Continuous symmetrized Sobolev inner products of order N (I)

Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials {Qn} satisfies that Q2n(x)=Pn(x²), Q2n+1(x)=xRn(x²) for certain sequences of monic polynomials {Pn} and {Rn}. In this paper, we deduce the integral representation of the inner products such that {Pn} and {Rn} are the corresponding sequences of orthogonal polynomials. Moreover, we state a relation between both inner products which extends the classical result for symmetric linear functionals.