A Multiple Orthogonal Polynomial? Andrei Martínez-Finkelshtein and Walter Van Assche Communicated by Cesar E

THEGRADUATESTUDENTSECTION

WHATIS… A Multiple Orthogonal Polynomial? Andrei Martínez-Finkelshtein and Walter Van Assche Communicated by Cesar E. Silva

Multiple orthogonal polynomials are polynomials of one variable that satisfy orthogonality conditions with respect to several measures. They are a very useful extension of orthogonal polynomials and recently received renewed interest because tools have become available to investi- gate their asymptotic behavior. They appear in rational approximation, number theory, random matrices, inte- grable systems, and geometric function theory. Various families of special multiple orthogonal polynomials have been found, extending the classical orthogonal polynomials but also giving completely new special functions [1, Ch. 23].

Deﬁnition

The classical Legendre polynomials 푃푛(푥), 1, 푥, (1/2)(3푥2 − 1), (1/2)(5푥3 − 3푥), … , are orthogonal on [−1, 1] with respect to the Lebesgue measure: 1 Figure 1. Zeros of 퐴푛,1⃗ , 퐴푛,2⃗ and 퐵푛⃗, 푛⃗ = (350, 350), ∫ 푃푚(푥)푃푛(푥)푑푥 = 0, for 푚 ≠ 푛. −1 corresponding to Type I Hermite-Padé approxima- 2 Equivalently, tion to the functions 푓1 = 푤, 푓2 = 푤 , where 푤(푧) 1 is a solution of the algebraic equation 푚 3 2 3 ∫ 푥 푃푛(푥)푑푥 = 0, for 푚 < 푛. 푤 + 3(푧 − 3) 푤 + 2푖(3푧 − 1) = 0. −1 Their direct generalization comprises the Jacobi polynomials, orthogonal on the same interval with respect to 훼 훽 the beta density (1 − 푥) (1 + 푥) , 훼, 훽 > −1. Similarly, the classical Laguerre polynomials are orthogonal with the classical Hermite polynomials are orthogonal with respect to the gamma density 푥훼푒−푥, 훼 > −1, on [0, ∞). −푥2 respect to normal density 푒 on the whole real line, and Multiple orthogonal polynomials satisfy orthogonality relations with respect to several measures, 휇 , 휇 , … , 휇 , Andrei Martínez-Finkelshtein is professor of applied mathematics 1 2 푟 at University of Almería. His email address is [email protected]. on the real line. Given a multi-index 푛⃗ = (푛1, 푛2, … , 푛푟) of positive integers of length |푛|⃗ = 푟 푛 , type I multiple Walter Van Assche is professor of mathematics at the Univer- ∑푖=1 푖 sity of Leuven (Belgium). His email address is walter.vanassche orthogonal polynomials are a vector (퐴푛,1⃗ , … , 퐴푛,푟⃗ ) of @wis.kuleuven.be. polynomials such that 퐴푛,푗⃗ has degree at most 푛푗 − 1 and For permission to reprint this article, please contact: 푟 [email protected]. 푘 (1) ∑ ∫ 푥 퐴푛,푗⃗ (푥)푑휇푗(푥) = 0, 0 ≤ 푘 ≤ |푛|⃗ − 2. DOI: http://dx.doi.org/10.1090/noti1430 푗=1

