Kernel polynomials from L-orthogonal polynomials

A positive measure ψ defined on [ a , b ] such that its moments μ n = ∫ a b t n d ψ ( t ) exist for n = 0 , ± 1 , ± 2 , … , is called a strong positive measure on [ a , b ] . If 0 ⩽ a < b ⩽ ∞ then the sequence of (monic) polynomials { Q n } , defined by ∫ a b t − n + s Q n ( t ) d ψ ( t ) = 0 , s = 0 , 1 , … , n − 1 , is known to exist. We refer to these polynomials as the L-orthogonal polynomials with respect to the strong positive measure ψ. The purpose of this manuscript is to consider some properties of the kernel polynomials associated with these L-orthogonal polynomials. As applications, we consider the quadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered.