- Email: [email protected]
Sharp parameter intervals for interlacing of zeros of equal degree Laguerre polynomials
Available online at www.sciencedirect.com
ScienceDirect Journal of Approximation Theory 248 (2019) 105303 www.elsevier.com/locate/jat
Full Length Article
Sharp parameter intervals for interlacing of zeros of equal degree Laguerre polynomials Kathy Drivera ,∗, Martin E. Muldoonb a
Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town 7708, South Africa b Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada Received 4 April 2018; received in revised form 30 August 2019; accepted 13 September 2019 Available online 26 September 2019 Communicated by E. Koelink
Abstract We prove that, for each α > −1, the t-interval (0, 2] is sharp for interlacing to hold for every n ∈ N (α) (α+t) between the zeros of the equal degree Laguerre polynomials L n (x) and L n (x). We use a recent result proved by Pálmai (2013) that, for µ, ν > −1, the positive zeros of the Bessel functions Jν (x) and Jµ (x) are interlacing if and only if |µ − ν| ≤ 2. c 2019 Elsevier Inc. All rights reserved. ⃝ Keywords: Laguerre polynomials; Zeros; Interlacing; Bessel functions; Sturm Comparison Theorem
1. Introduction It is well known that for α > −1, the Laguerre polynomial L (α) n (x) has n distinct positive (α) zeros and the zeros of L (α) (x) interlace with the zeros of L (x) for every n ∈ N, n ≥ 2 n n−1 [8, Chapter V]. In [3, Theorem 4.4], we proved that the t-interval [0, 2k] is sharp in order (α) for the zeros of two Laguerre polynomials L (α+t) n−k (x) and L n (x), α > 0, of degree n − k and n respectively, to be interlacing in the Stieltjes sense [8, Theorem 3.3.3] for each k, n ∈ N, (α) 1 ≤ k ≤ n − 1, excluding any values of t for which L (α+t) n−k (x) and L n (x) have a common zero. A slightly stronger result [3, Theorem 3.1] holds when k = 1, namely that, for α > −1, (α) L (α+t) n−1 (x) and L n (x) have no common zeros for n ∈ N, n ≥ 2, and 0 < t ≤ 2; their zeros are interlacing for each n ∈ N, n ≥ 2, and each t with 0 < t ≤ 2, and the t-interval [0, 2] is sharp in the sense of being maximal for interlacing to hold for every n ∈ N. ∗ Corresponding author.
E-mail addresses: [email protected] (K. Driver), [email protected] (M.E. Muldoon). https://doi.org/10.1016/j.jat.2019.105303 c 2019 Elsevier Inc. All rights reserved. 0021-9045/⃝
2
K. Driver and M.E. Muldoon / Journal of Approximation Theory 248 (2019) 105303
(0)
(t)
Fig. 1. The dotted lines are the x-zeros of L 4 (x) and the broken lines are the three smallest x-zeros of L 4 (x) (t) as functions of t, 0 ≤ t < 10. The three large dots indicate the points at which the ith positive zero of L 4 (x) (0) intersects the (i + 1)th positive zero of L 4 (x), i = 1, 2, 3.
The interlacing results proved in this paper involve zeros of polynomials of the same degree n. We recall that if x1 < x2 < · · · < xn and y1 < y2 < · · · < yn denote the zeros of two polynomials of degree n, the zeros are said to interlace if either x1 < y1 < x2 < y2 < · · · xn < yn or y1 < x1 < y2 < x2 < · · · yn < xn . For two Laguerre polynomials of equal degree n, the Markov monotonicity theorem [8, Theorem 6.12.1] and mixed three-term recurrence relations were used in [2, Theorem 3.1] to prove that for each n ∈ N and each α > −1, the zeros of L (α+t) (x) and L (α) n n (x) are interlacing for 0 < t ≤ 2. We use the Sturm Comparison Theorem (α+t) to prove interlacing results for the real zeros of L (α) (x). We recall the work of n (x) and L n Lorch [5], where the Sturm Theorem is a major tool in the proofs of results on the zeros of Laguerre polynomials. In [1], Dimitrov, Ismail and Rafaeli provide an alternative approach to the question of interlacing of zeros of orthogonal polynomials of equal degree by viewing changes in the parameter(s) as modifications of the orthogonality measure. Numerical examples given in [2, Theorem 2.3] show that interlacing between the zeros of L (α+t) (x) and those of L (α) n n (x) breaks down for some integer values of n and some values of t > 2, α > −1. We use the limiting relation to Bessel functions [8, §6.31] to prove that the interval 0 < t ≤ 2 cannot be enlarged if Laguerre polynomials L (α+t) (x) and L (α) n n (x) are to retain interlacing of their n zeros for every n ∈ N. In other words, for fixed t > 2, it is not true that the zeros of L (α+t) (x) interlace with the zeros of L (α) n n (x) for every n ∈ N. Our main result is the following theorem: (α+t) Theorem 1.1. Let α > −1 be a fixed real number. The zeros of L (α) (x) are n (x) and L n interlacing for each fixed t in (0, 2]. The interval (0, 2] cannot be enlarged to (0, 2 + ϵ] for any ϵ > 0 and interlacing of zeros retained for every n ∈ N.
