Trajectories of quadratic differentials for Jacobi polynomials with complex parameters
Motivated by the study of the asymptotic behavior of Jacobi polynomials with and we establish the global structure of trajectories of the related rational quadratic differential on . As a consequence, the asymptotic zero distribution (limit of the root-counting measures of ) is described. The support of this measure is formed by an open arc in the complex plan (critical trajectory of the aforementioned quadratic differential) that can be characterized by the symmetry property of its equilibrium measure in a certain external field.
AMS Mathematics Subject Classifications (2010): 33C45; 42C05; 30C15; 30C85; 30F15; 30F30; 31A15
Keywords: Asymptotics, zeros, orthogonal polynomials, equilibrium measure, -property, quadratic differential, Riemann surface.
The motivation of this work is the large-degree analysis of the behavior of the Jacobi polynomials , when the parameters , are complex and depend on the degree linearly. Recall that these polynomials can be given explicitly by (see [14, 21])
or, equivalently, by the well-known Rodrigues formula
Clearly, polynomials are entire functions of the complex parameters .
If we fix and allow , the zeros of will cluster on and distribute there according to the well-known arcsine law. A non-trivial asymptotic behavior can be obtained in the case of varying coefficients and . Namely, we will consider sequences
where both and are fixed. The case can be studied by the already standard techniques from the potential theory  or by the saddle point method applied to their integral representation, see e.g. . The general situation was analyzed in [8, 11, 12].
In this paper we are interested in the situation when at least one of the parameters, or , is non-real, see e.g. Figure 1. To be more precise, we assume that
Clearly, results for the case and can be easily deduced by reversing the roles of and .
The key feature of the Jacobi polynomials that we use in their asymptotic analysis is their orthogonality property. It is well known that when , the Jacobi polynomials are orthogonal on with respect to the weight function . But as it was shown in , for general we can associate with a complex, non-hermitian orthogonality, where the integration goes along some contour in the complex plane. This is the key, at least in theory, to the study of the limit root location, as well as of the so-called weak (or -th root) asymptotics of these polynomials (via the Gonchar-Rakhmanov-Stahl (GRS) theory [5, 19]), and of their strong uniform asymptotics on the whole plane (by means of the Riemann-Hilbert (RH) steepest descent method of Deift-Zhou ).
The GRS method, in its general form, allows to reduce the description of the cluster set of the zeros of the sequence (1.2)–(1.3) to the problem of finding in a given homotopic class of curves the one with the so-called -property in the associated external field (see the definition (4.5) in Section 4), which in our case is
The most general known existence theorem for the -curves is contained in  under the assumption that external field is harmonic in a complement to a finite set (see also  for the polynomial external fields). This is the case of in (1.4) for , see  for the analysis of the weak asymptotics (via the GRS theory) and [8, 12] for the strong uniform asymptotics on (by the RH technique).
However, is not harmonic (it is not even single valued) in if we assume (1.3), which adds a new essential feature to the problem. In this case, the explicit construction of the curve with the -property is a consequence of the analysis of the structure of trajectories of the following quadratic differential on the Riemann sphere :
Although the local structure of such trajectories is well known, the global topology of the so-called critical graph is usually much more difficult to analyze. Thus, one of the central results of this paper is this description, carried out in Section 3, Theorem 3.1. As a result, we claim that for every pair of parameters satisfying (1.3) there exists an analytic Jordan arc , homotopic in the punctured plane to a Jordan arc connecting both zeros of the polynomial in , and given by the equation
This curve is the limiting set for the zeros of the Jacobi polynomials. Namely, with each we associate its normalized zero-counting measure , such that for any compact set in ,
Here the zeros are counted with their multiplicities.
In Section 5 we show that the sequence converges (as ) in the weak-* topology to a measure , supported on , absolutely continuous with respect to the linear Lebesgue measure on , and given by the formula
It is worth noticing that the fact has an alternative interpretation from the point of view of the hypergeometric differential equation corresponding to : for each , is a discrete critical measures in the external field , and is the continuous critical measure in the same field. A general convergence theorem of this kind was proved in . However, the reduction of the case analyzed here to the results of  is not direct. In particular, the existence and uniqueness of continuous critical measures for external fields with complex parameters is in general an open problem.
