Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials

We study the sequence of polynomials { S n } n ≥ 0 that are orthogonal with respect to the general discrete Sobolev-type inner product ⟨ f , g ⟩ s = ∫ f ( x ) g ( x ) d μ ( x ) + ∑ j = 1 N ∑ k = 0 d j λ j , k f ( k ) ( c j ) g ( k ) ( c j ) , where μ is a finite Borel measure whose support supp μ is an infinite set of the real line, λ j , k ≥ 0 , and the mass points c i , i = 1 , … , N are real values outside the interior of the convex hull of supp μ ( c i ∈ R \ C h ( supp ( μ ) ) ∘ ) . Under some restriction of order in the discrete part of ⟨ · , · ⟩ s , we prove that S n has at least n − d * zeros on C h ( supp μ ) ∘ , being d * the number of terms in the discrete part of ⟨ · , · ⟩ s . Finally, we obtain the outer relative asymptotic for { S n } in the case that the measure μ is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of ⟨ · , · ⟩ s .