Monotonicity of zeros of Jacobi–Sobolev type orthogonal polynomials

Consider the inner product 〈 p , q 〉 = Γ ( α + β + 2 ) 2 α + β + 1 Γ ( α + 1 ) Γ ( β + 1 ) ∫ − 1 1 p ( x ) q ( x ) ( 1 − x ) α ( 1 + x ) β d x + M p ( 1 ) q ( 1 ) + N p ′ ( 1 ) q ′ ( 1 ) + M ˜ p ( − 1 ) q ( − 1 ) + N ˜ p ′ ( − 1 ) q ′ ( − 1 ) where α , β > − 1 and M , N , M ˜ , N ˜ ⩾ 0 . If μ = ( M , N , M ˜ , N ˜ ) , we denote by x n , k μ ( α , β ) , k = 1 , … , n , the zeros of the n-th polynomial P n ( α , β , μ ) ( x ) , orthogonal with respect to the above inner product. We investigate the location, interlacing properties, asymptotics and monotonicity of x n , k μ ( α , β ) with respect to the parameters M , N , M ˜ , N ˜ in two important cases, when either M ˜ = N ˜ = 0 or N = N ˜ = 0 . The results are obtained through careful analysis of the behavior and the asymptotics of the zeros of polynomials of the form p n ( x ) = h n ( x ) + c g n ( x ) as functions of c.