ZERO DISTRIBUTION OF POLYNOMIALS ORTHOGONAL ON THE RADIAL RAYS IN THE COMPLEX PLANE* G. V. Milovanović, P. M. Rajković and Z. M.

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1 FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. 12 (1997), ZERO DISTRIBUTION OF POLYNOMIALS ORTHOGONAL ON THE RADIAL RAYS IN THE COMPLEX PLANE* G. V. Milovanović, P. M. Rajković and Z. M. Marjanović This paper is dedicated to Professor D. S. Mitrinović Abstract. In this paper we continue our investigations on polynomials orthogonal on the radial rays in the complex plane introduced and discussed in [1 4]. Here, we study zero distribution of these polynomials and locate the regions in which these zeros are contained. We also analyze the cases when the zeros are on the rays. Several numerical examples are included. 1. Introduction One of us (see [1 4]) defined a new inner product as follows: For chosen lengths and angles let (1.1) (f, g) = l s (, + ], θ s [, 2π), s =, 1,..., m 1, ε 1 s f(z)g(z) w s (z) dz, L s ε s = e iθ s, where w s (z) is a weight function on the radial ray L s which connects the origin z = and the point z s = l s ε s ( s m 1). This can be rewritten in the form (f, g) = ls f(xε s )g(xε s ) w s (xε s ) dx, Received February 25, Mathematics Subject Classification. Primary 33C45. *This work was partly supported by the Serbian Scientific Foundation under grant #4M3. 127

2 128 G. V. Milovanović, P. M. Rajković and Z. M. Marjanović or (1.2) (f, g) = Because of f 2 = (f, f) = l s f(l s ε s x)g(l s ε s x) w s (l s ε s x) dx. ls f(xε s ) 2 w s (xε s ) dx > except for f(z), we conclude that this inner product is positive-definite. Thus, one can construct the corresponding orthogonal polynomial sequence {π N (z)} + N=, for example by using Gram-Schmidt orthogonalizing process. It implies the existence and uniqueness of such polynomial sequence. In papers [1 4], we discussed the recurrence relations, associated matrix polynomials for these sequences of the polynomials, and in some special cases, we found generating functions, differential equations, some representations and connections with some standard polynomials orthogonal on the real line. If we rotate the whole figure of the rays, we can notice some interesting properties. Theorem 1.1. Let α be an angle in ( π, π] and let the rays L, L 1,..., L, after a rotation for the angle α, become L α, L α 1,..., L α, respectively. Then, the sequence {πn α (z)}+ N= orthogonal with respect to the inner product ( ) f, g = e iα ε 1 α s f(z)g(z) w s (ze iα ) dz L α s can be expressed by π α N(z) = π N (ze iα ), where the polynomials {π N (z)} + N= are orthogonal with respect to (1.1). Proof. Let {π N (z)} + N= Then, we have ( πk (e iα z), π N (e iα z) ) α = be orthogonal polynomials with respect to (1.1). e iα ε 1 s π K (e iα z)π N (e iα z) w s (ze iα ) dz. L α s

3 Zero Distribution of Polynomials Orthogonal on the Radial Rays After changing variable u = ze iα, we get ( πk (ze iα ), π N (ze iα ) ) α = ε 1 s L s π K (u)π N (u) w s (u) du, i.e., ( πk (ze iα ), π N (ze iα ) ) α = ( π K (u), π N (u) ). Because of the uniqueness of orthogonal polynomials (up to a multiplicative constant), we conclude that the statement is valid. Corollary 1.2. The zeros of π α N (z) are obtained from the zeros of π N(z) by the rotation for the angle α. Proof. Let ζ be a zero of the polynomial π N (z), i.e., π N (ζ) =. According the previous theorem, we find π α N(e iα ζ) = π N (ζ) =, i.e, ζe iα is a zero of the polynomial π α N (z). Zeros of polynomials π N (z) and π α N (z) Example 1.1. Consider two rays in the complex plane whose ends are at the points z = 3 e iπ/6 and z 1 = 3 2 ei5π/18 (see Figure 1.1). The zeros of polynomials π N (z) orthogonal on these rays, with respect to the weights w s (z) = 1 (s =, 1), for N = 1, 2,..., 1, are in the sector between these rays (see Figure 1.1). The case after a rotation of the rays for α = π/2 is also presented in the same figure. The corresponding zeros of polynomials π α N (z), for N = 1, 2,..., 1, can be obtained from the zeros of π N (z) by a rotation by the same angle α = π/2. The previous simple statements will be very useful in the next section.

