Journal of Approximation Theory 162 (2010) 2243–2277 www.elsevier.com/locate/jat On orthogonal polynomials obtained via polynomial mappings M.N. de Jesusa, J. Petronilhob,∗ a Departamento de Matematica,´ Escola Superior de Tecnologia, Campus Politecnico´ de Repeses, 3504-510 Viseu, Portugal b CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal Received 1 February 2010; received in revised form 12 July 2010; accepted 22 July 2010 Available online 29 July 2010 Communicated by Francisco Marcellan Abstract Let .pn/n be a given monic orthogonal polynomial sequence (OPS) and k a fixed positive integer number such that k ≥ 2. We discuss conditions under which this OPS originates from a polynomial mapping in the following sense: to find another monic OPS .qn/n and two polynomials πk and θm, with degrees k and m (resp.), with 0 ≤ m ≤ k − 1, such that pnkCm.x/ D θm.x/qn(πk.x// .n D 0; 1; 2; : : :/: In this work we establish algebraic conditions for the existence of a polynomial mapping in the above sense. Under such conditions, when .pn/n is orthogonal in the positive-definite sense, we consider the corresponding inverse problem, giving explicitly the orthogonality measure for the given OPS .pn/n in terms of the orthogonality measure for the OPS .qn/n. Some applications and examples are presented, recovering several known results in a unified way. ⃝c 2010 Elsevier Inc. All rights reserved. Keywords: Orthogonal polynomials; Polynomial mappings; Recurrence relations; Stieltjes transforms; Inverse problems; Jacobi operators 1. Introduction and preliminaries Let .pn/n be a given monic orthogonal polynomial sequence (OPS) and k a fixed positive integer number such that k ≥ 2. The purpose of this work is to analyze conditions under which ∗ Corresponding author. E-mail addresses: [email protected] (M.N. de Jesus), [email protected] (J. Petronilho). 0021-9045/$ - see front matter ⃝c 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jat.2010.07.012 2244 M.N. de Jesus, J. Petronilho / Journal of Approximation Theory 162 (2010) 2243–2277 this OPS originates from a polynomial mapping in the following sense: to find another monic OPS .qn/n and two polynomials πk and θm, with degrees k and m (resp.), with 0 ≤ m ≤ k − 1, such that pnkCm.x/ D θm.x/qn(πk.x//; n D 0; 1; 2;:::: (1.1) The study of the theoretical aspects as well as applications of the OPS’s .pn/n and .qn/n which can be related by a polynomial transformation as in (1.1) has been a subject of considerable activity in the past few decades, especially due to the interesting applications that we can find in several domains (e.g., in physics, chemistry, operator theory, potential theory, and matrix theory). Far from giving an exhaustive list, we mention here specially the works by Bessis and Moussa [8], Geronimo and Van Assche [15,14], Charris and Ismail [9], Charris et al. [10], Peherstorfer [27,26], and Totik [34]. From the algebraic point of view, the problem was first considered for .k; m/ D .3; 0/ by Barrucand and Dickinson [7], and also for the cases k D 2 and k D 3 by Marcellan´ and Petronilho [21–23]. We notice that most of the works in the literature deal with situations where m D 0 or m D k − 1, i.e., the polynomial mapping is such that θm is either a constant or a polynomial of degree k − 1. From the analytical point of view, when we have orthogonality in the positive-definite sense, one of the most important properties satisfied by OPS’s fulfilling (1.1) is that .pn/n is orthogonal with respect to a positive measure whose support is contained in a union of at most k intervals, and mass points may appear in between these intervals (as expected, according to the results in the above references). Let us recall the definition of some determinants which will play a fundamental roleinthe sequel. According to the so-called Favard theorem (also known as the spectral theorem for orthogonal polynomials), any given monic OPS, .pn/n, can be characterized by a three-term recurrence relation. Fixing an integer number k ≥ 2, this recurrence relation can be given by general blocks (with k equations) of recurrence relations of the type [9,10] . j/ . j/ .x − bn /pnkC j .x/ D pnkC jC1.x/ C an pnkC j−1.x/; j D 0; 1;:::; k − 1I n D 0; 1; 2;:::; (1.2) . j/ . j/ with an 2 C n f0g and bn 2 C for all n and j, and satisfying initial conditions p−1.x/ D 0; p0.x/ D 1: (1.3) .0/ D ≤ − Without