We consider the sequence of polynomials $P_{s,n} (x)$ having the following property: in the interval [-1, 1] the polynomial $P_{s,n}^{2s+1}$ is orthogonal to polynomials $P_{s,k}$ with k < n. Ossicini has already demonstrated the existence of such polynomials. Other properties of the $P_{s,n} (x)$ are given here. Then a series expansion of the type $$\sum_{n=0}^{\infty} c_{n} \, P_{s,n}^{2s+1}(x)$$ is associated to every $f(x)$, with certain values of the coefficients $c_{n}$ that are determined by the property of orthogonality mentioned above. The Bessel inequality and the Fischer-Riesz theorem are extended to such expansions.