2. Sturm Comparison Theorem and Bessel functions One of our main tools is the Sturm Comparison Theorem, which gives information on interlacing of zeros of a solution of one second order linear ordinary differential equation with zeros of a solution of a related equation: Theorem 2.1 ([8, Theorem 1.82.1]). Let f (x) and F(x) be functions continuous in x0 < x < X 0 with f (x) ≤ F(x). Let the functions y(x) and Y (x), both not identically zero, satisfy the
K. Driver and M.E. Muldoon / Journal of Approximation Theory 248 (2019) 105303
3
differential equations y ′′ + f (x)y = 0, Y ′′ + F(x)Y = 0,
(2.1)
respectively. Let x ′ and x ′′ , x ′ < x ′′ , be two consecutive zeros of y(x). Then the function Y (x) has at least one variation of sign in the interval x ′ < x < x ′′ , provided f (x) ̸≡ F(x) in [x ′ , x ′′ ]. statement holds ]also for x ′ = x0 [y(x0 + 0) = 0] if the additional condition [ The ′ limx→x0 + y (x)Y (x) − y(x)Y ′ (x) = 0 is satisfied (similarly for x ′′ = X 0 ). As n → ∞, the Laguerre polynomials behave like Bessel functions [8, §6.31] so it can be expected that interlacing properties of zeros of Bessel functions will throw some light on interlacing of zeros of Laguerre polynomials of large degree. Our next result is a special case of a more general result proved by P´almai in [7, Theorem 3] for the positive zeros of cylinder functions Cν (x) and Cµ (x), ν, µ > 0. In [7, Theorem 3], P´almai proves that for ν, µ > 0, the positive zeros of Cν (x) and Cµ (x) are interlacing if and only if |µ − ν| ≤ 2. He remarks that, in the special case where the cylinder functions Cν (x) and Cµ (x) reduce to the Bessel functions Jν (x) and Jµ (x) respectively, the necessary and sufficient condition |µ − ν| ≤ 2 holds for interlacing of the positive zeros of Jν (x) and Jµ (x) when ν, µ > −1. P´almai’s proof uses the conditional transitivity of interlacing relations and a connection between interlacing and Wronskians. Our proof is based on Theorem 2.1 and offers an alternative approach to that of P´almai. Theorem 2.2. Let ν > −1. (i) The positive zeros of Jν (x) and Jν+t (x) interlace for 0 < t ≤ 2. (ii) The interlacing breaks down for sufficiently large zeros when t > 2. Proof of Theorem 2.2. The function y(ν, x) = x 1/2 Jν (x) satisfies the differential equation [6, eqn.(10.13.1)] y ′′ + f (ν, x)y = 0,
(2.2)
where f (ν, x) = 1 + (1/4 − ν 2 )/x 2 , and hence x 2 [ f (ν + 2, x) − f (ν, x)] = −4(ν + 1) < 0, ν > −1. Thus, an application of Theorem 2.1 shows that for ν > −1, there is a zero of Jν (x) between every pair of consecutive positive zeros of Jν+2 (x). We use the notation j(ν, k) for the kth positive zero of Jν (x). Since [9, p. 508] the zeros of Jν+t (x) are increasing functions of t, t > 0, we see that, for fixed ν (> −1), and sufficiently small t > 0, j(ν, 1) < j(ν + t, 1) < j(ν, 2) < j(ν + t, 2) < j(ν, 3) < j(ν + t, 3) < · · · .
