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Journal of Computational and Applied. Mathematics journal homepage: www.elsevier.com/locate/cam. Monotonicity of zeros of Laguerre polynomials. Dimitar K.
Journal of Computational and Applied Mathematics 233 (2009) 699–702

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Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam

Monotonicity of zeros of Laguerre polynomials Dimitar K. Dimitrov a,∗ , Fernando R. Rafaeli b a

Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, Brazil

b

Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Brazil

article

abstract

info

(α)

Denote by xnk (α), k = 1, . . . , n, the zeros of the Laguerre polynomial Ln (x). We establish monotonicity with respect to the parameter α of certain functions involving xnk (α). As a (α) consequence we obtain sharp upper bounds for the largest zero of Ln (x). © 2009 Elsevier B.V. All rights reserved.

Article history: Received 31 October 2007 Keywords: Zeros Laguerre polynomials Monotonicity

1. Introduction and statement of results The zeros of the classical orthogonal polynomials have been studied since 1886 when Markov [1] and Stieltjes [2] published their thorough pioneering papers. Topic of special interest is the behaviour of the zeros of Jacobi, Gegenbauer and Laguerre polynomials as functions of the corresponding parameters. The monotonicity of these zeros is intuitively clear from the beautiful electrostatic interpretation they possess (see [3, Section 6.83]). The facts that the zeros of the Jacobi polynomial (α,β) Pn (x) are increasing functions of β and decreasing functions of α and that the positive zeros of the Gegenbauer polynomial Cnλ (x) are decreasing functions of λ were formally established by Markov and Stieltjes, respectively. In what follows we assume that the zeros of the orthogonal polynomials under discussion are arranged in decreasing order. Then the natural question of studying the quantitative bahaviour of those zeros arose. An interesting approach to this problem was suggested by Laforgia who proved in [4] that the products λxn,k (λ), k = 1, . . . , [n/2], increase for λ ∈ (0, 1) and conjectured in [5] that the same holds for all λ > 0. Thus, instead of trying to directly investigate the speed of the corresponding zeros, the question of determining the extremal function f (λ) that forces the functions f (λ)xn,k (λ), k = 1, . . . , [n/2], to increase was naturally posed. Ismail and Letessier [6] and later Askey (see [7]) conjectured √ that an appropriate choice of the above function could be f (λ) = λ + 1. After various partial contributions [8–12] Elbert and Siafarikas [13]extended a previous result of Ahmed, Muldoon and Spigler [9] proving that, for any n ∈ N and k = 1, . . . , [n/2],

(λ + (2n2 + 1)/(4n + 2))1/2 xnk (λ) increase for λ > −1/2,

(1.1)

thus settling the conjecture of Ismail, Letessier and Askey. (α,β) A similar result about the zeros xnk (α, β) of Pn (x) was obtained recently in [14]. It was proved that



β+

2n2 + (2n + 1)(α + 1) 2n + α + 1



(1 − xnk (α, β)) increase for β ∈ (−1, ∞),

where the latter product is considered as a function of β .



Corresponding author. E-mail address: [email protected] (D.K. Dimitrov).

0377-0427/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2009.02.038

(1.2)

700

D.K. Dimitrov, F.R. Rafaeli / Journal of Computational and Applied Mathematics 233 (2009) 699–702

The sharpness of (1.1) and (1.2) was established in [15] and [14], respectively. √ It is worth mentioning that (1.1) and (1.2) were motivated by the asymptotic formulae limλ→∞ λxnk (λ) = hnk , where hnk denote the zeros of the Hermite polynomial Hn (x), and by limβ→∞ β(1−xnk (α, β)) = 2xn,n−k+1 (α), respectively. Observe that these formulae show that the functions (1.1) and (1.2) possess horizontal asymptotes when the variables λ and β go to infinity. (α) In this paper we study the corresponding problem for the zeros xnk (α) of the Laguerre polynomial Ln (x). Our results are inspired by the asymptotic formula xnk (α) = α +



