2023/04/26 by Luis Verde‐Star, Verde-Star, Luis
Computer Science · Mathematics · Physics and Astronomy · #15A30 #33C45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Quantum Mechanics and Non-Hermitian Physics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2304.13248
openalex publication_date 2023/04/26 · openalex created_date 2023/04/28 · openalex updated_date 2026/07/28
For a sequence of polynomials \pk(t)\ in one real or complex variable, where pk has degree k, for k≥ 0, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients d(n,m,k), called linearization coefficients, that satisfy pn(t) pm(t)=∑k=0n+m d(n,m,k) pk(t). For any pair of polynomial sequences \uk(t)\ and \pk(t)\ we find infinite matrices whose entries are the coefficients e(n,m,k) that satisfy pn(t) pm(t)=∑k=0n+m e(n,m,k) uk(t). Such results are obtained using a matrix approach. We also obtain recurrence relations for the linearization coefficients, apply the general results to general orthogonal polynomial sequences and to particular families of orthogonal polynomials such as the Chebyshev, Hermite, and Charlier families.