(2.3)
For k = 1, 2, . . . , we use the notation tk for the unique value of t for which j(ν + t, k) = j(ν, k + 1). Clearly the interlacing (2.3) cannot extend to any of the values tk , k = 1, 2, . . . . We will show that the sequence {tk , k = 1, 2, . . . } is decreasing and approaches 2 as k → ∞. Suppose, on the contrary, that for some fixed k, we have tk < tk+1 . Then, for t lying in the open interval (tk , tk+1 ), the Bessel function Jν+t (x) would have two successive zeros of Jν+t (x) with no zero of Jν (x) between them, contradicting the result that for ν > −1, there is a zero of Jν (x) between every pair of consecutive positive zeros of Jν+t (x). Therefore, (i) holds. To prove (ii), we need to show that tk → 2 as k → ∞. We use McMahon’s expansion [6, eqn. (10.21.19)] j(ν, k) = (k + ν/2 − 1/4)π + O(k −1 ), k → ∞.
(2.4)
4
K. Driver and M.E. Muldoon / Journal of Approximation Theory 248 (2019) 105303
Then, as k → ∞ 0 = j(ν, k + 1) − j(ν + tk , k) = (1 − tk /2)π + O(k −1 ),
(2.5)
so tk → 2 as k → ∞. The increase of the zeros as functions of t shows that, for 0 < t < 2, (2.3) can be extended to j(ν, 1) < j(ν + t, 1) < j(ν + 2, 1) < j(ν, 2) < j(ν + t, 2) < j(ν + 2, 2) < · · · .
(2.6)
This completes the proof of Theorem 2.2. 3. Proof of Theorem 1.1 The first statement of Theorem 1.1 was proved in [2, Theorem 2.3]. Fig. 1 illustrates the interlacing in the case n = 4 and illustrates the important role played by the location of points (α+t) at which zeros of L (α) (x) coincide. n (x) and L n (α+t) To show that the interlacing of the zeros of L (α) (x) does not extend beyond n (x) and L n t = 2 for every n ∈ N, suppose on the contrary that there is a positive number ϵ such (α+t) that the zeros of L (α) (x) interlace for t = 2 + ϵ, ϵ > 0 and hence for n (x) and L n 0 < t ≤ 2 + ϵ, n = 1, 2, . . . . Then the limit [8, Theorem 8.1.3] lim n −α L n(α) (z/n) = z −α/2 Jα (2z 1/2 )
n→∞
(3.1)
or the limit [4, p. 8] lim νλn,k (α) = j 2 (α, k),
n→∞
(3.2)
where ν = 4n + α + 2; λn,k (α), k = 1, 2, . . . , n are the zeros of L (α) n (x); and j(α, k), k = 1, 2, . . . , n are the zeros of Jα (x) shows that, for the Bessel function zeros, we get the weak interlacing property j 2 (α, 1) < j 2 (α + t, 1) ≤ j 2 (α, 2) < j 2 (α + t, 2) ≤ j 2 (α, 3) < . . . ,
(3.3)
for 0 < t ≤ 2 + ϵ where we have to use ≤ in the even inequalities because of the limit process. The odd inequalities are strict because of the increase of j(α + t, k) with t. From (3.3) we have j(α, k) < j(α + t, k) ≤ j(α, k + 1) < j(α + t, k + 1), k = 1, 2, 3, . . . ,
(3.4)
for each t with 0 < t ≤ 2 + ϵ. We recall that tk is the unique value of t for which j(α + t, k) = j(α, k + 1) and we proved in Theorem 2.2 that tk decreases with k and tk → 2 as k → ∞. It follows that we may choose k sufficiently large so that 2 < tk+1 < tk < 2 + ϵ and we can choose t such that 2 < tk+1 < t < tk < 2 + ϵ. Then, for these choices of k ∈ N and t, we have j(α, k) < j(α, k + 1) < j(α + t, k), which contradicts (3.4). This concludes the proof of Theorem 1.1. 4. Additional sharpness questions An interesting discussion on the sharpness of parameter intervals within which the zeros of Jacobi polynomials of the same or consecutive degree are interlacing can be found in [1]. In [1, Section 5], Dimitrov, Ismail and Rafaeli conjecture that if α and β are large fixed positive (α,β) (α,β+τ ) numbers, there exists ϵ = ϵ(α, β) > 0 such that the zeros of Pn (x) and Pn (x) are
K. Driver and M.E. Muldoon / Journal of Approximation Theory 248 (2019) 105303
5
