Let us define the basis of polynomials given by:
$$
\begin{array}\
P\_0=1, \\
P\_1=x, \\
P\_2=x(x-1), \\
P\_3=x(x-1)(x-2), \\
P\_4=x(x-1)(x-2)(x-3), \ldots\\
\end{array}
$$
I would like to know if this basis is orthogonal with respect to some measure. Thank you very much!



Orthogonal basis of polynomials?
If a sequence of monic polynomials is orthogonal with respect a measure, it satisfies a three-term recurrence 
\[
 p\_{n+1}(t) = (t-a\_n)p\_n(t) - b\_n p\_{n-1}(t)
\]
where $b\_n>0$. From this it follows that consecutive terms in the sequence cannot have a common zero. Your sequence fails badly on this test.


