NEW CONNECTION FORMULAE BETWEEN (p, q)−FIBONACCI POLYNOMIALS AND CERTAIN JACOBI POLYNOMIALS

The main purpose of this article is to solve the connection problems between (p, q)−Fibonacci polynomials and the two polynomials, namely Chebyshev polynomials of third and fourth kinds which are considered as two nonsymmetric polynomials of the Jacobi polynomials. Moreover, the inversion connection formulae for the latter formulae are given. We show that all the connection coefficients are expressed in terms of hypergeometric functions of the type 2F1 of certain arguments which can be summed in some cases. As special cases of the derived formulae, the connection formulae between Fibonacci, Pell, Fermat, second kind Chebyshev, second kind Dickson polynomials and Chebyshev polynomials of third and fourth kinds are deduced. Moreover, as applications of the introduced connection formulae, some new expressions for the celebrated numbers of Fibonacci, Pell, Mersenne and Fermat numbers and their derivatives sequences are presented. As another application of the derived connection formulae, some new definite integrals involving products of (p, q)−Fibonacci polynomials and Chebyshev polynomials of third and fourth kinds are given.