Our final remark is that using the construction of the measure and the steepest descent method for the Riemann–Hilbert characterization of the Jacobi polynomials [8, 9] the strong asymptotic formula can be proved. For instance (see (4.2) below),
where we take the holomorphic branch of the square root in such that
is holomorphic in the same domain. If denote the two zeros of , let
Then there is a sequence such that
locally uniformly in . Constants are chosen to match the leading term of .
This result (as well as its analogues on the limiting curve and at ) is established following almost literally the arguments of , and we refer the interested reader to that paper for details.
2 Critical points of
A rational quadratic differential on the Riemann sphere is a form , where is a rational function of a local coordinate . If is a conformal change of variables then
represents in the local parameter . The critical points of are its zeros and poles; all other points of are called regular points. We refer the reader to [6, 15, 20, 22] for further definitions and properties of quadratic differentials.
In this section we focus on a specific rational quadratic differential on the Riemann sphere ,
It depends on two parameters, and , for which (1.3) holds. Since
it is sufficient to restrict our attention to the following case:
for any other combination of the parameters with and we can readily derive the conclusions by combining the mappings
The quadratic differential (2.1) has five critical points on ; three of them at and . Since
under assumptions (2.3) these are double poles of . The other two critical points are the zeros of , that we describe next.
Let . Fixed , we denote by
the branch of this function, as a function of , in the cut plane , such that for . Equivalently, is a conformal mapping of onto the upper half plane with a slit:
With this notation, the zeros of are
respectively. Since and it is obvious that for and satisfying (2.3), and are simple and different from . Furthermore, the following assertions hold:
Under the assumptions (2.3), and . In particular, with ,
where (resp., ) denotes the boundary values of from the upper (resp., lower) half plane.
The polynomial in (2.2) can be rewritten as
it is a quadratic polynomial in , whose discriminant is
In particular, if , then , so that with such the identity can hold only for . This proves that under assumptions (2.3) the roots of cannot belong to . Furthermore, if has a real root (hence, ), by (2.9),
and the assumption implies that .
From the results of  (actually, it is straightforward to check) we know that function
decreases monotonically from to as traverses from to , while
increases monotonically on , and decreases monotonically on . Since is locally conformal, (2.8) follows from the correspondence of boundary points.
Finally, by (2.7),
Assuming that for certain satisfying (2.3), the root , and thus, , it follows that
However, and since and ,
3 Domain configuration of
Recall that the horizontal trajectories (or just trajectories) of are the loci of the equation
while the vertical or orthogonal trajectories are obtained by replacing by in the equation above. The trajectories and the orthogonal trajectories of produce a transversal foliation of the Riemann sphere .
A trajectory of starting and ending at (if exists) is called finite critical or short; if it starts at one of the zeros but tends to either pole, we call it infinite critical trajectory of . In a slight abuse of terminology, we say that such an infinite critical trajectory, if it exists, joins the zero with the corresponding pole. Since has only three poles, Jenkins’ three pole Theorem  asserts that it cannot have any recurrent trajectory.
The set of both finite and infinite critical trajectories of together with their limit points (critical points of ) is the critical graph of .
According to [6, Theorem 3.5] (see also [20, §10]), the complement of the closure of in consists of a finite number of domains called the domain configuration of . Among the possible types of domains there are the so-called circle and strip domains. A circle domain of is a maximal simply connected domain swept out by regular closed trajectories of surrounding a double pole that is the only singularity of in . A strip domain or a digon of is a maximal simply connected domain swept out by regular trajectories of , each diverging to a double pole in both directions; these double poles must represent distinct boundary points of (see ).
The main result of this section is the following theorem, which describes the critical graph as well as the domain configuration of (see Figure 2).
Let and . Then there exists a short trajectory of , joining and . This trajectory is unique, homotopic in the punctured plane to a Jordan arc connecting in .
Furthermore, the structure of the critical graph of is as follows:
the short trajectory of , joining and ;
the unique finite critical trajectory of emanating from and forming a closed loop, encircling ;
the infinite critical trajectory , emanating from and diverging towards ;
the infinite critical trajectory , emanating from and diverging towards .
splits into two connected domains: the bounded circle domain with center at , and an unbounded strip domain , whose boundary points are and .
In other words, we claim that the critical graph of is made of 2 short and 2 infinite critical trajectories. Recall that it is sufficient to analyze the case when satisfy assumptions (2.3).
