Monotonicity of zeros of Laguerre–Sobolev-type orthogonal polynomials

Abstract

Denote by x n , k M , N ( α ) , k = 1 , … , n , the zeros of the Laguerre–Sobolev-type polynomials L n ( α , M , N ) ( x ) orthogonal with respect to the inner product 〈 p , q 〉 = 1 Γ ( α + 1 ) ∫ 0 ∞ p ( x ) q ( x ) x α e − x d x + M p ( 0 ) q ( 0 ) + N p ′ ( 0 ) q ′ ( 0 ) , where α > − 1 , M ⩾ 0 and N ⩾ 0 . We prove that x n , k M , N ( α ) interlace with the zeros of Laguerre orthogonal polynomials L n ( α ) ( x ) and establish monotonicity with respect to the parameters M and N of x n , k M , 0 ( α ) and x n , k 0 , N ( α ) . Moreover, we find N 0 such that x n , n M , N ( α ) < 0 for all N > N 0 , where x n , n M , N ( α ) is the smallest zero of L n ( α , M , N ) ( x ) . Further, we present monotonicity and asymptotic relations of certain functions involving x n , k M , 0 ( α ) and x n , k 0 , N ( α ) .