Let z 1, z 2,…, z n denote the zeros of the complex polynomial V n ( z). Put σ k=z k 1+z k 2+…+z k n,kϵ N;σ k is called the kth power sum of zeros (psz) of V n ( z). Laguerre (in 1880 and 1887) posed and solved the following problem. Construct a polynomial V n ( z) given its odd psz σ 1, σ 3,…, σ 2 n−1 . To ensure a unique solution he required V n (− z)/ V n ( z) to be irreducible. The solution is the ( n, n)-Padé denominator to exp{2(σ 1 z −1 + σ 3 z −3 + ⋯ + σ 2 n−1 z −2 n+1 )}. The special case where σ 1 = − 1 2 , σ 3 = σ 5 = ⋯ = σ 2 n−1 = 0 yields the ordinary Bessel polynomial of degree n. With exception of their orthogonality, Laguerre gave the first treatment of the orthogonal polynomial system (OPS), called Bessel polynomials, much earlier than Bochner (1929) and Burchnall and Chaundy (1931) did. In the present paper we generate OPS by prescribing the odd psz: an infinite sequence σ 1, σ 3,…,σ 2 n−1 ,…. Results on the differential equations and zeros are given. Finally, a generalization of ordinary Bessel polynomials is given, such that the defining property of being an OPS with prescribed psz is retained.