October 2016 Notices of the AMS 1029 THEGRADUATESTUDENTSECTION

The type II multiple orthogonal polynomial 푃푛⃗ is the monic More recently, Apéry proved in 1979 that 휁(3) is polynomial of degree |푛|⃗ that satisfies the orthogonality irrational. His proof can be seen as a problem of Hermite- conditions Padé approximation to three functions with a mixture of type I and type II multiple orthogonal polynomials. (2) ∫ 푃 (푥)푥푘 푑휇 (푥) = 0, 0 ≤ 푘 ≤ 푛 − 1, 푛⃗ 푗 푗 Similar Hermite-Padé constructions were used by Ball and for 1 ≤ 푗 ≤ 푟. Both relations (1), endowed with an Rivoal in 2001 to show that infinitely many 휁(2푛 + 1) are additional normalization condition, and (2) yield a cor- irrational, and somewhat later Zudilin was able to prove responding linear system of |푛|⃗ equations in the |푛|⃗ that at least one of the numbers 휁(5), 휁(7), 휁(9), 휁(11) unknowns: either the coefficients of the polynomials is irrational. (퐴푛,1⃗ , … , 퐴푛,푟⃗ ) or the coefficients of the monic polynomial 푃푛⃗. The matrix of each of these linear systems is the Random Matrices and Nonintersecting Random transpose of the other and contains moments of the 푟 Paths measures (휇1, … , 휇푟). A solution of these linear systems A research field where multiple orthogonal polynomials may not exist or may not be unique. In order for a solution have appeared more recently and turned out to be very to exist and be unique, one needs extra assumptions on useful is random matrix theory. It was well known that the measures (휇1, … , 휇푟). If a unique solution exists for a orthogonal polynomials play an important role in the multi-index 푛⃗, then the multi-index is said to be normal. If so-called Gaussian Unitary Ensemble of random matrices. all multi-indices are normal, then the system (휇1, … , 휇푟) Another random matrix model is one with an external is said to be a perfect system. source, i.e., a fixed (nonrandom) hermitian matrix 퐴 and probability density Hermite-Padé Approximation 1 −푛Tr(푀2−퐴푀) Multiple orthogonal polynomials originate from Hermite- 푒 푑푀. 푍 Padé approximation, a simultaneous rational approxima- 푛 If 퐴 has 푛 eigenvalues 푎 (1 ≤ 푗 ≤ 푟) and 푛 = |푛|⃗ , then tion to several functions 푓1, 푓2, … , 푓푟 given by their analytic 푗 푗 germs at infinity, the average characteristic polynomial ∞ 푐 푃푛⃗(푧) = 피 det(푧퐼푛 − 푀) 푓 (푧) = 푘,푗 , 1 ≤ 푗 ≤ 푟. 푗 ∑ 푘+1 푘=0 푧 is a type II multiple orthogonal polynomial with orthogo- A type I Hermite-Padé approximation consists of finding nality conditions ∞ polynomials (퐴 , … , 퐴 ) and 퐵 , with deg 퐴 ≤ 푛 − 1, 2 푛,1⃗ 푛,푟⃗ 푛⃗ 푛,푗⃗ 푗 푘 −푛(푥 −푎푗푥) ∫ 푃푛⃗(푥)푥 푒 푑푥 = 0, 0 ≤ 푘 ≤ 푛푗 − 1, such that −∞ 푟 1 퐴 (푧)푓 (푧) − 퐵 (푧) = 풪( ), 푧 → ∞; for 1 ≤ 푗 ≤ 푟. This is called a multiple Hermite polynomial. ∑ 푛,푗⃗ 푗 푛⃗ |푛|⃗ 푗=1 푧 Instead of Hermitian matrices, one can also use positive definite matrices from the Wishart ensemble and find see Figure 1. A type II Hermite-Padé approximation con- multiple Laguerre polynomials. More recently, products of sists of finding rational approximants 푄 /푃 with the 푛,푗⃗ 푛⃗ Ginibre random matrices were investigated by Akemann, common denominator 푃 in the sense that for 1 ≤ 푗 ≤ 푟, 푛⃗ Ipsen, and Kieburg in 2013. The singular values of such 1 matrices are described in terms of multiple orthogonal 푃푛⃗(푧)푓푗(푧) − 푄푛,푗⃗ (푧) = 풪( ), 푧 → ∞. 푧푛푗+1 polynomials for which the weight functions are Meijer If the function 푓푗 is the Cauchy transform of the measure G-functions. 휇푗, Eigenvalues and singular values of random matrices 푑휇푗(푥) are special cases of determinantal point processes. These 푓 (푧) = ∫ , 푗 푧 − 푥 are point processes for which the correlation function then (퐴푛,1⃗ , … , 퐴푛,푟⃗ ) are the type I multiple orthogonal can be written as a determinant of a kernel function 퐾: polynomials and 푃푛⃗ is the type II multiple orthogonal 휌푛(푥1, … , 푥푛) = det(퐾(푥푖, 푥푗))1≤푖,푗≤푛 polynomial for the measures (휇1, … , 휇푟). for every 푛 ≥ 1. For random matrices with an external Number Theory source the kernel is in terms of type I and type II The construction of the previous section goes back to the multiple orthogonal polynomials. This kernel extends the end of the nineteenth century, especially to Hermite and well-known Christoffel-Darboux kernel for orthogonal his student Padé, as well as to Klein and his scientific polynomials. descendants Lindemann and Perron. Hermite’s proof that Other determinantal point processes are related to non- 푒 is a transcendental number is based on Hermite-Padé intersecting random paths. As Kuijlaars and collaborators approximation. Using these ideas, Lindemann generalized have shown, nonintersecting Brownian motions leaving this result, proving that 푒훼1푧, … , 푒훼푟푧 are algebraically at 푟 points and arriving at 1 point can be described in independent over ℚ as long as the algebraic numbers terms of multiple Hermite polynomials, and the analysis 훼1, … , 훼푟 are linearly independent over ℚ (which, in turn, uses the same tools as random matrices with an external yields transcendence of 푒 and 휋). source. If the Brownian motions leave from 푟 points and