interlacing for all n ∈ N and for every τ ∈ (0, 2 + ϵ). In the context of Laguerre polynomials, their conjecture would suggest that if α is a large, fixed number, there exists ϵ = ϵ(α) > 0 such (α+τ ) that the zeros of L (α) (x) are interlacing for all n ∈ N and for every τ ∈ (0, 2+ϵ). n (x) and L n Dedication This paper is dedicated to a wonderful man and mathematician, my friend and co-author Martin E. Muldoon, 28 February 1939–1 August 2019, who sadly passed away prior to the publication of this article. Acknowledgements Many thanks to the referees whose suggestions have improved the paper in several ways. Kathy Driver’s research is supported by the National Research Foundation of South Africa under Grant Number 466367. References [1] D.K. Dimitrov, M.E.H. Ismail, F.R. Rafaeli, Interlacing of zeros of orthogonal polynomials under modification of the measure, J. Approx. Theory 175 (2013) 64–76. [2] K. Driver, K. Jordaan, Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials, Numer. Math. 107 (2007) 615–624. [3] K. Driver, M.E. Muldoon, Common and interlacing zeros of families of Laguerre polynomials, J. Approx. Theory 193 (2015) 89–98. [4] L. Gatteschi, Asymptotics and bounds for the zeros of laguerre polynomials: a survey, J. Comput. Appl. Math. 14 (2002) 7–27. [5] L. Lorch, Elementary comparison techniques for certain classes of Sturm–Liouville equations, in: G. Berg, M. Essén, Å. Pleijel (Eds.), Differential Equations (Proc. Internat. Conf. Uppsala, 1977), Almqvist & Wiksell, Stockholm, 1977, pp. 125–133. [6] F.W.J. Olver, et al. (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, dlmf.nist.gov. [7] T. Pálmai, On the interlacing of cylinder functions, Math. Inequal. Appl. 16 (2013) 241–247. [8] G. Szeg˝o, Orthogonal Polynomials, Vol. 23, forth ed., American Mathematical Society Colloquium Publications, 1975. [9] G.N. Watson, A Treatise on the Theory of Bessel Functions, second ed., Cambridge University Press, 1944.
ScienceDirect Journal of Approximation Theory 248 (2019) 105303 www.elsevier.com/locate/jat
Full Length Article
Sharp parameter intervals for interlacing of zeros of equal degree Laguerre polynomials Kathy Drivera ,∗, Martin E. Muldoonb a
Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town 7708, South Africa b Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada Received 4 April 2018; received in revised form 30 August 2019; accepted 13 September 2019 Available online 26 September 2019 Communicated by E. Koelink
Abstract We prove that, for each α > −1, the t-interval (0, 2] is sharp for interlacing to hold for every n ∈ N (α) (α+t) between the zeros of the equal degree Laguerre polynomials L n (x) and L n (x). We use a recent result proved by Pálmai (2013) that, for µ, ν > −1, the positive zeros of the Bessel functions Jν (x) and Jµ (x) are interlacing if and only if |µ − ν| ≤ 2. c 2019 Elsevier Inc. All rights reserved. ⃝ Keywords: Laguerre polynomials; Zeros; Interlacing; Bessel functions; Sturm Comparison Theorem
1. Introduction It is well known that for α > −1, the Laguerre polynomial L (α) n (x) has n distinct positive (α) zeros and the zeros of L (α) (x) interlace with the zeros of L (x) for every n ∈ N, n ≥ 2 n n−1 [8, Chapter V]. In [3, Theorem 4.4], we proved that the t-interval [0, 2k] is sharp in order (α) for the zeros of two Laguerre polynomials L (α+t) n−k (x) and L n (x), α > 0, of degree n − k and n respectively, to be interlacing in the Stieltjes sense [8, Theorem 3.3.3] for each k, n ∈ N, (α) 1 ≤ k ≤ n − 1, excluding any values of t for which L (α+t) n−k (x) and L n (x) have a common zero. A slightly stronger result [3, Theorem 3.1] holds when k = 1, namely that, for α > −1, (α) L (α+t) n−1 (x) and L n (x) have no common zeros for n ∈ N, n ≥ 2, and 0 < t ≤ 2; their zeros are interlacing for each n ∈ N, n ≥ 2, and each t with 0 < t ≤ 2, and the t-interval [0, 2] is sharp in the sense of being maximal for interlacing to hold for every n ∈ N. ∗ Corresponding author.
