The recurrence $f_{n+1} = (t-2)f_n - f_{n-1}$ with $f_0=1$ and $f_1=t-1$ suggests that $f_n$ can be expressed in terms of [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials) as:
\begin{split}
f_n(t) &= T_n(\tfrac{t}2-1) + \frac{t}{2} U_{n-1}(\tfrac{t}2-1) \\
&=U_n(\tfrac{t}2-1) + U_{n-1}(\tfrac{t}2-1),
\end{split}
and so their orthogonality and other properties should follow from those of Chebyshev polynomials.