Studies in Sums of Finite Products of the Second, Third, and Fourth Kind Chebyshev Polynomials

Taekyun Kim,Jongkyum Kwon,Hyunseok Lee,Dae San Kim

doi:10.3390/math8020210

Abstract

In this paper, we consider three sums of finite products of Chebyshev polynomials of two different kinds, namely sums of finite products of the second and third kind Chebyshev polynomials, those of the second and fourth kind Chebyshev polynomials, and those of the third and fourth kind Chebyshev polynomials. As a generalization of the classical linearization problem, we represent each of such sums of finite products as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. These are done by explicit computations and the coefficients involve terminating hypergeometric functions 2 F 1 , 1 F 1 , 2 F 2 , and 4 F 3 .

Highlights

Introduction and PreliminariesThe classical linearization problem consists of determining the coefficients cn,m (k ) in the expansion of the product of two polynomials qn ( x ) and rm ( x ) in terms of arbitrary polynomial sequence{ pk ( x )}k≥0
In [20], sums of finite products of Chebyshev polynomials of the second kind were expressed in terms of the same orthogonal polynomials
As one motivation of the present research, we noticed that this problem can be viewed as a generalization of the classical linearization problem

Summary

Introduction and Preliminaries

The classical linearization problem consists of determining the coefficients cn,m (k ) in the expansion of the product of two polynomials qn ( x ) and rm ( x ) in terms of arbitrary polynomial sequence. As one motivation for the present research, we would like to generalize the linearization problem in (1) and consider the following three sums of finite products of Chebyshev polynomials of two different kinds: αn,r,s ( x ) =. I1 +···+ir + j1 +···+ js =n where Un ( x ), Vn ( x ), and Wn ( x ) are respectively Chebyshev polynomials of the second, third, and fourth kinds, and the sums are over all nonnegative integers, i1 , . Let us first recall that the Bernoulli polynomials are given by. In terms of generating functions, the above mentioned orthogonal polynomials are given as follows: Un ( x )tn ,.

Statements of Results

Proofs of Theorems 1 and 2

Proof of Theorem 3

Further Remarks

Conclusions