In order to prove Theorem 3.1 we start from the local structure of the trajectories of at its critical points (see e.g. [6, 15, 20, 22]). Recall that at any regular point the trajectories are locally simple analytic arcs passing through this point, and through every regular point of passes a uniquely determined horizontal and uniquely determined vertical trajectory, mutually orthogonal at this point [20, Theorem 5.5]. Furthermore, there are trajectories emanating from under equal angles .
By (2.4) we conclude that the trajectories are closed Jordan curves in a neighborhood of , and the radial or the log-spiral form at and . The radial structure at occurs if , and at infinity, when .
Let be a Jordan arc in joining and . Then in we can fix a single-valued branch of by requiring that
determine uniquely the homotopy class of in the punctured plane . We have,
where is the boundary value on one of the sides of .
By the properties of the square root, the integral in the left hand side of (3.3) can be written as
which can be calculated using the residues of the integrand at and . Thus,
As it will be seen in Section 5, the short trajectory , joining the zeros , beings the carrier of the asymptotic zero distribution of the Jacobi polynomials, must satisfy
By the proof of Proposition 3.2, this is equivalent to conditions (3.2). So, we need to establish the homotopic class of curves for which conditions (3.2) are satisfied. According to Proposition 3.4 below, there cannot exist a trajectory passing through either pole and joining both zeros . This shows that the homotopic class of curves within the domain given by assumptions (2.3) remains invariant, and it is sufficient to analyze the limit case , , for which, by Lemma 2.1, . By (3.1)–(3.2),
which shows that cuts at some point . We conclude that
Another tool needed to finish the proof of Theorem 3.1 is the following result:
Under assumptions (2.3),
There cannot exist two infinite critical trajectories emanating from the zeros of and diverging to the pole at .
There cannot exist two infinite critical trajectories emanating from the same zero of and diverging to .
Its proof is based on the so-called Teichmüller lemma (see [20, Theorem 14.1]) and follows literally the arguments that have been used in [1, Lemma 4]. We omit repeating them here for the sake of brevity.
Let us establish the structure of the critical graph . Under the assumptions (2.3), is the center of a circle domain , whose boundary, , is made of critical trajectories. Since , we conclude that is made of short critical trajectories. Hence, a priori there are two possibilities:
either is made of two short trajectories, both connecting and , or
is a single closed critical trajectory passing either through or .
For a fixed let be the closure of the domain defined by the conditions (2.3) in the -plane. Observe that the origin does not belong to the image of by the mapping (2.5)–(2.6), which means that are simple in the whole . A consequence of this fact and of Proposition 3.4 is that the homotopic class in of the curves comprising the critical graph is invariant for . For the structure is well-known (see e.g. ): , and is comprised of the interval and of two loops, one emanating from and encircling , and another one emanating from and encircling . In other words, it corresponds to the condition (b) above. Hence, we may discard the possibility (a) for the whole set of parameters satisfying the assumptions (2.3).
In the case (b), let be the zero of on the boundary of . Then the third trajectory, emanating from the same zero, cannot diverge to or : it would oblige two critical trajectories, coming from the other zero of , to diverge to the same pole, contradicting Proposition 3.4.
Thus, we conclude that there exists a short trajectory, , connecting and . Since we have discarded the case (a) mentioned above, this settles automatically the rest of the structure of the critical graph .
Finally, the fact that it is the zero on the boundary of (and in consequence, that is connected with both and by critical trajectories) can be established by the deformation arguments, like in the proof of Proposition 3.3.
The distinguished short trajectory plays an essential role in what follows. For the rest of the paper we use a notation for the holomorphic branch of in :
We introduce in the following analytic function,
Let be the orthogonal trajectory of emanating from that is the analytic continuation of the horizontal trajectory that joins and . Function in (3.5) is defined in such a way that
Under assumptions (2.3), function is a conformal mapping of the domain onto the vertical strip .
We fix the orientation of the critical graph as follows: both and are emanating from , is entering , and is oriented clockwise. This orientation induces the and (that is the right and left) sides of each curve, that we indicate with superscripts. For convenience, we reproduce again the Figure 2 in Figure 3, indicating now the corresponding sides of the curves.
and in consequence, establishes a bijection of the boundary
of the strip domain , oriented from 1 to , and the imaginary axis , oriented from to . By orientation preservation, lies in the right half-plane.
