra , 400056283202316129411911202226220232023Abstract
Objectives: This study will introduce some new identities for sums of finite products of the Pell, Fibonacci, and Chebyshev polynomials in terms of derivatives of Pell polynomials. Similar identities for Fibonacci and Lucas numbers will be deduced. Methods: Results are obtained by using differential calculus, combinatory computations, and elementary algebraic computations. Findings: In terms of derivatives of Pell polynomials, identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of third and fourth kinds are obtained. Novelty:Existing research has identified identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of the third and fourth kinds in terms of derivatives of Fibonacci polynomials or Chebyshev polynomials; identities on sums of finite products in terms of Pell polynomials, however, have not been investigated, so identities primarily in terms of Pell polynomials are obtained.
This section will introduce the basic definitions and symbols that are going to be used throughout the paper and the central theme of the manuscript through the citation of previously deduced results by various authors on sums of products of some special polynomials.
To start with, the Fibonacci numbers (Fα) and Lucas numbers (ℒα) are respectively defined by the linear recursive relations
ℱα=ℱα-1+ℱα-2,α≥2,ℱ0=0 and ℱ1=1
and
ℒα=ℒα-1+ℒα-2,α≥2,ℒ0=2 and ℒ1=1ℱα(z)=zℱα-1(z)+ℱα-2(z),α≥2,ℱ0(z)=0 and ℱ1(z)=1
The Pell numbers (Pα) and Pell polynomials(Pαz)1 are defined recursively as
Pα=2Pα-1+Pα-2,α≥2,P0=0 and P1=1
and
Pα(z)=2Pα-1(z)+Pα-2(z),α≥2,P0(z)=0 and P1(z)=1
The Chebyshev polynomials of the first (Tαz), second (Uαz), third (Vαz), and fourth kind (Wαz)2, 3, 4 are defined recursively for α≥1, as
Tα(z)=2zTα-1(z)-Tα-2(z),T0(z)=1 and T1(z)=zUα(z)=2zUα-1(z)-Uα-2(z),U0(z)=1 and U1(z)=2zVα(z)=2zVα-1(z)-Vα-2(z),V0(z)=1 and V1(z)=2z-1Wα(z)=2zWα-1(z)-Wα-2(z),W0(z)=1 and W1(z)=2z+1
These linear recurrence sequences will in turn lead to the following general formulae 5, 6
Many authors have studied the properties of Chebyshev polynomials. For example, Zhang investigated the sums of finite products of the second kind of Chebyshev polynomials 7 and derived many identities, particularly
whereUαrz denotes the rth derivative of Uαz w.r.t z and the sum runs over all the r+1- dimensional non-negative integral coordinates d1,d2,⋯,dr+1 such thatd1+d2+⋯+dr+1=α. Similar results were observed by T. Kim, D. S. Kim, D.V. Dolgy, and J. Kwon8 for finite products of Chebyshev polynomials of the first kind and Lucas polynomials.
In 9, T. Kim, D. S. Kim, D.V. Dolgy, and D. Kim have observed the sums of finite products of Chebyshev polynomials of the third Vαzand fourth kindWαz as follows:
where all sums in eqns. (6)-(7) runs over all non-negative integers d1,d2,⋯,dr+1 such thatd1+d2+⋯+dr+1=αwith r+1j for j > r+1. Here the authors have developed the results on sums of finite products of Chebyshev polynomials of the third and fourth kind in terms of Chebyshev polynomials of the second kind and hypergeometric functions. Similar results were studied by D. Han and L. Xinging10 for sums of finite products of Chebyshev polynomials of the first and second kind, Lucas and Fibonacci polynomials in terms of Chebyshev polynomials of the first and second kind, and Lucas polynomials. A. Patra and G.K. Panda11 also developed similar identities on sums of finite products of Pell polynomials in terms of orthogonal polynomials, including Chebyshev polynomials.
According to the preceding literature, previous work has been done to develop identities on the sums of finite products of Fibonacci and Lucas numbers, Pell, Lucas, and Fibonacci polynomials, and Chebyshev polynomials of third and fourth kinds in terms of derivatives of Fibonacci polynomials, Lucas polynomials, or Chebyshev polynomials, but the identities on the sums of finite products in terms of Pell polynomials have not been investigated. So, in this paper, some more identities on sums of finite products of the Fibonacci and Lucas numbers and Pell, Fibonacci, and third and fourth kinds of Chebyshev polynomials, primarily in terms of derivatives of the Pell polynomials, are obtained.
Methodology
Results are obtained by using differential calculus, combinatory computations, and elementary algebraic computations.
Results and Discussion
In this section, the main results of the paper on the sums of finite products of the Pell polynomials, Chebyshev polynomials of third and fourth kind, Fibonacci and Lucas numbers, and the derivative of the Pell polynomials are obtained using elementary computations. These results are established along the lines of the sums of finite products in Eqns (5)-(7) and are encapsulated in the following theorems.
Lemma 1
For any non-negative integer α, the following identities hold:
i). Pα+1-32i=i-αF2α+1.
ii). Pα+132i=iαF2α+1.
iii). Pα+1-2=iα2F3α+1.
iv). Vα32=F2α+1.
v). Wα32=L2α+1.
Proof
The lemma can be easily established by taking z=-3i2,3i2,-2 , in eq. (4), z=32 in Eq.1 and Eq. 2 and using the fact Uα32=F2α+1,Uα-32=-1αF2α+1,Uα-2i=-1α2F3α+1,Uα32=F2α+2 respectively.
where all sum runs over all non-negative integers (d_1,d_2,\cdots,d_{r+1}) such that\;d_1+d_2+\cdots+d_{r+1}=\alpha\;with \left(\begin{array}{c}
r+1 \\
j
\end{array}\right) for j > r+1 and i=\sqrt{-1}.
Proof
Using Eq. 7 and proceeding as above in Theorem 5 establishes the theorem.
where sum runs over all non-negative integers (d_1,d_2,\cdots,d_{r+1}) such that\;d_1+d_2+\cdots+d_{r+1}=\alpha\;with \left(\begin{array}{c}
r+1 \\
j
\end{array}\right)for j > r+1 and i=\sqrt{-1}.
where all sum runs over all non-negative integers (d_1,d_2,\cdots,d_{r+1}) such that\;d_1+d_2+\cdots+d_{r+1}=\alpha\;with \left(\begin{array}{c}
r+1 \\
j
\end{array}\right) for j > r+1 and i=\sqrt{-1}.
Proof
In theorem 6, replacing z with z=-\frac32i and proceeding as in theorem 7 yields the desired result.
Corollary 1
For any non-negative integer \alpha, the following identities hold:
Takingr=2,3 in theorem 6, and proceeding as in corollary 5, establishes this corollary.
Conclusion
In this paper, the sums of the finite products of the Fibonacci and Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of the third and fourth kind were considered as a linear combination of the derivatives of the Pell polynomials. Here, various lemmas using Pell polynomials are developed, and certain relations involving the Fibonacci and Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of the third and fourth kind are deduced and utilised to achieve the objectives. These results will certainly serve as a possibility for the prospective researchers to express such sums of finite products in terms of other orthogonal polynomials. Moreover, there is a possibility of studying sums of finite products of other orthogonal polynomials as well. The control theory is an intriguing area in which these findings are useful.
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