E-mail addresses: [email protected] (K. Driver), [email protected] (M.E. Muldoon). https://doi.org/10.1016/j.jat.2019.105303 c 2019 Elsevier Inc. All rights reserved. 0021-9045/⃝
2
K. Driver and M.E. Muldoon / Journal of Approximation Theory 248 (2019) 105303
(0)
(t)
Fig. 1. The dotted lines are the x-zeros of L 4 (x) and the broken lines are the three smallest x-zeros of L 4 (x) (t) as functions of t, 0 ≤ t < 10. The three large dots indicate the points at which the ith positive zero of L 4 (x) (0) intersects the (i + 1)th positive zero of L 4 (x), i = 1, 2, 3.
The interlacing results proved in this paper involve zeros of polynomials of the same degree n. We recall that if x1 < x2 < · · · < xn and y1 < y2 < · · · < yn denote the zeros of two polynomials of degree n, the zeros are said to interlace if either x1 < y1 < x2 < y2 < · · · xn < yn or y1 < x1 < y2 < x2 < · · · yn < xn . For two Laguerre polynomials of equal degree n, the Markov monotonicity theorem [8, Theorem 6.12.1] and mixed three-term recurrence relations were used in [2, Theorem 3.1] to prove that for each n ∈ N and each α > −1, the zeros of L (α+t) (x) and L (α) n n (x) are interlacing for 0 < t ≤ 2. We use the Sturm Comparison Theorem (α+t) to prove interlacing results for the real zeros of L (α) (x). We recall the work of n (x) and L n Lorch [5], where the Sturm Theorem is a major tool in the proofs of results on the zeros of Laguerre polynomials. In [1], Dimitrov, Ismail and Rafaeli provide an alternative approach to the question of interlacing of zeros of orthogonal polynomials of equal degree by viewing changes in the parameter(s) as modifications of the orthogonality measure. Numerical examples given in [2, Theorem 2.3] show that interlacing between the zeros of L (α+t) (x) and those of L (α) n n (x) breaks down for some integer values of n and some values of t > 2, α > −1. We use the limiting relation to Bessel functions [8, §6.31] to prove that the interval 0 < t ≤ 2 cannot be enlarged if Laguerre polynomials L (α+t) (x) and L (α) n n (x) are to retain interlacing of their n zeros for every n ∈ N. In other words, for fixed t > 2, it is not true that the zeros of L (α+t) (x) interlace with the zeros of L (α) n n (x) for every n ∈ N. Our main result is the following theorem: (α+t) Theorem 1.1. Let α > −1 be a fixed real number. The zeros of L (α) (x) are n (x) and L n interlacing for each fixed t in (0, 2]. The interval (0, 2] cannot be enlarged to (0, 2 + ϵ] for any ϵ > 0 and interlacing of zeros retained for every n ∈ N.
2. Sturm Comparison Theorem and Bessel functions One of our main tools is the Sturm Comparison Theorem, which gives information on interlacing of zeros of a solution of one second order linear ordinary differential equation with zeros of a solution of a related equation: Theorem 2.1 ([8, Theorem 1.82.1]). Let f (x) and F(x) be functions continuous in x0 < x < X 0 with f (x) ≤ F(x). Let the functions y(x) and Y (x), both not identically zero, satisfy the
K. Driver and M.E. Muldoon / Journal of Approximation Theory 248 (2019) 105303
3
differential equations y ′′ + f (x)y = 0, Y ′′ + F(x)Y = 0,
(2.1)
