Some Formulas and Recurrences of Certain Orthogonal Polynomials Generalizing Chebyshev Polynomials of the Third-Kind

This paper investigates certain Jacobi polynomials that involve one parameter and generalize the well-known orthogonal polynomials called Chebyshev polynomials of the third-kind. Some new formulas are developed for these polynomials. We will show that some of the previous results in the literature can be considered special ones of our derived formulas. The derivatives of the moments of these polynomials are derived. Hence, two important formulas that explicitly give the derivatives and the moments of these polynomials in terms of their original ones can be deduced as special cases. Some new expressions for the derivatives of different symmetric and non-symmetric polynomials are expressed as combinations of the generalized third-kind Chebyshev polynomials. Some new linearization formulas are also given using different approaches. Some of the appearing coefficients in derivatives and linearization formulas are given in terms of different hypergeometric functions. Furthermore, in several cases, the existing hypergeometric functions can be summed using some standard formulas in the literature or through the employment of suitable symbolic algebra, in particular, Zeilberger's algorithm.