2α hnk +

1 3

√  (1 + 2n + 2h2nk ) + O 1/ α

as α → ∞,

(1.3)

due to Calogero [16]. Note that Ifantis and Siafarikas [17] proved that xn1 (α)/(α + 1) decreases for α > −1. This and (1.3) imply that xn1 (α) > α+ 1, which is, unfortunately, a quite straightforward inequality which follows also from x11 (α) = α+ 1 and the interlacing property √of zeros of orthogonal polynomials. Natalini and Palumbo [18] proved that for any admissible n and k the functions xnk (α)/ 2n + α + 1 increase when α increases in the interval (−1, ∞). They established two additional monotonicity properties of quotients of the form xnk (α)/α p , where p is fixed, 2 ≤ p < 2n + 1. In this paper we prove the following result: Theorem 1. For every n ≥ 2 and each k, k = 1, . . . , n, the quantities xnk (α) − (2n + α − 1)



2(n + α − 1)

are increasing functions of α , for α ≥ −1/(n − 1). Moreover, when k = 1 the above function increases in the entire range α ∈ (−1, ∞). Since from (1.3) we have the asymptotics xnk (α) − (2n + α − 1)



2(n + α − 1)

−→ hnk as α → ∞,

then we immediately obtain: Corollary 1. The inequalities xnk (α) ≤ 2n + α − 1 +

p

2(n + α − 1)hnk ,

hold for every n ≥ 2 and k = 1, . . . , n, and for any α ≥ −1/(n − 1). Moreover, xn1 (α) ≤ 2n + α − 1 +

p

2(n + α − 1)hn1 ,

for all n ∈ N and α > −1. Then the upper bounds for the zeros of Hermite polynomials obtained in [19] yield: Corollary 2. The inequalities

v ( ) u r u 2 π π xn1 (α) ≤ 2n + α − 1 + t2(n + α − 1) n − 2 + 1 + (n − 2)2 cos2 cos n+2 2n hold for every positive integer n ≥ 2 and α > −1. Moreover, if n ≥ 3 is odd, then the sharper inequalities

v ( ) u r u 2π π t 2 xn1 (α) ≤ 2n + α − 1 + 2(n + α − 1) n − 2 + 1 + (n − 1)(n − 3) cos cos n+1 2n hold.

√ It was proved in [20] that hn1 ≤

2n − 4 when n ≥ 2. This implies:

Corollary 3. The inequalities

p

xn1 (α) ≤ 2n + α − 1 + 2 (n − 2)(n + α − 1) hold for every n ≥ 2 and for any α > −1. This upper bound is sharper than the one obtained in [21].

D.K. Dimitrov, F.R. Rafaeli / Journal of Computational and Applied Mathematics 233 (2009) 699–702

701

2. Proof of the main result and a conjecture Let {pj (x; τ )} be a sequence of orthonormal polynomials generated by the recurrence relation p−1 (x; τ ) = 0, p0 (x; τ ) = 1,

(2.1)

xpj (x; τ ) = aj (τ )pj+1 (x; τ ) + bj (τ )pj (x; τ ) + aj−1 (τ )pj−1 (x; τ ), j = 0, 1, . . . , where with aj (τ ) > 0 and bj (τ ) are smooth functions of τ in (τ1 , τ2 ). Then the Perron–Frobenius theorem (see Theorem 8.4.5 in Horn and Johnson [22]) yields that the largest zero ξn1 (τ ) of pn (x; τ ) is an increasing function of τ ∈ (τ1 , τ2 ) if b0j (τ ) ≥ 0, j = 0, . . . , n − 1, and a0j (τ ) ≥ 0, j = 0, . . . , n − 2, for τ ∈ (τ1 , τ2 ). A result which guarantees that all the zeros ξnk (τ ), k = 1, . . . , n, of pn (x; τ ) are increasing functions of τ ∈ (τ1 , τ2 ) can be obtained by combining the Hellmann–Feynman theorem (see [23]), the Wall–Wetzel theorem [24] and the fact that the constant sequence whose terms are all equal to a quarter is a chain sequence. The result can be formulated as follows: Theorem A. Let the orthonormal polynomials pj (x; τ ) be generated by (2.1). Then all the zeros ξnk (τ ), k = 1, . . . , n, of pn (x; τ ) are increasing functions of τ ∈ (τ1 , τ2 ) if

[a0j−1 (τ )]2 0 bj−1 (τ )b0j (τ )



1 4

,

for τ ∈ (τ1 , τ2 )

and

j = 1, . . . , n − 1.