More precisely, let be a simple Jordan arc, from to , and intersecting only once, in . Using again the arguments from the proof of Proposition 3.2,
Thus, under assumptions (2.3),
which shows that the other boundary of the strip domain is mapped by onto the vertical line .
In the next section we will need one more technical result, related to the domain configuration of . Let be a Jordan curve joining and , lying entirely (except for its endpoints) in , passing through in such a way that , and otherwise disjoint with the critical graph . We denote the open arc of joining with , and by the open arc of joining with .
With the notations above,
First, observe that by (3.2),
where we choose an appropriate branch of the logarithm. This shows that the inequalities hold in a neighborhood of . On the other hand, assume there is a point , , such that
By assumptions, . Let be the horizontal trajectory of passing through ; it must intersect at least one of the vertical trajectories of emanating from . Hence, deforming the path from to into the union of an arc and an arc from we run into contradiction with (3.7). ∎
4 An equilibrium problem for the logarithmic potential
On the short trajectory we define the following measure, absolutely continuous with respect to the arc-length measure:
with defined in (3.4), and the boundary values are with respect to the chosen orientation of . Since is a horizontal trajectory of , and using (3.6) we conclude that is a positive probability measure defined on this arc. Straightforward calculations using residues, similar to those performed in (3.6), show that
For measure on , its logarithmic potential is defined by
By (4.2), there exists a constant such that for ,
is a multivalued analytic function in with a single-valued real part.
Let us define
Equation (4.3) can be rewritten as
Since is a trajectory of , we see that
Let be a Jordan curve joining and , lying entirely (except for its endpoints) in , passing through in such a way that , and otherwise disjoint with the critical graph . From Lemma 3.6 we conclude that
This property characterizes the fact that is actually the equilibrium measure of in the external field , and is the corresponding equilibrium constant (see [5, 17]). Furthermore, for defined in (4.4) the trivial identity on111Here we understand by the open arc without its endpoints . yields the so-called -property in the external field : for every ,
where are the normals to .
5 Relation to the asymptotics of Jacobi polynomials with varying parameters
Let us return to the Jacobi polynomials considered in Section 1, and consider the case of varying coefficients and and study the asymptotic behavior of the zeros of the sequences of polynomials given in (1.2), where the constants and satisfy the assumptions (1.3). As it was mentioned, it is sufficient to restrict our attention to the case (2.3).
Our main goal now is to study the convergence of the sequence of the zero counting measures (1.5) in the weak- topology and, if the limit exists, to find it explicitly.
The main result of this section is the following theorem:
The measure is supported on the short trajectory , is absolutely continuous with respect to the linear Lebesgue measure on , and is given by the formula (4.1).
The main property satisfied by polynomials is the non-hermitian orthogonality conditions. Integrating by parts successively the Rodrigues formula (1.1), it is straightforward to obtain the following result, proved in :
Under assumptions (2.3), let be a Jordan curve joining and , and lying entirely (except for its endpoints) in . Then, for all sufficiently large ,
Here the integral is understood in terms of the analytic continuation of any branch of the integrand along .
The main tools for the study of the weak asymptotic behavior of polynomials satisfying a non-hermitian orthogonality have been developed in the seminal works of Stahl  and Gonchar and Rakhmanov . They showed that when the complex analytic weight function depends on the degree of the polynomial, the limit zero distribution is characterized by an equilibrium problem on a compact set in the presence of an external field and satisfying the -property described in Section 4. In fact, Theorem 5.1 is a direct consequence of Proposition 5.2, the properties of established in Section 4, and the original work  (see also ).
Finally, as it was mentioned in the Introduction, measure and the structure of the trajectories of are also the main ingredients of the steepest descent method for the Riemann–Hilbert characterization of the Jacobi polynomials. The analysis follows almost literally the calculations of , so we refer the reader to that paper for the details.
The first and second authors (AMF and PMG) were partially supported by MICINN of Spain and by the European Regional Development Fund (ERDF) under grants MTM2011-28952-C02-01 and MTM2014-53963-P, by Junta de Andalucía (the research group FQM-229), and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almería. Additionally, AMF was supported by Junta de Andalucía through the Excellence Grant P11-FQM-7276. Part of this work was carried out during the visit of AMF to the Department of Mathematics of the Vanderbilt University. He acknowledges the hospitality of the hosting department, as well as a partial support of the Spanish Ministry of Education, Culture and Sports through the travel grant PRX14/00037.
