The authors studies in detail the family of orthogonal polynomials \(\left\{B_n(x;\omega)\right\}_{n\geq 0}\) defined by the recurrence \[ \begin{cases} B_0(x;\omega) = 1, \quad B_1(x;\omega) = x, \\ B_{n+2}(x;\omega) = xB_{n+1}(x;\omega) - \gamma_{n+1}(\omega)B_{n}(x;\omega), \quad n \geq 0, \end{cases} \] with \[ \gamma_{n+1}(\omega) = \frac{1}{4}\omega, \quad \gamma_{n+1}(\omega)= \frac{1}{4}, \quad n \geq 1, \quad \omega \in \mathbb C-\{0\}. \] These polynomials generalize the Chebyshev polynomials of the first kind and the Chebyshev polynomials of the second kind, which are obtained for \(\omega = 1\) and \(2\), respectively.
After an introductory section and preliminary results, the semi-classical character of \(\left\{B_n(x;\omega)\right\}_{n\geq 0}\) is explained; in particular it is shown that this sequence is of class two for \(\omega\) different from \(1\) and \(2\). Moreover, an integral representation of the corresponding form is given by means of its link with the second kind Chebyshev form. Finally, the authors give a structure relation and a second-order differential equation satisfied by this sequence in a very explicit way.