1030 Notices of the AMS Volume 63, Number 9 THEGRADUATESTUDENTSECTION

Credits Figure 1 courtesy of S. P. Suetin. Photo of Andrei Martínez-Finkelshtein and Walter Van Assche courtesy of Andrei Martínez-Finkelshtein.

ABOUT THE AUTHORS Andrei Martí nez- Finkelshtein’s research interests include approximation theory, orthogonal polynomials, asymptotic and numerical analysis, Walter Van Assche (l) and mathematical mod- and Andrei Martínez- elling in ophthalmology Finkelshtein (r). and the life sciences. He likes gadgets, literature, and photography. Figure 2. Numerical simulation of 50 rescaled nonintersecting squared Bessel paths, starting at Walter Van Assche’s main interests are orthogonal 푎 = 2 and ending at the origin. The statistics of the polynomials, special functions, and their applications in position of the paths at each time 푡 is described in approximation theory, number theory, and probability. terms of multiple orthogonal polynomials with He is an active supporter of the ecological movement and likes to hike and bike. respect to weights related to modiﬁed Bessel functions.

arrive at 푟 points, then one needs a mixture of type I and type II multiple orthogonal polynomials. Brownian motions in 푑-dimensional space, in which one tracks the square of the distance of the particle from the origin, give rise to nonintersecting squared Bessel paths, whose determinantal structure is written now in terms of multiple orthogonal polynomials related to modiﬁed

Bessel functions 퐼휈 and 퐼휈−1, where 휈 = 푑/2; see Figure 2.

Analytic Theory of MOPs As the pioneering work of Nikishin and others showed, various tools from complex analysis and geometric function theory illuminate the study of multiple orthogonal polynomials. For example, one needs to consider ex- tremal problems for the logarithmic energy for a vector of measures. In addition, algebraic functions solving a polynomial equation of order 푟 + 1 often come into play. The related Riemann surfaces typically have 푟 + 1 sheets, and matrices of order 푟 + 1 arise in a matrix Riemann-Hilbert characterization. One sees here not mere technical complications but rather the richness of the asymptotic behavior of multiple orthogonal polynomials. At least for the near future, it is diﬃcult to envision a general theory.

Reference [1] M. E. H. Ismail, Classical and Quantum Orthogonal Poly- nomials in One Variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, 2005 (paperback edition 2009). MR2191786 (2007f:33001)

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