ON THE KRALL-TYPE POLYNOMIALS

where (Ai)i=1 and (Bj) N j=1 are real numbers, δ(x− y) and δ′(x− y) mean the Dirac linear functional and its derivative, respectively, defined by 〈δ(x− y), p(x)〉 = p(y), 〈δ′(x− y), p(x)〉 = −p′(y), for all p ∈ P. If at most one Bi is nonzero and Ai = 0, i = 1, . . . ,m, then the analysis of the quasi-definiteness of the linear functional ũ is presented in [9]. Special important examples appear when u is a classical linear functional and the mass points are located at the ends of the interval of orthogonality. We have studied the corresponding sequences of orthogonal polynomials in this case in several papers: for the Laguerre linear functional see [2, 3], for the Bessel linear functional see [7], for the Jacobi linear functional see [8], and for the Hermite case see [1]. In particular


Introduction
Let u : P → R be a quasidefinite linear functional in the vector space P of polynomials with complex coefficients, then there exists a sequence of monic polynomials (P n ) n with degP n = n, such that [12] u, P n P m = k n δ n,m , k n = 0. ( Special cases of quasidefinite linear functionals are the classical ones (those of Jacobi, Laguerre, Hermite, and Bessel).
In recent years perturbations of a quasidefinite linear functional u via the addition of Dirac delta functionals and their derivatives have been extensively studied (see, e.g., [1,5,14,15,16,17,18,19,23] and the references therein), that is, the linear functional where (A i ) M i=1 and (B j ) N j=1 are real numbers, δ(x − y) and δ (x − y) mean the Dirac linear functional and its derivative, respectively, defined by δ(x − y), p(x) = p(y), δ (x − y), p(x) = −p (y), for all p ∈ P. If at most one B i is nonzero and A i = 0, i = 1,...,m, then the analysis of the quasi-definiteness of the linear functional u is presented in [9].
Special important examples appear when u is a classical linear functional and the mass points are located at the ends of the interval of orthogonality.We have studied the corresponding sequences of orthogonal polynomials in this case in several papers: for the Laguerre linear functional see [2,3], for the Bessel linear functional see [7], for the Jacobi linear functional see [8], and for the Hermite case see [1].In particular the quasi-definiteness of u, relative asymptotics, the representation as a hypergeometric function, and the location of their zeros have been obtained.This kind of perturbations can be seen as the simplest (lower-order) discretecontinuous version of the "bispectral" property.The continuous "bispectral problem" consists of describing all Schrödinger-type differential operators of second order such that their eigenfunctions should satisfy a differential equation (of arbitrary finite order) in the spectral variable.The basic tools are based in the Darboux factorization method [13].In the discrete-continuous case, this factorization is related to the LU and UL factorization of the Jacobi matrix associated with the sequence of orthogonal polynomials which are the eigenfunctions of a second-order linear differential operator of hypergeometric type.Thus the so-called Krall polynomials appear in a natural way [15,16,17,18,19].
Very recently, in [11], the study of sequences of orthogonal polynomials and their associated Hankel determinants when we add an extra term to a weight function is considered using the Heine formula.In particular, the authors analyze separately the addition of Aδ(x − a) and Aδ (x − a).Notice that they do not discuss the quasidefinite character of the linear functional u given by either This is a very important gap, specially when derivatives of Dirac delta functionals are considered, because necessary and sufficient conditions for the existence of the corresponding sequence of orthogonal polynomials are needed.They obtain particular cases of our previous results [2,3,4,5,8] using a different approach based in the above-mentioned Heine formula.The aim of the present contribution is to cover the above gap about the quasidefiniteness when we add two extra terms to a linear functional u in such a way that the perturbed linear functional is given by (1.4) The case a = b has been considered in some previous work by the authors and coworkers [2,3,5,7].
We use an innovative and algebraic approach taking into account the orthogonality properties of the sequence of polynomials.
Section 2 deals with the analysis of necessary and sufficient conditions for the quasidefiniteness of u given by (1.4).In such a case, the coefficients of the three-term recurrence relation that the corresponding sequence of monic orthogonal polynomials satisfies is obtained.Furthermore, if we assume that the u is a semiclassical linear functional, that is, it satisfies a Pearson equation D[φu] = Ψu, with φ, ψ polynomials such that deg(ψ) ≥ 1, see [24], then the linear functional u is also semiclassical.This can be read in terms of the holonomic second-order linear differential equation that such polynomials satisfy.The explicit expression for their coefficients is given.Finally, as an application, the WKB method is used for the analysis of the location of their zeros.

General theory
For the sake of simplicity we will consider a linear functional u given by u = u + Aδ(x − a) − Bδ (x − b).Here we assume that a is different from b.This situation has been extensively considered in the literature for a = b and was found to yield a simplification of the computation in the representation formulas (see, e.g., [1]).Thus we will recover the result in [11] in a very simple way (for the general case we refer the reader to [1]).

