Explicit polynomial expansions of regular real functions by means of Bernoulli polynomials and boundary values

For a function $f\in C^{2n+1}([a,b])$ an explicit polynomial interpolant in $a$ and in the even derivatives up to the order $2n-1$ at the end-points of the interval is derived. Explicit Cauchy and Peano representations and bounds for the error are given and the analysis of the remainder term allows to find sufficient conditions on $f$ so that the polynomial approximant converges to $f$. These results are applied to derive a new summation formula with application to rectangular quadrature rule. The polynomial interpolant is related to a fairly interesting boundary value problem for ODEs. We will exhibit solutions for this problem in some special situations.