On Orthogonal Polynomials of Sobolev Type: Algebraic Properties and Zeros

Bases of the space of solutions of some fourth-order linear difference equations: applications in rational approximation

“…In fact, this three term recurrence relation characterizes the orthogonality with respect to a measure. This result is known in the literature as Favard theorem (see [5]) and states that if a sequence of monic polynomials {P n } ∞ n=0 satisfies a three-term recurrence relation as (3), with η n ∈ R and γ n ∈ R + , then there exists a positive Borel measure µ such that {P n } ∞ n=0 is orthogonal with respect to the inner product (1).…”

“…Another linearly independent solution, is the (not necessarily monic) polynomial p [1] n−1 (deg(p [1] n−1 ) = n − 1) that can be obtained from (4) with the initial conditions Y −1 = 1 and Y 0 = 0. In the theory of orthogonal polynomials the sequence {p [1] n } ∞ n=0 is often called sequence of first associated, numerator or second kind polynomials with respect to the sequence of monic orthogonal polynomials {P n } ∞ n=0 . The recursion (3) plays an important role in the study of analytic and algebraic properties of orthogonal polynomials.…”

“…Let us define the inner product f, g µ := I f (x)g(x)dµ(x), (1) and the corresponding norm f µ = f, f µ . Obviously, (1) satisfies the identity xf (x), g(x) µ = f (x), xg(x) µ ,…”

Inner products involving differences: the meixner—sobolev polynomials

“…From the equality SpJ(x)=h(x)p(x) we obtain a system of (k+ 1) linear equations with (k + 1) unknowns. It has a unique solution which can be obtained using the forward substitution method.…”

“…the leading coefficient of Pn(x) is one. In these conditions, the sequence {Pn{x)}n is cailed a Monic Orthogonal Polynomial Sequence (MOPS) with respect to the inner product (1). Such a MOPS {Pn(x)}n satisfies a three-term recurrence relation Po(X) = 1, P-I(x) = 0, en =f.…”