4 13 G. V. Milovanović, P. M. Rajković and Z. M. Marjanović 2. Location of the Zeros We start with a general case of orthogonality on the rays. Theorem 2.1. The polynomial π N (z) (N > ) orthogonal with respect to (1.1) has all zeros in the minimal rectangular cover of the radial rays with edges parallel with the coordinate axes, where and R = { z C : a 1 < Re(z) < a 2 b 1 < Im(z) < b 2 }, a 1 = b 1 = min l s cos θ s, a 2 = max l s cos θ s, cos θ s cos θ s min l s sin θ s, b 2 = max l s sin θ s. sin θ s sin θ s Proof. Suppose that ζ is a zero of π N (z). Then we can write π N (z) = (z ζ)r N 1 (z), r N 1 (z) P N 1. Because of the orthogonality, we have = ( π N (z), r N 1 (z) ) = i.e., Using notation as in (1.2), we yield ε 1 s (z ζ)r N 1 (z)r N 1 (z) w s (z) dz =, L s ε 1 s (z ζ) r N 1 (z) 2 w s (z) dz =. L s l s (l s ε s x ζ) r N 1 (l s ε s x) 2 w s (l s ε s x) dx =. Since the real and imaginary part of the integral on the left must be equal to zero, we have (xl s cos θ s A)l s r N 1 (xl s ε s ) 2 w s (xl s ε s ) dx = and (xl s sin θ s B)l s r N 1 (xl s ε s ) 2 w s (xl s ε s dx =,

5 Zero Distribution of Polynomials Orthogonal on the Radial Rays where A = Re(ζ) and B = Im(ζ). It means that the functions and F (x) = (xl s cos θ s A)l s r N 1 (xl s ε s ) 2 w s (xl s ε s ) G(x) = (xl s sin θ s B)l s r N 1 (xl s ε s ) 2 w s (xl s ε s ) change their signs in x (, 1). In order to conclude that the zeros are in R, we suppose contrary. At first, we suppose that A a 2 = max { l s cos θ s : cos θ s }. Then, for every s and x (, 1), we have A l s cos θ s > xl s cos θ s. But, because of A xl s cos θ s x >, we conclude that F (x) <, for x (, 1), which gives a contradiction (F (x) changes the sign in (, 1)). Thus, A < a 2. In the same way we prove A > a 1 and b 1 < B < b 2. Example 2.1. Consider a case with four rays determined by z = 2e iπ/3, z 1 = 5 2 ei5π/18, z 2 = 2e i4π/3, z 3 = 3e i11π/6. The zeros of polynomials π N (z) orthogonal on these rays, with respect to the weights w s (z) = 1 (s =, 1, 2, 3), for N = 1, 2,..., 1, are in a rectangular cover (see Figure 2.1). Zeros of polynomials π N (z), for N = 1, 2,..., 1.

