What Is ... a Multiple Orthogonal Polynomial

WHAT IS . . . A MULTIPLE ORTHOGONAL POLYNOMIAL?

Andrei Mart´ınez-Finkelshteinand Walter Van Assche

Multiple orthogonal polynomials (MOPs) are wj(x) dµ(x), then the function polynomials of one variable which satisfy orthog- r X onality conditions with respect to several mea- Q~n(x) = A~n,j(x)wj(x) sures. They should not be confused with multi- j=1 variate of multivariable orthogonal polynomials, is orthogonal to all polynomials of degree ≤ |~n|−2 which are polynomials of several variables. Other with respect to the measure µ. The type II mul- terminology is also used, e.g., Hermite-Padépoly- tiple orthogonal polynomial P is the monic polynomials, polyorthogonal polynomials (Nikishin ~n nomial of degree |~n| that satisfies the orthogonal- and Sorokin [15]), d-orthogonal polynomials (the ity conditions latter primarily by the French-Tunisian school of Z Maroni, Doauk, Ben Cheikh and collaborators). k (3) P~n(x)x dµj(x) = 0, 0 ≤ k ≤ nj − 1, They are a very useful extension of orthogonal polynomials, and recently received renewed inter- for 1 ≤ j ≤ r. Both conditions (1)–(2) and con- est because tools have become available to inves- ditions (3) yield a corresponding linear system tigate their asymptotic behavior and they do ap- of |~n| equations in the |~n| unknowns, either co- pear in a number of interesting applications. Var- efficients of the polynomials (A~n,1,...,A~n,r) or ious families of special MOPs have been found, coefficients of the monic polynomial P~n. The extending the classical orthogonal polynomials matrices of these linear systems are each other’s but also giving completely new special functions transpose and contain moments of the r measures [1], [8, Ch. 23]. (µ1, . . . , µr). A solution of these linear systems may not exist or may not be unique. One needs 1. Definition extra assumptions on the measures (µ1, . . . , µr) Let r be a positive integer and µ1, µ2, . . . , µr be for a solution to exist and be unique. If a unique positive measures on the real line for which all the solution exists for a multi-index ~n then the multi- r arXiv:1707.09511v1 [math.CA] 29 Jul 2017 moments are finite. Let ~n = (n1, n2, . . . , nr) ∈ N index is said to be normal. If all multi-indices are be a multi-index of size |~n| = n1 + n2 + ··· + nr. normal, then the system (µ1, . . . , µr) is said to be There are two types of multiple orthogonal poly- a perfect system. nomials. Type I multiple orthogonal polynomials One can also define MOPs of mixed type, com- are given as a vector (A~n,1,A~n,2,...,A~n,r) of r bining both types of orthogonality conditions (1) polynomials, where A~n,j has degree ≤ nj − 1, for and (3). which (1) 2. Hermite-Pade´ approximation r Z X k Multiple orthogonal polynomials originate x A~n,j(x) dµj(x) = 0, 0 ≤ k ≤ |~n|−2, j=1 from Hermite-Padéapproximation, a simultane- ous rational approximation to several functions. and one usually adds the normalization Suppose f , f , . . . , f are analytic functions in a r 1 2 r X Z neighborhood of infinity, with series expansions (2) x|~n|−1A (x) dµ (x) = 1. ~n,j j of the form j=1 ∞ X ck,j If all the measures are absolutely continuous f (z) = , 1 ≤ j ≤ r. j zk+1 with respect to one measure µ, i.e., dµj(x) = k=0 1 2

-1

-2

-3 -3 -2 -1 0 1 2 3

Figure 1. Zeros of A~n,1, A~n,2 and B~n, ~n = (350, 350), corresponding to Type I 2 Hermite-Pad´eapproximation to the functions f1 = w, f2 = w , where w(z) is a solution of the algebraic equation w3 +3(z −3)2w −2i(3z −1)3 = 0 (picture courtesy of S.P. Suetin).

