Summary: We define a certain class of polynomials denoted by $P_{n,m,p}(x)$, and give the combinatorial meaning of the coefficients. Chebyshev polynomials are special cases of $P_{n,m,p}(x)$. We first show that $P_{n,m,p}(x)$ can be expressed in terms of $P_{n,0,p}(x)$. From this we derive that $P_{n,2,2}(x)$ can be obtained in terms of trigonometric functions, from which we obtain some of its important properties. Some questions about orthogonality are also addressed. Furthermore, it is shown that $P_{n,2,2}(x)$ fulfills the same three-term recurrence as the Chebyshev polynomials. We also obtain some other recurrences for $P_{n,m,p}(x)$ and its coefficients. Finally, we derive a formula for the coefficients of Chebyshev polynomials of the second kind.

05A10, 33C99

Chebyshev polynomials, binomial coefficients, recurrence relations