6 132 G. V. Milovanović, P. M. Rajković and Z. M. Marjanović Theorem 2.2. Let z = l e iθ, z 1 = l 1 e iθ 1 be the endpoints of the radial rays L i L 1, respectively, and let (2.1) (f, g) = e iθ L f(z)g(z) w (z) dz + e iθ 1 L 1 f(z)g(z) w 1 (z) dz, where w (z) and w 1 (z) are weight functions on these rays. Then all zeros of the polynomial π N (z), orthogonal with respect to the inner product (2.1), are in the triangle Oz z 1. Proof. Using notation from Corollary 1.2 we find three rectangles which contain all zeros of π N (z). (a) If we rotate the whole figure of the rays L and L 1 for the angle α, such that z (α ) belongs to the positive part of the real axes. Now, all zeros of the polynomial π (α ) N (z) orthogonal on the new rays L(α ), L (α ) 1 belong to the minimal rectangular cover of the rays P (α ), whose one of the edges contains L (α ). (b) If we rotate the whole figure of the rays L and L 1 for the angle α 1, such that z (α 1) 1 belongs to the positive part of the real axes. Now, all zeros of the polynomial π (α 1) N (z) orthogonal on the new rays L(α 1), L (α 1) 1 belong to the minimal rectangular cover of the rays P (α 1) 1, whose one of the edges contains L (α 1) 1. (c) Finally, if we rotate the whole figure of the rays L and L 1 for the angle α 2, such that z (α 2) z (α 2) 1 belongs to the line parallel to the real axes. Now, all zeros of the polynomial π (α 2) N (z) orthogonal on the new rays L(α 2), L (α 2) 1 belong to the minimal rectangular cover of the rays P (α 2) 2, whose one of the edges contains z (α 2) z (α 2) 1. Rotating figures obtained in (1), (2) and (3) back to the starting position, we rotate also the zeros of the polynomials π (α k) N (z), k =, 1, 2, and their covers to the corresponding rectangles P, P 1, P 2, which contain all zeros of the polynomial π N (z). An intersection of these rectangles is the interior of the triangle Oz z 1. Example 2.2. We consider again the case of rays from Example 1.1. The rectangles P, P 1, P 2, mentioned in the proof of Theorem 2.2, are presented in Figures 2.2a, 2.2b, and 2.2c.

7 Zero Distribution of Polynomials Orthogonal on the Radial Rays a2.2b2.2c Theorem 2.3. All zeros of the polynomial π N (z), N N, orthogonal with respect to (1.1) lie in the convex hull which contains the rays. Proof. Denote the rays by L k, k =, 1,..., m 1 and their endpoints by z k, k =, 1,..., m 1. Let L k be the longest ray. If exists, the point z k1 (k 1 > k ) such that the line z k z k1 does not have an intersection with any ray from the set L j, j =, 1,..., m 1, we rotate the whole ray-figure for the angle α such that the line z (α ) k z (α ) k 1 is parallel with the real axes. Then, all zeros of the polynomial π (α ) N (z) lie in rectangle P (α ) whose one edge contains z (α ) k z (α ) k 1. Also, if exists the endpoint z k2 (k 2 < k ) such that the line z k z k2 does not have an intersection with any ray from the set L j, j =, 1,..., m 1, we rotate the whole ray-figure for the angle α 1 such that the line z (α 1) k z (α 1) k 2 be parallel with the real axes. Then, all zeros of the polynomial π (α 1) N (z) lie in rectangle P (α 1) 1 whose one edge contains z (α 1) k z (α 1) k 2. We repeat the whole procedure for the points z k1 and z k2, and the next points, respectively. So, we find rectangles P (α j) j, j =, 1,..., which contain all zeros of the polynomials π (α j) (z), j =, 1,.... By a rotation to the N starting position of the figure, we yield the rectangles P j, j =, 1, 2,..., which contain all zeros of the polynomial π N (z). Their intersection is a convex hull, containing all zeros of π N (z). The vertices of this convex hull are the tops of those rays which hold on its convexity. Example 2.3. Consider the case of the radial rays L s determined the points z s = l s e iθ s (s =, 1,... 7), where l s θ s π/18 π/9 π/6 2π/9 5π/18 π/3 7π/18 4π/9

8 134 G. V. Milovanović, P. M. Rajković and Z. M. Marjanović Some zeros of the polynomials π N (z), N = 1, 2,..., 1, are out of the regions which are determined by tops of successive rays (see Figure 2.3). But, all zeros lie in the convex hull over the rays. We consider now the case of m rays on the real line determined by z s = l s (s =, 1,..., ν 1) and z s = l s (s = ν,..., m 1), where < ν < m 1. Let w(x) be a weight function on the real line. The polynomials π N (x) orthogonal with respect to the inner product ν 1 (f, g) = ls f(x)g(x)w(x) dx + s=ν l s f(x)g(x)w(x) dx can be treated in an usual sense. Namely, taking the intervals on the real line R, s = [, l s ] (s =, 1,..., ν 1) and s = [ l s, ] (s = ν,..., m 1), and the characteristic function of a set s, defined by { 1, t s, χ( s ; t) =, t / s, the previous inner product can be represented as (f, g) = f(x)g(x) Ω(x) dx, R