Then, to prove our main result, recall first the recurrence relation for the orthonormal Laguerre polynomials in the symmetric form L−1 (x; α) = 0,

L0 (x; α) = 1,

xLj (x; α) = aj (α)Lj+1 (x; α) + bj (α)Lj (x; α) + aj−1 (α)Lj−1 (x; α),

(2.2)

where bj (α) = 2j + α + 1,

j = 0, 1, . . . ,

and aj−1 (α) =

p

j(j + α) > 0,

j = 1, 2, . . . .

Motivated by Calogero’s asymptotic formula (1.3), our principal idea is to determine the best constants c, d1 and d2 such that the quantities znk (α) =

xnk (α) − (2n + α + c )



d1 α + d2

,

k = 1, . . . , n,

(2.3)



converge monotonically, as functions of α , to 2hnk . √ Performing the change x = d1 α + d2 z + 2n + α + c in (2.2) we obtain the new sequence of orthogonal polynomials {e Lj (z )}, j = 1, . . . , n, generated by the recurrence relation

e L−1 (z ) = 0,

e L0 (z ) = 1,

ze Lj (z ) = e aje Lj+1 (z ) + e bje Lj (z ) +e aj−1e Lj−1 (z ),

j = 0, 1, . . . , n − 1,

where 2(j − n) + 1 − c e bj = , j = 0, 1, . . . , n − 1, √ d1 α + d2 s j(j + α) e aj − 1 = , j = 1, 2, . . . , n − 1, d1 α + d2 and such that the zeros of e Ln (z ) are the quantities znk (α), given in (2.3). Observe that

∂e bj d1 {2(n − j) + c − 1} e b0j = = , ∂α 2(d1 α + d2 )3/2 and

e a0j−1 =

j = 0, 1, . . . , n − 1,

√ j(d2 − jd1 ) ∂e aj − 1 = , ∂α 2(j + α)1/2 (d1 α + d2 )3/2

j = 1, . . . , n − 1.

(2.4)

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D.K. Dimitrov, F.R. Rafaeli / Journal of Computational and Applied Mathematics 233 (2009) 699–702

If we choose c = −1, d1 = 1 and d2 = n − 1, then obviously e b0j ≥ 0 and e a0j−1 ≥ 0 when α > −1 and j ≤ n − 1. Then the (α)

Perron–Frobenius theorem implies the statement of our theorem for the largest zero of Ln (x). Then again for c = −1, d1 = 1, d2 = n − 1 and j = 1, . . . , n − 1, we see that the requirements 4[˜a0j−1 ]2 ≤ b˜ 0j−1 b˜ 0j of Theorem A are equivalent to α ≥ −j/(n − j) and the latter obviously holds for j = 1, . . . , n − 1, provided α ≥ −1/(n − 1). This completes the proof of the theorem. It seems that the correct approach to the question of the quantitative behaviour of the zeros of Laguerre polynomials is to establish sharp monotonicity properties based on the asymptotic formula (1.3). We have performed a vast amount of numerical experiments so we dare to state the following: Conjecture 1. We conjecture that for every n ≥ 2, and k = 1, . . . , n, the quantities xnk (α) − α + 13 (1 + 2n + h2nk )







2(α + n − 1)hnk

are decreasing functions while xnk (α) − α + 13 (1 + 2n + h2nk )







2(α + n + 1)hnk

are increasing functions of α ∈ (−1, ∞), with the only exception when n = 2m + 1 is odd and k = m, when these functions are not defined. Acknowledgements The first author’s research was supported by the Brazilian Science Foundations CNPq under Grant 304830/2006-2 and FAPESP under Grant 03/01874-2. The second author was supported by the Brazilian Science Foundation FAPESP under Grant 07/02854-6. References [1] [2] [3] [4] [5]

