Prove a recurrence relation for Dolph-Chebyshev function

Question

The Chebyshev polynomials of the first kind can be obtained from the recurrence relation $$ T_n(x)=2xT_{n-1}(x)-T_{n-2}(x) $$ starting from $ T_0(x)=1, \ T_1(x)=x $. Now suppose $ L>0 $ is an odd integer, $0<\gamma<1$ and let $$ \varphi_n:=2\ \mathrm{arccot}\left( \tan(\pi\frac{n}{L}) \sqrt{1-\gamma^2} \right). $$ Consider the "generalized" recurrence relation $$ a_{n+1}(x)=x \left( 1-e^{i\varphi_{n}} \right)a_{n}(x)+e^{i\varphi_{n}}\ a_{n-1}(x) \quad (1\leq n \leq L-1) $$ starting from $ a_0(x)=1, \ a_1(x)=x $. How to prove the following equation? $$ a_L(x)=\frac{T_L(\frac{x}{\gamma})}{T_L(\frac{1}{\gamma})} $$ The equation is stated without proof in Fixed-point quantum search with an optimal number of queries.