9 where Zero Distribution of Polynomials Orthogonal on the Radial Rays Ω(x) = w(x) χ( s ; x). It is easy to see that polynomials π N (x) has all zeros in the interval (a, b), where a = max l s, b = max l s. ν s s ν 1 Remark 2.1. Using the characteristic function of a set, the case of several rays with the same angle can be replaced only with one ray by the same angle. 3. The Case of Equidistant Angles In this section, we study the case of equal angles between successive rays, i.e., ε s = e i2πs/m, s =, 1,..., m 1, with an inner product defined by (3.1) (f, g) = l s f(xl s ε s )g(xl s ε s ) w s (xl s ε s ) dx. Theorem 3.1. Let m = pq (p, q N) and let the lengths and the weights of the radial rays be p-periodical, i.e., (3.2) l s+kp = l s, w s+kp (xl s+kp ε s+kp ) = w s (xl s ε s ), where s p 1, k = 1, 2,..., q 1. Then the polynomial π N (z) (N > ) has the property π N (zε p ) = ε N p π N (z), N =, 1,.... Proof. Let π N (z) be the (monic) polynomial of degree N orthogonal with respect to the inner product (3.1). Then (π N, g) = for any g(z) P N 1. l s π N (xl s ε s )g(xl s ε s ) w s (xl s ε s ) dx =,

10 136 G. V. Milovanović, P. M. Rajković and Z. M. Marjanović For a given p, we put Q N,p (z) = ε N p π N (zε p ) and H p (z) = g(zε p ). Evidently, the polynomial Q N,p (z) is monic. We have (Q N,p, H p ) = = = ε N p because of ε m+k = ε k. l s Q N,p (xl s ε s )H p (xl s ε s ) w s (xl s ε s ) dx l s +p k=p = ε N p (π N, g) =, ε N p π N (xl s ε p ε s )g(xl s ε s ε p ) w s (xl s ε s ) dx l k π N (xl k ε k )g(xl k ε k ) w s (xl s ε s ) dx Since H p (z) can be every polynomial in P N 1, we conclude that Q N,p (z) is an orthogonal (monic) polynomial with respect to (3.1). Finally, from the uniqueness of π N (z) it follows that ε N p π N (zε p ) = π N (z). Theorem 3.2. Under the assumptions (3.2), if ξ is a zero of π N (z), then its zeros are also ξε kp, k = 1, 2,..., q 1. Example 3.1. Consider a case of m = 16 radial rays L s : z s = l s e iθ s (s =, 1,..., 15) in the complex plane, whose lengths are given by l 4k+ν = 2ν + 1, ν, k =, 1, 2, 3, and arguments by θ s = sπ/8, s =, 1,..., 15 (Figure 3.1). As we can see, the complete figure can be obtained by using only 4 successive rays. Namely, such a sub-figure constituted by 4 rays, should be rotated 3 times by the angles kπ/2, k = 1, 2, 3. Also, we see that for a zero in the first sub-figure there exist the corresponding zeros in the other sub-figures obtained by rotations. Now, we discuss the cases when the zeros stay on the rays. Theorem 3.3. Let the conditions (3.2) hold for p = 1 or p = 2. Then the polynomial π N (z) (N > ) orthogonal on the radial rays L s with respect to the inner product (1.2) has all zeros on the rays L s (s =, 1,..., m 1). Proof. Let ζ = ρe iα be a zero of π N (z). According to the previous theorem, its zeros are also ζ k = ζ ε kp, k = 1,..., q 1, where q = m/p. Hence q 1 (z ζ k ) = z q ζ q. k=

11 Zero Distribution of Polynomials Orthogonal on the Radial Rays Zeros of polynomials π N (z) orthogonal on equidistant rays Then the polynomial π N (z) we can write in the form π N (z) = (z q ζ q )r N q(z), where r N q (z) is a polynomial from P N q. Because of orthogonality, we have i.e., = (π N, r N q ) = = l s ε 1 s (z q ζ q )r N q(z)r N q (z) w s (z) dz, L s [( xls ε s ) q ζ q ] rn q (xl s ε s ) 2 w s (xl s ε s ) dx. The real and imaginary part of the previous integral become and l s l s [ (xl s ) q cos ( 2π ] m sq) ρ q cos(αq) r N q (xl s ε s ) 2 w s (xl s ε s ) dx = [ (xl s ) q sin ( 2π ] m sq) ρ q sin(αq) r N q (xl s ε s ) 2 w s (xl s ε s ) dx =,