Representation formulas.
If u is quasidefinite, then there exists a sequence of monic polynomials ( P n ) n orthogonal with respect to u, and therefore, we can consider the Fourier expansion (2.1) . ( As a consequence, (2.1) becomes where, as usual, As a convention, K (0,0) n (x, y) := K n (x, y) denotes the reproducing kernel associated with the linear functional u.It is very well known (reproducing property of the kernels) that u,K n (x, y)p(x) = p(y) for every polynomial p(x) of degree less than or equal to n. Evaluating (2.3) for x = a, we get

.5)
A similar evaluation in (2.3) for x = b gives Finally, taking derivatives in (2.3) and evaluating the resulting expression for x = b, we obtain Thus we get a system of three linear equations in the variables P n (a), P n (b), and P n (b): where . (2.9) Notice that if the matrix I + K n−1 D, where I is the identity matrix, is nonsingular, then we get the existence and uniqueness for the solution of (2.8).Therefore, (2.3) becomes (2.10) All the above lead to the following theorem.
In such a case, the sequence ( P n ) n of monic orthogonal polynomials with respect to u is given by (2.10).
Formula (2.11) constitutes the first representation formula for the polynomials ( P n ) n .
Remark 2.2.If the entries of the matrix D are nonzero, then D is a nonsingular Hermitian matrix, and where D −1 + K n−1 is a Hermitian matrix.Therefore (2.11) means that , (2.13) a i = a,b, and ε n = sign( u, P 2 n (x) ).
We now obtain another representation formula.Taking into account the Christoffel-Darboux formula as well as inserting these two expressions in (2.10), and denoting φ(x where A(x;n) and B(x;n) are polynomials of degrees independent of n, of at most 3 and 2, respectively.On the other hand, from the three-term recurrence relation that the sequence (P n ) n satisfies and taking into account (2.16), for n ≥ 1, we get (2.18) Now, from any of the above expressions a very useful representation formula follows.In fact, if u is a semiclassical linear functional [24], then there exist a polynomial ψ(x) and two polynomials M 1 (x;n), N 1 (x;n) with degree independent of n such that or, equivalently, using the three-term recurrence relation (TTRR) (2.17), we get where a, b, and q are polynomials with degree independent of n.The last formula plays a fundamental role in finding the second-order linear differential equation that the polynomials ( P n ) n satisfy, as shown in Section 2.3.

Three-term recurrence relation and lowering and raising operators.
In the following we assume that u is quasidefinite.Then, the sequence ( P n ) n of monic polynomials orthogonal with respect to u satisfies a TTRR: with the initial conditions P −1 (x) = 0, P 0 (x) = 1.
Here we will obtain the coefficients β n and γ n of the TTRR (2.22) for the polynomials P n orthogonal with respect to u, in terms of the coefficients β n and γ n of the TTRR (2.17) of the monic polynomials orthogonal with respect to u.Since then, taking into account (2.11) as well as Remark 2.2 of Theorem 2.1, for n > 1, we get as well as, for n = 1, where u 0 = u,1 is the first moment of the linear functional u.
On the other hand, (2.26) Thus, for n ≥ 1, (2.27) Finally, for n = 0, we have Notice that in order to compute the values b n we can also use (2.21).Following the ideas in [1] it is easy to show that in this case the polynomials P n admit both lowering and raising operators of the form where M 1 (x;n), N 1 (x;n), M 2 (x;n), and M 2 (x;n) are polynomials depending on n and of bounded degree independent of n, respectively.These two formulas are very useful.In particular, from them we can easily derive the second-order linear differential equation that the polynomials P n satisfy (see [1] for more details).Nevertheless, from formula (2.21), it is possible to find it by a simpler method as we will show in the next section.In particular this leads to the second-order linear differential equation that the polynomials considered in [2,3,7,8], and therefore [11], satisfy.

Second-order linear differential equation for
Here we will describe an algorithm for finding the second-order differential equations (SODE) which satisfy the polynomials P n (x) defined by (2.21), where (P n ) n is a solution of the SODE σ(x)P n (x) + τ(x)P n (x) + λ(x)P n (x) = 0. (2.30) A special case is when (P n ) n constitutes a classical or semiclassical orthogonal family so (2.30) takes place.Taking derivatives in (2.21), we get q (x;n) P n (x) + q(x;n) P n (x) = a (x;n)P n (x) + a(x;n)P n (x) + b (x;n)P n (x) + b(x;n)P n (x).

Theorem 2 . 1 . 1 Pn= 0
The linear functional u is a quasidefinite linear functional if and only if(i) the matrix I + K n−1 D is nonsingular for every n ∈ N,(ii) u, P 2 n (x) + (P n (a),P n (b),P n (b))D(I + K n−1 D) −for every n ∈ N, (iii) the norm u, P 2 n (x) is given by u, P 2 n (x) = u, P 2 n (x) + P n (a),P n (b),P n (b) way.If b n denote the coefficient of x n−1 for P n and b n the corresponding coefficient of x n−1 for P n , then we have, β n = b n − b n+1 .To obtain b n we use (2.10): b n = b n − ε n ε n−1 γ n 1/2 P n−1 (a), P n−1 (b), P n−1 (b) M n−1 P n (a), P n (b), P n (b) T .