View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathematics 99 (1998) 423-447 Multiple orthogonal polynomials A.I. Aptekarev * The Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, PI. Miusskaya 4, 125047 Moscow, Russia Received 1 December 1997; received in revised form 20 June 1998 Abstract Results on multiple orthogonal polynomials will be surveyed. Multiple orthogonal polynomials are intimately related to Hermite-Pad6 approximants and often they are also called Hermite-Pad6 polynomials. Special attention will be paid to an application of multiple orthogonal polynomials and to analytic theory of two model families of general multiple orthogonal polynomials, referred to as Angelesco and Nikishin systems. Among the applications the number theory, special functions and spectral analysis of nonsymmetric band operators will be highlighted. In the analytic theory results and methods for the study of multiple orthogonal polynomials asymptotics will be reviewed. New results on strong asymptotics of multiple orthogonal polynomials for Nikishin system will be presented. @ 1998 Elsevier Science B.V. All rights reserved. Keywords." Multiple orthogonality; Hermite-Pad6 approximants; asymptotics of orthogonal polynomials 1. Introduction During the past 15 years there was a stable interest in the study of multiple orthogonal polynomials. There are several survey papers about the topic (see [16, 22, 46, 57]). Here we review the progress of the subject for the past five years, particularly in its application to various fields of mathematics and in the study of their analytic, asymptotic properties. In the introduction below we present the basic definitions, show a connection with Hermite- Pad6 approximants, introduce examples of general multiple orthogonal polynomials such as Angele- sco and Nikishin systems and consider polynomials orthogonal with respect to a varying weight, which play the key role in the study of the analytic properties of general multiple orthogonal polynomials. * E-mail: [email protected]. 0377-0427/98/S-see front matter (~) 1998 Elsevier Science B.V. All rights reserved. PII: S0377-0427(98)00175-7 424 A.I. Aptekarev / Journal of Computational and Applied Mathematics 99 (1998) 423~447 1.1. Sequences of multiple orthoyonal polynomials Definition 1.1. A polynomial Q,(x) is called multiple orthoyonal polynomial of a vector index n=(nl ..... np) E Np (1.1) with respect to a vector of positive Borel measures, supported on the real axis /~=(#t .... ,#p), supp/~E~, ~=l,...,p (1.2) if it satisfies the following conditions: p (*) deg Q, ~< In I := ~-~ n~, (1.3) 1 Q,: f (**) JQ.(x)x'd~(x)=O, v:O,...,n~- 1, ~=l,...,p. When p = 1 the multiple orthogonal polynomial becomes the standard orthogonal polynomial Qn(x): degQ,=n and fQ°(x)x'du(x)=O,v=0,1,...,n- 1. (1.3') The notion of multiple orthogonal polynomial can be generalized if we consider the non-Hermitian complex orthogonality with respect to a complex valued vector function f(z) = (fl(z) ..... L(z)) = (~-~f',izj+, .... , ~) (1.4) j=0 i=0 on some contours F~ of the complex plane C. Thus Definition 1.1 becomes Definition 1.1'. A polynomial Q, is called multiple orthogonal polynomial of index (1.1) with respect to a complex weight (1.4) if it satisfies (,) in (1.3) and ~rQn(z)zVZ(z)d(z)=O, v=O,...,n~- 1, ~=1 p. (1.5) Usually multiple orthogonal polynomials are considered with the following sequences of indexes (a) diagonal: nEID <=~ n=(n ..... n), nE{~, Inl=P'n. (1.6) (b) Step-line: nEIsL ¢=~ n=(m÷ 1,...,m + 1,m ..... m), mEN, [nl :=hE ~. (1.7) Multiple orthogonal polynomials along the step-line are also called vector orthogonal polynomials (see [60]). A.I. Aptekarev/Journal of Computational and Applied Mathematics 99 (1998) 423~147 425 1.2. Hermite-PadO (simultaneous) rational approximants From Definition 1.1' it follows that if the components of the vector function (1.4) are analytic functions in a neighbourhood of infinity containing contours F~, then there exists a set of polynomials P,~), e = 1,..., p such that degP.~')~< Inl - 1, Q,(z)f~(z) - P2:)(z) = 0 (1) (1.8) and therefore f~(z) P.~)(Z)-o( 1 ) O.(z) 1 ..... p. (1.8') Definition 1.2. A set of rational function (1.9) satisfying (,) in (1.3) and (1.8) is called Hermite-PadO approximants of type II or simultaneous rational approximants of index n, (1.1) for the vector-function (1.4). Thus, multiple orthogonal polynomials are also called Hermite-Pad6 Polynomials or simultaneous orthogonal polynomials. If we consider the simultaneous rational approximants for the vector-function in which the com- ponents are the so-called Markov functions then the common denominator of approximants (1.9) becomes the polynomial introduced by Definition 1.1. 