The present paper is a study of modied Hermitepolynomials Hn(x; a) which reduces to Hermite polynomialsHn(x) for a = e.

Two problems related to orthogonal polynomials and special functions are considered. For q greater than 1 it is known that continuous q-Jacobi polynomials are orthogonal on the imaginary axis. The first problem is to find proper normalization to form a system of polynomials that are orthogonal on the real line. By introducing a degree reducing operator and a scalar product one can show that the normalized continuous q-Jacobi polynomials satisfies an eigenvalue equation. This implies orthogonality of the normalized continuous q-Jacobi polynomials. As a byproduct, different results related to the normalized system of polynomials, such as its closed form,three-term recurrence relation, eigenvalue equation, Rodrigues formula and generating function will be computed. A discriminant related to the normalized system is also obtained. The second problem is related to recent results of Dilcher and Stolarky on resultants of Chebyshev polynomials. They used algebraic methods to evaluate the resultant of two combinations of Chebyshev polynomials of the second kind. This work provides an alternative method of computing the same resultant and also enables one to compute resultants of more general combinations of Chebyshev polynomials of the second kind. Resultants related to combinations of Chebyshev polynomials of the first kind are also considered.

The present paper is a study of modied Hermite polynomials of two variables Hn(x; y; a) which for a = e reduces to Hermite polynomials of two variables Hn(x; y) due to M.A. Khan and G.S. Abukhammash. / El presente artculo se estudian polinomios modicados de Hermite de dos variables Hn(x; y; a) que para a = e se reducen a los polinomios de Hermite de dos variables Hn(x; y) introducidos por M.A. Khan y G.S.Abukhammash.