respectively. Let x ′ and x ′′ , x ′ < x ′′ , be two consecutive zeros of y(x). Then the function Y (x) has at least one variation of sign in the interval x ′ < x < x ′′ , provided f (x) ̸≡ F(x) in [x ′ , x ′′ ]. statement holds ]also for x ′ = x0 [y(x0 + 0) = 0] if the additional condition [ The ′ limx→x0 + y (x)Y (x) − y(x)Y ′ (x) = 0 is satisfied (similarly for x ′′ = X 0 ). As n → ∞, the Laguerre polynomials behave like Bessel functions [8, §6.31] so it can be expected that interlacing properties of zeros of Bessel functions will throw some light on interlacing of zeros of Laguerre polynomials of large degree. Our next result is a special case of a more general result proved by P´almai in [7, Theorem 3] for the positive zeros of cylinder functions Cν (x) and Cµ (x), ν, µ > 0. In [7, Theorem 3], P´almai proves that for ν, µ > 0, the positive zeros of Cν (x) and Cµ (x) are interlacing if and only if |µ − ν| ≤ 2. He remarks that, in the special case where the cylinder functions Cν (x) and Cµ (x) reduce to the Bessel functions Jν (x) and Jµ (x) respectively, the necessary and sufficient condition |µ − ν| ≤ 2 holds for interlacing of the positive zeros of Jν (x) and Jµ (x) when ν, µ > −1. P´almai’s proof uses the conditional transitivity of interlacing relations and a connection between interlacing and Wronskians. Our proof is based on Theorem 2.1 and offers an alternative approach to that of P´almai. Theorem 2.2. Let ν > −1. (i) The positive zeros of Jν (x) and Jν+t (x) interlace for 0 < t ≤ 2. (ii) The interlacing breaks down for sufficiently large zeros when t > 2. Proof of Theorem 2.2. The function y(ν, x) = x 1/2 Jν (x) satisfies the differential equation [6, eqn.(10.13.1)] y ′′ + f (ν, x)y = 0,
(2.2)
where f (ν, x) = 1 + (1/4 − ν 2 )/x 2 , and hence x 2 [ f (ν + 2, x) − f (ν, x)] = −4(ν + 1) < 0, ν > −1. Thus, an application of Theorem 2.1 shows that for ν > −1, there is a zero of Jν (x) between every pair of consecutive positive zeros of Jν+2 (x). We use the notation j(ν, k) for the kth positive zero of Jν (x). Since [9, p. 508] the zeros of Jν+t (x) are increasing functions of t, t > 0, we see that, for fixed ν (> −1), and sufficiently small t > 0, j(ν, 1) < j(ν + t, 1) < j(ν, 2) < j(ν + t, 2) < j(ν, 3) < j(ν + t, 3) < · · · .
(2.3)
For k = 1, 2, . . . , we use the notation tk for the unique value of t for which j(ν + t, k) = j(ν, k + 1). Clearly the interlacing (2.3) cannot extend to any of the values tk , k = 1, 2, . . . . We will show that the sequence {tk , k = 1, 2, . . . } is decreasing and approaches 2 as k → ∞. Suppose, on the contrary, that for some fixed k, we have tk < tk+1 . Then, for t lying in the open interval (tk , tk+1 ), the Bessel function Jν+t (x) would have two successive zeros of Jν+t (x) with no zero of Jν (x) between them, contradicting the result that for ν > −1, there is a zero of Jν (x) between every pair of consecutive positive zeros of Jν+t (x). Therefore, (i) holds. To prove (ii), we need to show that tk → 2 as k → ∞. We use McMahon’s expansion [6, eqn. (10.21.19)] j(ν, k) = (k + ν/2 − 1/4)π + O(k −1 ), k → ∞.
(2.4)
4
K. Driver and M.E. Muldoon / Journal of Approximation Theory 248 (2019) 105303
Then, as k → ∞ 0 = j(ν, k + 1) − j(ν + tk , k) = (1 − tk /2)π + O(k −1 ),
(2.5)
so tk → 2 as k → ∞. The increase of the zeros as functions of t shows that, for 0 < t < 2, (2.3) can be extended to j(ν, 1) < j(ν + t, 1) < j(ν + 2, 1) < j(ν, 2) < j(ν + t, 2) < j(ν + 2, 2) < · · · .
