### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:
Matthew Russell, Rutgers University, russell2 {at} math [dot] rutgers [dot] edu)
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Let $n$ be any nonnegative integer. Let $V=P_n$ be the vector space of polynomials of degree at most $n$, equipped with the inner product $\langle f, g \rangle = \int_0^1 f(x)g(x)dx$. Let $D : V \longrightarrow V$ be the differentiation operator, $D(f) = f^\prime$. Then $D$ has an adjoint $D^*$. We have closed form expressions for $D^*$, which were conjectured by computing $D^*$ for small values of $n$ and finding a pattern. (If $f(x)$ is a polynomial of degree $k \leq n$, then, while the value of $D(f(x))$ is independent of $n$, the value of $D^*(f(x))$ depends on $n$.) We also find formulas for $D^*$ in terms of classical Legendre polynomials, shifted to the interval $[0,1]$. Using these formulas it is easy to prove that our closed form expressions are correct.