as stated above, I want to check wether $\{(1-t)^{\lambda}(1+t)^{2n-1-\lambda}, \lambda=0,1,...,2n-1\}$ forms a basis in $P_{2n-1}$, where $P_{2n-1}$ is the vector space of polynomials of degree less than or equal $2n-1$.
It has been used without any further comment in a paper of Walter Gautschi on Gaussian quadratur formulae, https://www.cs.purdue.edu/homes/wxg/selected_works/section_07/128.pdf, third page between (2.5) and (2.6).
I thought about showing the linear independence of these vectors, i.e. showing that $$\sum_{\lambda=0}^{2n-1} a_{\lambda}(1-t)^{\lambda}(1+t)^{2n-1-\lambda}=0 $$ implies that $a_{\lambda}=0$, $\lambda=0,1,\dots,2n-1$.
Didn't seem like a big problem, but induction for $n \in \mathbb{N}$ didn't work for me. Neither did I manage to get anything done with the binomial theorem.
I'd be grateful for any kind of help.
Hint: The set $\{1,t,\dots,t^{2n-1}\}$ is a basis of $P_{2n-1}$, and the map $\Phi:P_{2n-1} \to P_{2n-1}$ defined by $$ \Phi(f(t)) = (1-t)^{2n - 1} f\left(\frac{1+t}{1-t} \right) $$ is linear and invertible.