Type I Hermite-Padé approximation is to (and his scientific descendants, Lindemann and find polynomials (A~n,1,...,A~n,r) and B~n, with Perron). Hermite’s proof that e is a transcenden- deg A~n,j ≤ nj − 1, such that tal number is based on Hermite-Padéapproxima- r tion. Using these ideas, Lindemann generalized X 1 α z α z A (z)f (z) − B (z) = O , z → ∞. this result proving that e 1 , . . . , e r are alge- ~n,j j ~n z|~n| j=1 braically independent over Q as long as the al- Type II Hermite-Padéapproximation is to find gebraic numbers α1, . . . , αr are linearly indepen- rational approximants Q /P with a common dent over Q (which, in turn, yields transcendence ~n,j ~n of e and π). denominator for the functions f1, . . . , fr in the sense that Apéry’smore recent proof from 1979 that ζ(3) is irrational can be seen as a problem of Hermite- 1 P~n(z)fj(z) − Q~n,j(z) = O , z → ∞, Padéapproximation of the functions znj +1 Z 1 Z 1 for 1 ≤ j ≤ r. If 1 log x f1(z) = dx, f2(z) = − dx, Z 0 z − x 0 z − x dµj(x) fj(z) = , z − x 1 Z 1 log2 x f3(z) = dx, then (A~n,1,...,A~n,r) are the type I multiple or- 2 0 z − x thogonal polynomials, and P~n is the type II in the sense that one wants to find polynomials multiple orthogonal polynomial for the measures An, Bn, Cn, Dn of degree ≤ n for which (as (µ1, . . . , µr). z → ∞) In the case when r = 1, both approximations 1 coincide, and the rational function Q~n,1/P~n is A (z)f (z) − B (z)f (z) − C (z) = O , n 1 n 2 n zn+1 known simply as the Padéapproximant to f1 at z = ∞. 1 A (z)f (z) − 2B (z)f (z) − D (z) = O , n 2 n 3 n zn+1 3. Number theory with the extra condition An(1) = 0. The polyno- The construction of the previous section goes mials (An,Bn) then are a vector of mixed type back to the end of the 19th century, especially, I–type II multiple orthogonal polynomials, which to Hermite (and his student Padé)and to Klein can be obtained explicitly in terms of Legendre 3 polynomials on [0, 1]. The relevant quantity is for 1 ≤ j ≤ r, and this is a so-called multiple f3(1) = ζ(3), and the construction gives rational Hermite polynomial [6]. Instead of Hermitian approximants to ζ(3) which are better than pos- matrices, one can also use positive definite ma- sible if ζ(3) would be rational, implying that this trices from the Wishart ensemble and find mul- number is irrational. tiple Laguerre polynomials [4]. More recently, Similar Hermite-Padéconstructions were used products of Ginibre random matrices were inves- by Ball and Rivoal in 2001 to show that infin- tigated by Akemann, Ipsen and Kieburg in 2013, itely many ζ(2n + 1) are irrational, and some- and the singular values of such matrices are de- what later Zudilin was able to prove that at least scribed in terms of multiple orthogonal polyno- one of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irra- mials for which the weight functions are Meijer tional [7]. Krattenthaler, Rivoal and Zudilin also G-functions [11]. obtained irrationality results for a q-extension of the zeta function in 2006, and the rational ap- 5. Non-intersecting random paths proximation now involves multiple little q-Jacobi polynomials. Eigenvalues and singular values of random matrices are special cases of determinantal point 4. Random matrices processes. These are point processes for which the correlation function can be written as a de- A research field where multiple orthogonal terminant of a kernel function K: polynomials have appeared more recently and turned out to be very useful, is random matrix ρn(x1, . . . , xn) = det K(xi, xj) 1≤i,j≤n theory. It was well known that orthogonal polynomials play an important role in the so-called for every n ≥ 1. For the eigenvalues of a random Gaussian Unitary Ensemble (GUE) of random matrix from GUE the kernel function is matrices. These are random n × n hermitian ma- n−1 −n(V (x)+V (y))/2 X trices M for which the probability distribution of Kn(x, y) = e pk(x)pk(y), the matrix entries is given by a density k=0

1 −nTrV (M) where the (pk)k are the orthonormal polynomi- e dM, −V (x) Zn als for the weight e on R. This kernel is where V is of polynomial growth at infinity and the well-known Christoffel-Darboux kernel for orthogonal polynomials. For random matrices with Zn is a normalizing constant that makes this a probability density. The average characteristic an external source the kernel is in terms of type polynomial I and type II multiple orthogonal polynomials n−1 Pn(z) = E det(zIn − M) X Kn(x, y) = P~nk (x)Q~nk+1 (y), is a monic polynomial of degree n that satisfies k=0 Z k −V (x) r ~ Pn(x)x e dx = 0, k = 0, 1, . . . , n − 1, where (~nk)0≤k≤n is a path in N from 0 to ~n, and the sum turns out to be independent of this path. and hence it is the nth degree orthogonal poly- Another determinantal point process is related nomial for the weight e−V (x). Another random to non-intersecting random paths, in particu- matrix model is one with an external source [9], lar non-intersecting Brownian motions [5] and i.e., a fixed (non-random) hermitian matrix A, squared Bessel paths (BESQ) [10]. As Kuijlaars and the probability density now is and collaborators have shown, non-intersecting