We also wish to thank the anonymous referee for very useful remarks.
-  M. J. Atia, A. Martínez-Finkelshtein, P. Martínez-González, and F. Thabet, Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters, J. Math. Anal. Appl. 416 (2014), 52–80.
-  P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, New York University Courant Institute of Mathematical Sciences, New York, 1999. MR 2000g:47048
-  W. Gawronski and B. Shawyer, Strong asymptotics and the limit distribution of the zeros of Jacobi polynomials , Progress in approximation theory, Academic Press, Boston, MA, 1991, pp. 379–404. MR 1114785 (92j:33019)
-  A. A. Gonchar and E. A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sbornik 125 (1984), no. 2, 117–127, translation from Mat. Sb., Nov. Ser. 134(176), No.3(11), 306-352 (1987).
-  , Equilibrium distributions and degree of rational approximation of analytic functions, Math. USSR Sbornik 62 (1987), no. 2, 305–348, translation from Mat. Sb., Nov. Ser. 134(176), No.3(11), 306-352 (1987).
-  J. A. Jenkins, Univalent functions and conformal mapping, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 18. Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin, 1958. MR MR0096806 (20 #3288)
-  , A topological three pole theorem, Indiana University Mathematics Journal 21 (1972), no. 11, 1013–1018.
-  A. B. J. Kuijlaars and A. Martínez-Finkelshtein, Strong asymptotics for Jacobi polynomials with varying nonstandard parameters, J. Anal. Math. 94 (2004), 195–234. MR MR2124460 (2005k:33006)
-  A. B. J. Kuijlaars, A. Martinez-Finkelshtein, and R. Orive, Orthogonality of Jacobi polynomials with general parameters, Electron. Trans. Numer. Anal. 19 (2005), 1–17 (electronic). MR MR2149265 (2006e:33010)
-  A. B. J. Kuijlaars and G. L. F. Silva, S-curves in polynomial external fields, J. Approximation Theory 191 (2015), 1–37.
-  A. Martínez-Finkelshtein, P. Martínez-González, and R. Orive, Zeros of Jacobi polynomials with varying non-classical parameters, Special functions (Hong Kong, 1999), World Sci. Publ., River Edge, NJ, 2000, pp. 98–113. MR 1805976 (2002d:33017)
-  A. Martínez-Finkelshtein and R. Orive, Riemann-Hilbert analysis of Jacobi polynomials orthogonal on a single contour, J. Approx. Theory 134 (2005), no. 2, 137–170. MR MR2142296 (2006e:33013)
-  A. Martínez-Finkelshtein and E. A. Rakhmanov, Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials, Comm. Math. Phys. 302 (2011), no. 1, 53–111. MR MR2770010
-  NIST Digital Library of Mathematical Functions, url http://dlmf.nist.gov/18.3.
-  Ch. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
-  E. A. Rakhmanov, Orthogonal polynomials and -curves, in “Recent advances in orthogonal polynomials, special functions, and their applications”, Contemp. Math. 578 (2012), 195–239.
-  E. B. Saff and V. Totik, Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften, vol. 316, Springer-Verlag, Berlin, 1997.
-  A. Yu. Solynin, Quadratic differentials and weighted graphs on compact surfaces, Analysis and Mathematical Physics, Trends Math., Birkhäuser Verlag, 2009, pp. 473–505.
-  H. Stahl, Orthogonal polynomials with complex-valued weight function. I, II, Constr. Approx. 2 (1986), no. 3, 225–240, 241–251. MR 88h:42028
-  K. Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 86a:30072
-  G. Szegő, Orthogonal polynomials, fourth ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975.
-  A. Vasilev, Moduli of families of curves for conformal and quasiconformal mappings, Lecture Notes in Mathematics, vol. 1788, Springer-Verlag, Berlin, 2002. MR MR1929066 (2003j:30003)
A. Martínez-Finkelshtein (email@example.com)
Department of Mathematics
University of Almería, Spain, and
Instituto Carlos I de Física Teórica y Computacional
Granada University, Spain
P. Martínez-González (firstname.lastname@example.org)
Department of Mathematics
University of Almería, Spain
F. Thabet (email@example.com)
ISSAT, University of Gabes,