1.3. Examples of general multiple orthogonal polynomials: Uniqueness In the theory of orthogonal polynomials (see [59]) there are two subjects: classical orthogonal polynomials and general orthogonal polynomials. Properties of classical orthogonal polynomials de- pend very strongly on the special weights of orthogonality and many of the properties disappear under a small perturbation of the weight. In contrast, properties of general orthogonal polynomials (which include the classical ones) remain valid. Similarly, for the multiple orthogonal polynomial the study of the special weights of the polynomials is separated from the results of their general character. We will consider multiple orthogonal polynomials for the special weights in the next section. Here we introduce two families of general multiple orthogonal polynomials. First, we address the existence and uniqueness of the multiple orthogonal polynomials (1.3), (1.5). Existence. The polynomial Q. determined by Definition 1.1 always exists, because for its Inl + 1 unknown coefficients orthogonality relations (1.3) give a system of In I linear algebraic homogeneous equations, which always has a nontrivial solution. 426 A.I. Aptekarev/ Journal of Computational and Applied Mathematics 99 (1998) 423~447 Uniqueness. Unlike the usual orthogonal polynomial (p = 1) defined by (1.3') (a) Q, is determined uniquely up to a constant factor; (b) degQ,=n; (1.11) (c) Q, has n simple zeros which are all contained in the convex hull of supp. for the multiple orthogonal polynomials p > 1 these assertions are in general not true. The simplest example is when the vector of measures (1.2) has identical components. However, there exist two general systems of measures (1.2) for which multiple orthogonal poly- nomials inherit these basic properties (1.11) of orthogonal polynomials. They are called Angelesco and Nikishin systems. 1.3.1. Angelesco system Definition 1.3. A system of arbitrary measures (1.2) which satisfy the following conditions supp #~ = F~ = [a~, b~] C 0~, (1.12) P: F~AF/~=0, c~¢fi, c~,fl=l ..... p, is called Angelesco system. The multiple orthogonal polynomials with respect to the system (1.2), (3.2) have been intro- duced in 1918-23 by Angelesco [3-5] (in 1979 they have been rediscovered by Nikishin [42]). An immediate consequence of orthogonality relations (1.3) and conditions (1.12) is the following Theorem 1.4 (Angelesco [4]). The multiple orthogonal polynom&l Q, with respect to the Angelesco system has in the interior of each F~ exactly n~ simple zeros, ~ = 1..... p. Corollary 1.5. The multiple orthogonal polynomial Q, with respect to the Angelesco system is determined by Definition 1.1 in the unique way, up to a constant factor. 1.3.2. Nikishin system Another general system of measures which guaranties the uniqueness for the multiple orthogonal polynomials has been introduced by Nikishin in [43]. This system is generated by a general set of positive Borel measures a = (al,. , ~rp) (1.13) satisfying the condition supp~r~=F~=[a~,b~]c~, F~AF~+I=0, c~=l,...,p-1, (1.14) by the following procedure: d#l(x) = dal(x), am(x)= (f F da2(x2))da,(x), (1.15) 2 X--X2/ d"a x> = C - - .... A.I. Aptekarev / Journal of Computational and Applied Mathematics 99 (1998) 423~147 427 If we use a notation from [29] / da2(t) d(a,,a2) :-- da,(x), d(a,,a2,...,a,) :=d(a,,(a2 ..... ak)), x--t then the definition of the Nikishin system becomes more compact. Definition 1.6. A system of measures (1.2) which is generated by (1.13), (1.14) as ~: /~l=al, #~=(ol,...,a~), ~=2,3 ..... p, (1.15') is called a Nikishin system. We note that contrary to the Markov system, all measures of the Nikishin system are supported by the same interval F~. Theorem 1.7 (Nikishin [43]; Driver and Stahl [23]). If multiindex (1.1) satisfies n~<~n~_l+l, 0~=2,3,...,p, (1.16) then the multiple orthogonal polynomial Q, with respect to the Nikishin system has in the interior of F1 exactly In I simple zeros. Therefore, under condition (1.16) Q, is defined by (1.3) in the unique way. Remark. Nikishin proved this theorem for the special case n E IsL. For the indexes satisfying (1.16) the theorem has been proved by Driver and Stahl. The uniqueness of multiple orthogonal polynomials for Nikishin system with indexes not satisfying (1.16) is still an open problem. 1.4. Polynomials orthoyonal with varying weight related to the multiple orthogonal polynomials A system of polynomials q, = {q~.,}P satisfying usual orthogonality relations but with respect to varying weight depending on the polynomials themselves, plays the key role for proving results about the uniqueness of multiple orthogonal and their asymptotic behaviour. q,: q~,,(x)x'w~(q,,x)da~(x)=O," v=0,.