1 2 Brownian motions leaving at r points and ar- e−nTr(M −AM) dM. Zn riving at 1 point can be described in terms of multiple Hermite polynomials and the analysis If A has n eigenvalues a (1 ≤ j ≤ r) and j j uses the same tools as random matrices with an n = |~n|, then the average characteristic polyno- external source. If the Brownian motions leave mial from r points and arrive at r points, then one P (z) = det(zI − M) ~n E n needs a mixture of type I and type II multi- is a type II multiple orthogonal polynomial with ple orthogonal polynomials. BESQ are Brown- orthogonality conditions ian motions in d-dimensional space, in which one ∞ Z 2 k −n(x −aj x) tracks the distance of the particle from the ori- P~n(x)x e dx = 0, 0 ≤ k ≤ nj−1, −∞ gin. This distance squared is distributed as a 4

3.5

2.5

2 x

1.5

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t

Figure 2. Numerical simulation of 50 rescaled non-intersecting BESQ, starting at a = 2 and ending at the origin. The statistics of the position of the paths at each time t is described in terms of MOPs with respect to weights related to modified Bessel functions Iν and Iν−1. non-central χ2 random variable, and the depen- many properties, there are several available tools dence structure (repulsion of the particles, since to study their analytic behavior (zeros, asymp- the paths may not intersect) gives a determinan- totics, etc.), which are absent when consider- tal structure with multiple orthogonal polynomi- ing other families. The first “universal” tool in als related to modified Bessel functions Iν and this sense was logarithmic potential theory: it is Iν−1, where ν = d/2, see Figure 2. known (this knowledge can be traced all the way back to Gauss, but was fully developed at the 6. Integrable systems end of 20th century in the works of Saff, Mhaskar, Gonchar, Rakhmanov, Stahl and others) that the Multiple orthogonal polynomials satisfy a sys- global zero (or weak) asymptotics of general fam- tem of nearest neighbor recurrence relations, ex- ilies of orthogonal polynomials can be described tending the well-known three term recurrence in terms of solutions of extremal problems for relation for orthogonal polynomials. For this the logarithmic energy of positive measures (with reason, MOPs are related to the spectral the- multiple variations, such as the presence of an ex- ory of non-symmetric operators (see the work of ternal field, upper constraints, symmetry condi- Aptekarev, Kalyagin and collaborators). For the tions, etc.). In many cases, these solutions are same reason, MOPs are also appearing in the the- related to harmonic functions on compact Rie- ory of integrable systems [2]. To mention a few mann surfaces. For a more detailed description examples, multiple Jacobi polynomials were used of the asymptotics, especially at some particu- by Mukhin and Varchenko [14] to give a coun- lar points on the plane, we have to recur to spe- terexample to the Bethe Ansatz conjecture for cial functions (both classical and modern, such as the Gaudin model, some non-Hermitian oscilla- Painlevétranscendents). tor Hamiltonians were constructed using multi- Recently a new method, developed by Deift ple Charlier [13] and multiple Meixner polynomi- and Zhou, based on the matrix Riemann-Hilbert als [12], multiple Meixner-Pollaczek polynomials characterization of orthogonal polynomials (due appear in the six-vertex model [3], and discrete to Fokas, Its, and Kitaev) has emerged, which integrable systems can be generated by multiple encompasses these previous tools and, when suc- orthogonal polynomials. cessful, gives detailed information about the asymptotic behavior, uniformly in the whole com- 7. Analytic theory of MOPs plex plane. Since the classical families of orthogonal poly- As the pioneering work of Nikishin, followed nomials (Jacobi, Laguerre, Hermite) satisfy so by Aptekarev, Gonchar, Kalyagin, Kuijlaars, 5

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Andrei Mart´ınez-Finkelshtein Universidad de Almer´ıa,Spain [email protected] Walter Van Assche KU Leuven, Belgium [email protected]