12 138 G. V. Milovanović, P. M. Rajković and Z. M. Marjanović respectively. Now, we consider two cases: p = 1 and p = 2. Case p = 1. The previous relations become ( ) l s (xl s ) m ρ m cos(αm) r N m (xl s ε s ) 2 w s (xl s ε s ) dx = and ρ m sin(αm) l s r N m (xl s ε s ) 2 w s (xl s ε s ) dx =. From the second one we find α = πν/m, ν N. But, if ν is an odd number, the first relation reduces to ) l s ((xl s ) m + ρ m r N m (xl s ε s ) 2 w s (xl s ε s ) dx =, what is impossible because of positivity of the integrand. Thus, ν must be an even number (ν = 2s), so that α = 2πs/m, s N. Case p = 2. The second relation immediately gives sin(αm/2) =, i.e., α = 2πs/m, s N. Thus, we can conclude that an arbitrary zero ζ = ρe iα of π N (z) may have the argument α from the set of the arguments of the rays only. Also, using Theorem 2.3 on the convex hull of the zeros, we conclude that the zeros of π N (z) are on the rays.

13 Zero Distribution of Polynomials Orthogonal on the Radial Rays Example 3.2. Consider radial rays in the complex plane determined by z s = l s e iθ s, s =, 1,... 5, where l 2s = 1, l 2s+1 = 2, s =, 1, 2, and θ s = sπ/3, s =, 1, Let w(z) 1. Zeros of the corresponding orthogonal polynomials π N (z), N = 1, 2,..., 1, are on the radial rays (see Figure 3.2). Example 3.3. Let again w(z) 1. The zeros of polynomials π N (z), N = 1, 2,..., 1, orthogonal on the radial rays in the complex plane, whose ends are at the points z s = l s e iθ s (s =, 1, 2, 3, where l = l 2 = 3, l 1 = l 3 = 2, and θ s = πs/2, s = 1, 2, 3, 4, are also on the rays (see Figure 3.3). Some zeros of π N (z) are outside the rays Example 3.4. Now, we consider three equidistant rays L s : z s = e 2isπ/3, s =, 1, 2, but with different weights w s (xε s ) = x α s (1 x) β s, s =, 1, 2, where (α, β ) = (1, 2), (α 1, β 1 ) = (3, 4), (α 2, β 2 ) = (5, 6). The zeros of polynomials π N (z), N = 1, 2,..., 11, orthogonal on these radial rays are displayed in Figure 3.4. Evidently, in this case, there are zeros outside the rays.

14 14 G. V. Milovanović, P. M. Rajković and Z. M. Marjanović 4. A Complete Symmetric Case Now we consider a complete symmetric case, i.e., when l s = l, ε s = e i2πs/m, s m 1, and w s (xε s ) = w(x), x (, l), s =, 1,..., m 1. Then the inner product (1.1) reduces to (4.1) (f, g) = l ( For the zeros of π N (z) we can prove: f(xε s )g(xε s ) ) w(x) dx. Theorem 4.1. All zeros of the polynomial π N (z), orthogonal with respect to (4.1), are simple and located on the radial rays, with possible exception of a multiple zero in origin z = of the order ν ( < ν < m), if N ν (mod m). Proof. In Theorem 3.3, we proved that the zeros of π N (z) are on the rays. In the paper [1], it was proved that the polynomial π N (z) can be expressed in the form π N (z) = z ν q n (ν) (z m ), ν {, 1,..., m 1}, where q n (ν) (t) is orthogonal on (, l m ) with respect to a positive weight. It is well known that the zeros of q n (ν) (t) are real and distinct and are located in (, l m ). Let, k = 1,..., n, denote the zeros of q n (ν) (t) in an increasing order τ (n,ν) k Each zero τ (n,ν) k τ (n,ν) 1 < τ (n,ν) 2 < < τ (n,ν) n. generates m zeros z (n,ν) k,s of π N (z). On every ray we have = m τ (n,ν) k e i2πs/m, s =, 1,..., m 1, z (n,ν) 1,s < z (n,ν) 2,s < < z (n,ν) n,s, s =, 1,..., m 1. Also, π N+1 (z) and π N (z) separate their zeros on the rays.