loss of generality, we will take a0 1, and polynomials p j of degree j 1 will always be defined as the zero polynomial. Next introduce determinants ∆n.i; jI·/ as in [9,10], such that 8 j i − <0 if < 2 ∆n.i; jI x/ VD 1 if j D i − 2 (1.4) : .i−1/ x − bn if j D i − 1 and, if j ≥ i ≥ 1, .i−1/ x − bn 1 0 ::: 0 0 a.i/ x − b.i/ n n 1 ::: 0 0 C C 0 a.i 1/ x − b.i 1/ ::: 0 0 I VD n n ∆n.i; j x/ : : : : : : ; (1.5) : : : :: : : − . j−1/ 0 0 0 ::: x bn 1 . j/ . j/ 0 0 0 ::: an x − bn M.N. de Jesus, J. Petronilho / Journal of Approximation Theory 162 (2010) 2243–2277 2245 for every n D 0; 1; 2;::: . Taking into account that ∆n.i; jI·/ is a polynomial whose degree may . j/ . j/ exceed k, and since in (1.2) the an ’s and bn ’s were defined only for 0 ≤ j ≤ k − 1, we adopt the convention .kC j/ VD . j/ .kC j/ VD . j/ bn bnC1; an anC1 (1.6) for all i; j D 0; 1; 2;::: and n D 0; 1; 2;::: , and so the useful equality ∆n.k C i; k C jI·/ D ∆nC1.i; jI·/ (1.7) holds for every i; j D 0; 1; 2;::: and n D 0; 1; 2;::: . The structure of the paper is as follows. In Section 2 algebraic conditions for the existence of a polynomial mapping in the sense of (1.1) are established. Under such conditions, in Section 3 the positive-definite case is analyzed, so that we show that the orthogonality measure for the given OPS .pn/n can be explicitly obtained from the one for the OPS .qn/n. In Sections 4 and 5 some examples of application of the previous results are presented, recovering several known results in an unified way. This includes a description of the so called sieved ultraspherical polynomials of the second kind introduced by Al-Salam et al. [1], as well as a classical result of Geronimus [16,17] concerning orthogonal polynomials such that the coefficients in the corresponding three-term recurrence relation are periodic sequences of period k. In both cases .qn/n is a sequence of classical Chebyshev polynomials of the second kind (suitably shifted and rescaled). In Section 6 an example in a situation corresponding to k D 5 and m D 1 is analyzed in detail. Finally, in Section 7, we discuss a result of A. Mat´ e,´ P. Nevai, and W. Van Assche concerning the support of the measure associated with an OPS with limit-periodic recurrence coefficients and the spectrum of the related self-adjoint Jacobi operator. At this point we would like to point out that the subject “OPS’s obtained via polynomial mappings” is nowadays a classical topic in the theory of OP’s, and so it is not surprising that many of the results presented here (specially in the last sections) have been discovered previously by many authors, but we believe that the unified approach presented here may be of interest. Besides, in most cases our proofs of the known results are quite different from the original ones. 2. OPS’s obtained via polynomial mappings The following proposition gives necessary and sufficient conditions for the existence ofa polynomial mapping in the sense of (1.1). Theorem 2.1. Let .pn/n be a monic OPS characterized by the general block of recurrence relations (1.2). Fix r 2 C, k 2 N and m 2 N0, with 0 ≤ m ≤ k −1. Then, there exist polynomials πk and θm of degrees k and m (resp.) and a monic OPS .qn/n such that q1.0/ D −r and pknCm.x/ D θm.x/qn(πk.x//; n D 0; 1; 2;::: (2.1) if and only if the following four conditions hold: .m/ (i) bn is independent of n for n D 0; 1; 2;::: ; (ii) ∆n.m C 2; m C k − 1I x/ is independent of n for n D 0; 1; 2;::: ; (iii) ∆0.m C2; m Ck −1I·/ is divisible by ∆0.1; m −1I·/, i.e., there exists a polynomial ηk−1−m of degree k − 1 − m such that ∆0.m C 2; m C k − 1I x/ D ∆0.1; m − 1I x)ηk−1−m.x/I 2246 M.N. de Jesus, J. Petronilho / Journal of Approximation Theory 162 (2010) 2243–2277 (iv) rn.x/ is independent of x for all n D 1; 2;::: , where VD .mC1/ C C − I − .mC1/ C C − I rn.x/ an ∆n.m 3; m k 1 x/ a0 ∆0.m 3; m k 1 x/ C .m/ C C − I − .m/ − I an ∆n−1.m 2; m k 2 x/ a0 ∆0.1; m 2 x)ηk−1−m.x/: Under such conditions, the polynomials θm and πk are explicitly given by θm.x/ D ∆0.1; m − 1I x/ ≡ pm.x/; (2.2) D I − .mC1/ C C − I C πk.x/ ∆0.1; m x)ηk−1−m.x/ a0 ∆0.m 3; m k 1 x/ r; and the monic OPS .qn/n is generated by the three-term recurrence relation qnC1.x/ D .x − rn/qn.x/ − snqn−1.x/; n D 0; 1; 2;::: (2.3) with initial conditions q−1.x/ D 0 and q0.x/ D 1, where VD VD C VD .m/ .mC1/ .mCk−1/ r0 r; rn r rn.0/; sn an an−1 ::: an−1 (2.4) for all n D 1; 2;::: .