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

A. Markov, Sur les racines de certaines équations (second note), Math. Ann. 27 (1886) 177–182. T.J. Stieltjes, Sur les racines de l’equation Xn = 0, Acta Math. 9 (1886) 385–400. G. Szegő, Orthogonal Polynomials, 4th edition, Amer. Math. Soc. Coll. Publ., vol. 23, Providence, RI, 1975. A. Laforgia, A monotonic property for the zeros of ultraspherical polynomials, Proc. Amer. Math. Soc. 83 (1981) 757–758. A. Laforgia, Monotonicity properties for the zeros of orthogonal polynomials and Bessel function, in: Polynomes Orthogonaux et Applications, Proceedings of the Laguerre Symposium, Bar-de-Duk, Spain, 1984, in: Lecture Notes in Mathematics, vol. 1171, Springer-Verlag, Berlin, 1985, pp. 267–277. M.E.H. Ismail, J. Letessier, Monotonicity of zeros of ultraspherical polynomials, in: M. Alfaro, J.S. Dehesa, F.J. Marcellán, J.L. Rubio de Francia, J. Vinuesa (Eds.), Orthogonal Polynomials and Their Applications, in: Lecture Notes in Mathematics, vol. 1329, Springer-Verlag, Berlin, 1988, pp. 329–330. M.E.H. Ismail, Monotonicity of zeros of orthogonal polynomials, in: D. Stanton (Ed.), q-Series and Partitions, Springer-Verlag, New York, 1989, pp. 177–190. R. Spigler, On the monotonic variation of the zeros of ultraspherical polynomials with the parameter, Canad. Math. Bull. 27 (1984) 472–477. S. Ahmed, M.E. Muldoon, R. Spigler, Inequalities and numerical bound for zeros of ultraspherical polynomials, SIAM J. Math. Anal 17 (1986) 1000–1007. A. Elbert, A. Laforgia, A note to the paper of Ahmed Muldoon Spigler, SIAM J. Math. Anal. 17 (1986) 1008–1009. E.K. Ifantis, P.D. Siafarikas, Differential inequalities on the greatest zero of Laguerre and ultraspherical polynomials, in: Actas del VI Simposium sobre Polinomios Ortogonales y Aplicaciones, Gijon, 1989, pp.187–197. D.K. Dimitrov, On a conjecture concerning monotonicity of zeros of ultraspherical polynomials, J. Approx. Theory 85 (1996) 88–97. A. Elbert, P.D. Siafarikas, Monotonicity properties of the zeros of ultraspherical polynomials, J. Approx. Theory 97 (1999) 31–39. D.K. Dimitrov, F.R. Rafaeli, Monotonicity of zeros of Jacobi polynomials, J. Approx. Theory 149 (2007) 15–29. D.K. Dimitrov, R.O. Rodrigues, On the behaviour of zeros of Jacobi polynomials, J. Approx. Theory 116 (2002) 224–239. (α) F. Calogero, Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomials Ln (x) as the index α → ∞ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials, Nuovo Cimento 23 (1978) 101–102. E.K. Ifantis, P.D. Siafarikas, Differential inequalities and monotonicity properties of the zeros of associated Laguerre and Hermite polynomials, Ann. Num. Math. 2 (1995) 79–91. P. Natalini, B. Palumbo, Some monotonicity results on the zeros of the generalized Laguerre polynomials, J. Comput. Appl. Math. 153 (2003) 355–360. I. Area, D.K. Dimitrov, E. Godoy, A. Ronveaux, Zeros of Gegenbauer and Hermite polynomials and connection coefficients, Math. Comp. 73 (2004) 1937–1951. D.K. Dimitrov, G. Nikolov, Sharp bounds for the extreme zeros of classical orthogonal polynomials (manuscript). M.E.H. Ismail, X. Li, Bound on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 115 (1992) 131–140. R.A. Horn, C.A. Johnson, Matrix Analysis, Cambridge Univ. Press, 1985. M.E.H. Ismail, M.E. Muldoon, A discrete approach to monotonicity of zeros of orthogonal polynomials, Trans. of the Amer. Math. Soc. 323 (1991) 65–78. H.S. Wall, M. Wetzel, Quadratic forms and convergence regions for continued fractions, Duke Math. J. 11 (1944) 272–297.