Some Müntz orthogonal systems

Let Λ = ∗λ0, λ1,…∗ be a given sequence of complex numbers such that ¦λi¦ > 1 and Re(λiλj − 1) > 0 for each i and j. For Müntz polynomials P(x) = ∑i=0npixλi and Q(x) = ∑j=0mqjxλj we define an inner product [P, Q] by [P,Q]=∫01 (PQ)(x)dxx2 where (P⊙Q)(x) = ∑i=0n∑j=0mpiqjxλiλj. We obtain the Müntz polynomials (Qn(x)) orthogonal with respect to this inner product and connect them with the Malmquist rational functions Wν(z)=∏v=0n−1(z−1λv)∏v=0n(z−λv) orthogonal on the unit circle with respect to the inner product (u,v)=12τi∮|| =1 u(z)v(z)dzzSeveral interesting properties of polynomials Qn(x) are given.