(2.6)
This completes the proof of Theorem 2.2. 3. Proof of Theorem 1.1 The first statement of Theorem 1.1 was proved in [2, Theorem 2.3]. Fig. 1 illustrates the interlacing in the case n = 4 and illustrates the important role played by the location of points (α+t) at which zeros of L (α) (x) coincide. n (x) and L n (α+t) To show that the interlacing of the zeros of L (α) (x) does not extend beyond n (x) and L n t = 2 for every n ∈ N, suppose on the contrary that there is a positive number ϵ such (α+t) that the zeros of L (α) (x) interlace for t = 2 + ϵ, ϵ > 0 and hence for n (x) and L n 0 < t ≤ 2 + ϵ, n = 1, 2, . . . . Then the limit [8, Theorem 8.1.3] lim n −α L n(α) (z/n) = z −α/2 Jα (2z 1/2 )
n→∞
(3.1)
or the limit [4, p. 8] lim νλn,k (α) = j 2 (α, k),
n→∞
(3.2)
where ν = 4n + α + 2; λn,k (α), k = 1, 2, . . . , n are the zeros of L (α) n (x); and j(α, k), k = 1, 2, . . . , n are the zeros of Jα (x) shows that, for the Bessel function zeros, we get the weak interlacing property j 2 (α, 1) < j 2 (α + t, 1) ≤ j 2 (α, 2) < j 2 (α + t, 2) ≤ j 2 (α, 3) < . . . ,
(3.3)
for 0 < t ≤ 2 + ϵ where we have to use ≤ in the even inequalities because of the limit process. The odd inequalities are strict because of the increase of j(α + t, k) with t. From (3.3) we have j(α, k) < j(α + t, k) ≤ j(α, k + 1) < j(α + t, k + 1), k = 1, 2, 3, . . . ,
(3.4)
for each t with 0 < t ≤ 2 + ϵ. We recall that tk is the unique value of t for which j(α + t, k) = j(α, k + 1) and we proved in Theorem 2.2 that tk decreases with k and tk → 2 as k → ∞. It follows that we may choose k sufficiently large so that 2 < tk+1 < tk < 2 + ϵ and we can choose t such that 2 < tk+1 < t < tk < 2 + ϵ. Then, for these choices of k ∈ N and t, we have j(α, k) < j(α, k + 1) < j(α + t, k), which contradicts (3.4). This concludes the proof of Theorem 1.1. 4. Additional sharpness questions An interesting discussion on the sharpness of parameter intervals within which the zeros of Jacobi polynomials of the same or consecutive degree are interlacing can be found in [1]. In [1, Section 5], Dimitrov, Ismail and Rafaeli conjecture that if α and β are large fixed positive (α,β) (α,β+τ ) numbers, there exists ϵ = ϵ(α, β) > 0 such that the zeros of Pn (x) and Pn (x) are
K. Driver and M.E. Muldoon / Journal of Approximation Theory 248 (2019) 105303
5
interlacing for all n ∈ N and for every τ ∈ (0, 2 + ϵ). In the context of Laguerre polynomials, their conjecture would suggest that if α is a large, fixed number, there exists ϵ = ϵ(α) > 0 such (α+τ ) that the zeros of L (α) (x) are interlacing for all n ∈ N and for every τ ∈ (0, 2+ϵ). n (x) and L n Dedication This paper is dedicated to a wonderful man and mathematician, my friend and co-author Martin E. Muldoon, 28 February 1939–1 August 2019, who sadly passed away prior to the publication of this article. Acknowledgements Many thanks to the referees whose suggestions have improved the paper in several ways. Kathy Driver’s research is supported by the National Research Foundation of South Africa under Grant Number 466367. References [1] D.K. Dimitrov, M.E.H. Ismail, F.R. Rafaeli, Interlacing of zeros of orthogonal polynomials under modification of the measure, J. Approx. Theory 175 (2013) 64–76. [2] K. Driver, K. Jordaan, Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials, Numer. Math. 107 (2007) 615–624. [3] K. Driver, M.E. Muldoon, Common and interlacing zeros of families of Laguerre polynomials, J. Approx. Theory 193 (2015) 89–98. [4] L. Gatteschi, Asymptotics and bounds for the zeros of laguerre polynomials: a survey, J. Comput. Appl. Math. 14 (2002) 7–27. [5] L. Lorch, Elementary comparison techniques for certain classes of Sturm–Liouville equations, in: G. Berg, M. Essén, Å. Pleijel (Eds.), Differential Equations (Proc. Internat. Conf. Uppsala, 1977), Almqvist & Wiksell, Stockholm, 1977, pp. 125–133. [6] F.W.J. Olver, et al. (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, dlmf.nist.gov. [7] T. Pálmai, On the interlacing of cylinder functions, Math. Inequal. Appl. 16 (2013) 241–247. [8] G. Szeg˝o, Orthogonal Polynomials, Vol. 23, forth ed., American Mathematical Society Colloquium Publications, 1975. [9] G.N. Watson, A Treatise on the Theory of Bessel Functions, second ed., Cambridge University Press, 1944.