15 Zero Distribution of Polynomials Orthogonal on the Radial Rays Numerical Calculations In the previous examples we used a method for numerical determination of polynomial zeros. The method was based on the following facts. Let {π N (z)} + N= be a sequence of orthogonal polynomials with respect to the inner product (1.2). Such polynomials can be expressed by linear relations where zπ k (z) = k β kj π j (z) + π k+1 (z), k =, 1,..., N 1, j= β kj = (zπ k, π j ) (π j, π j ). In the matrix form, it can be represented as (5.1) zπ N (z) = B N π N (z) + π N (z)e N, where β 1 β 1 β 11 1 (5.2) B N =.... β N 1, β N 1,1 β N 1,N 1 and π N (z) = [π (z), π 1 (z),..., π N 1 (z)] T, e N = [,,...,, 1] T. For the zeros ξ j ( j n) of π N (z), (5.1) reduces to the eigenvalue problem ξ j π N (ξ j ) = B N π N (ξ j ). Thus, ξ j are eigenvalues of the matrix B N and π N (ξ j ) are the corresponding eigenvectors. In numerical evaluation of the inner product (1.2) we use, in general case, m Gaussian n-point quadrature rules with respect to the weight functions Ω s (x) = w(l s ε s x) on (, 1), i.e., F (x)ω s (x) dx = n i=1 W (s) i F ( x (s) ) i (s =, 1,..., m 1),

16 142 G. V. Milovanović, P. M. Rajković and Z. M. Marjanović in order to get a discretized approximation of this product. In this way, we obtain (5.3) (f, g) = = l s l s f(l s ε s x)g(l s ε s x) Ω s (x) dx n i=1 W (s) i f ( l s ε s x (s) i ) ( g ls ε s x (s) ) i. The number of nodes n should be taken so that the elements β kj in the matrix (5.2) can be computed exactly, except for rounding errors. For that, it is enough to take n = N. Of course, in the simplest case (Legendre case) we take only one Gaussian formula, i.e., Gauss-Legendre rule on (, 1). Then (5.3) becomes (f, g) = n i=1 W i l s f(l s ε s x i )g(l s ε s x i ), where x i and W i (i = 1,..., n) are Gauss-Legendre nodes and weights on (, 1), respectively. Finally, for computing the eigenvalues of the upper Hessenberg matrix B T N (zeros of π N (z)) we use the EISPACK routine COMQR [5, pp ]. R E F E R E N C E S 1. G. V. Milovanović: Some nonstandard types of orthogonality (A survey). FILOMAT 9 (1995), G. V. Milovanović: A class of orthogonal polynomials on the radial rays in the complex plane. J. Math. Anal. Appl. 26 (1997), G. V. Milovanović: Generalized Hermite polynomials on the radial rays in the complex plane. In: Theory of Functions and Applications, Collection of Works Dedicated to the Memory of M. M. Djrbashian (ed. H.B. Nersessian), Louys Publishing House, Yerevan, 1995, G. V. Milovanović, P. M. Rajković and Z. M. Marjanović: A class of orthogonal polynomials on the radial rays in the complex plane, II. Facta Univ. Ser. Math. Inform. 11 (1996), B. T. Smith et al.: Matrix Eigensystem Routines EISPACK Guide. Lect. Notes Comp. Science Vol. 6, Springer Verlag, Berlin Heidelberg New York, 1976.

17 Zero Distribution of Polynomials Orthogonal on the Radial Rays Faculty of Electronic Engineering Department of Mathematics, P.O. Box Niš, Serbia, Yugoslavia e mail: grade@elfak.ni.ac.yu Faculty of Mechanical Engineering Department of Mathematics 18 Niš, Serbia, Yugoslavia e mail: pecar@masfak.masfak.ni.ac.yu Faculty of Electronic Engineering Department of Mathematics, P.O. Box Niš, Serbia, Yugoslavia