ORTHOGONAL POLYNOMIALS

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ORTHOGONAL POLYNOMIALS

PLAMEN SIMEONOV Received 28 April 2003

We obtain the zero distribution of sequences of classical orthogonal polynomials associated with Jacobi, Laguerre, and Hermite weights. We show that the limit measure is the extremal measure associated with the corresponding weight.

1. Introduction

In this paper, we study the zero distribution of sequences of Jacobi, Laguerre, and Hermite polynomials. Our approach is based on identifying these orthogonal polynomials with certain Fekete polynomials defined below, and using monotonicity properties of the zeros of the polynomials.

LetE⊂Rbe a closed set that consists of finitely many intervals. Letw:E→ [0,∞) be a weight function such thatw(x)>0,x∈Int(E), and|x|w(x)→0 as

|x| → ∞,x∈E, ifEis unbounded. Consider the function V_nx1,...,x_n:=

1_≤i< j≤n

wx_iwx_jx_j−x_i, (1.1)

{xi}ⁿ_i₌1⊂E. It can be shown thatVn attains its maximum for some setᏲn= {xi}ⁿ_i₌1⊂Ecallednth weighted Fekete set or simply Fekete set.

We introduce the following notation: ifµis a measure, its logarithmic poten- tialU^µ(z) is defined by

U^µ(z) :=

log 1

|z−t|dµ(t), (1.2)

and if w is a weight as defined above, µw denotes the corresponding extremal measure [4], which is the unique measure that minimizes the weighted

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logarithmic energy Iw(µ) :=

log 1

w(z)w(t)|z−t|dµ(z)dµ(t) (1.3) over all probability Borel measures supported onE. The support of the measure µwwill be denoted bySw.

The asymptotic distribution of Fekete points is known (see [4, Chapter III, Theorem 1.3]).

Theorem1.1. LetνᏲn be the discrete measure that has mass 1/nat each Fekete pointxi∈Ᏺn. Then,

νᏲn

w^∗

−−→µ_w, n−→ ∞, (1.4)

that is,lim_n_→∞νᏲn=µwin the weak-star topology of measures. Furthermore, ifFn

is thenth degree monic polynomial with zero setᏲn,

nlim→∞F_n(z)¹^/n=exp−U^µ^w(z) (1.5) uniformly on compact subsets of C\S_w.

We will assume thatw(x)=0 whenx∈E\Int(E) andxis finite. This condi- tion implies that every Fekete setᏲn⊂Int(E). Consequently, the partial deriva- tives of log(V_n²) vanish at the Fekete points:

d dx_i

logV_n²=2(n−1)wx_i wx_i + 2

k=i

1 x_i−x_k

=2(n−1)wxi

wxi +F_nxi

F_nxi =0, i=1,...,n.

(1.6)

InSection 2, we study the zero distribution of Jacobi polynomialsP⁽n^αⁿ^,βⁿ⁾with parametersαn>0 andβn>0 that satisfy lim_n_→∞αn/n=2α>0 and limn→∞βn/n= 2β >0.

InSection 3, we consider Laguerre polynomialsL⁽n^αⁿ⁾with parametersα_n>0 that satisfy lim_n_→∞αn=2α >0.

InSection 4, we obtain the zero distribution of the Hermite polynomialsHn. Asymptotics and zero distribution of classical orthogonal polynomials have been studied in [1,2,3,5]. Here, we extend these results using a simple method that works for all classical orthogonal polynomials.

2. Zero distribution of Jacobi polynomials The Jacobi weightw_α,β(x) is defined by

wα,β(x)=(1−x)^α(1 +x)^β, x∈[−1,1], (2.1)

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with positiveαandβ. The corresponding extremal measure is given by [4, Chap- ter IV, Section 5]

dµ_wα,β(t)= 1 π

(1 +α+β) 1−t²

(t−a)(b−t)¹^/²dt, t∈S_wα,β, (2.2) with support [4, Chapter IV, Section 1]

Swα,β=[a,b]=

θ2²−θ²1− ¹^/²,θ²2−θ²1+ ¹^/², (2.3) whereθ1=α/(1 +α+β),θ2=β/(1 +α+β), and =(1−(θ1+θ2)²)(1−(θ1− θ2)²).

LetP⁽n^α,β⁾andqn,α,βdenote the orthonormal polynomial of degreenand the monic orthogonal polynomial of degree n, respectively, with respect to the weightw_α,β. Let

νn,α,β:=1 n

x:Pn^(α,β)(x)=0

δ(x) (2.4)

denote the discrete probability measure with mass 1/n at each zero of Pn⁽^α,β⁾. Here,δ(x) denotes the discrete probability measure with supportx(the point mass atx).

We first show that the Fekete polynomials for Jacobi weightsw_α,βwithα >0 andβ >0 are, in fact, Jacobi polynomials.

Letα >0 andβ >0 be fixed and setw=w¹_α,β^/⁽ⁿ⁻¹⁾in the functionV_ndefined with (1.1). Since

w(x) w(x) ⁼

1 (n−1)

w_α,β(x) wα,β(x)⁼

1 (n−1)

β−α−(α+β)x

1−x² , x∈(−1,1), (2.5) equations (1.6) yield

2β−α−(α+β)x_iF_nx_i+1−x_i²F_nx_i=0, i=1,...,n. (2.6) Thus, the polynomial (1−x²)F_n(x) + 2(β−α−(α+β)x)F_n(x) of degreenwith leading coeﬃcient−n(n+ 2α+ 2β−1) has zeros atx1,...,x_n, and therefore

1−x²F_n(x) + 2β−α−(α+β)xF_n(x) +n(n+ 2α+ 2β−1)F_n(x)=0.

(2.7) By [6, Theorem 4.2.1], the polynomialq_n,2α−1,2β−1 satisfies (2.7) as well. How- ever, (2.7) has a unique monic polynomial solution of degreen. Indeed, the polynomial

r:=F_n−q_n,2α−1,2β−1=ⁿ⁻ 1

j=0

c_jq_j,2α−1,2β−1 (2.8)

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satisfies (2.7). Substitutingrin (2.7), we obtain 0=

1−x²r(x) + 2β−α−(α+β)xr(x) +n(n+ 2α+ 2β−1)r(x)

=ⁿ⁻ 1

j=0

c_j 1−x²q_j,₂_α₋₁_,₂_β₋₁(x) + 2β−α−(α+β)xq_j,₂_α₋₁_,₂_β₋₁(x) +n(n+ 2α+ 2β−1)qj,2α−1,2β−1(x)

=

n−1 j=0

cj

−j²+n²+ (n−j)(2α+ 2β−1)qj,2α−1,2β−1(x),

(2.9)

where (2.7) was applied toqj,2α−1,2β−1,j=0,...,n−1. Since (n−j)(n+j+ 2α+ 2β−1)>0 for j=0,...,n−1, (2.9) implies cj=0, j=0,...,n−1, and the uniqueness of the polynomial solution of (2.7) follows.

We have shown that for positiveαandβ, thenth Fekete polynomialFn,α,βasso- ciated with the Jacobi weightwα,βis the Jacoby polynomialqn,2(n−1)α−1,2(n−1)β−1. Theorem2.1. Let{αn}and{βn}be sequences of positive numbers satisfying

αn

n ^−→2α >0, βn

n ^−→2β >0, n−→ ∞. (2.10) Ifαandβare finite, then

νn,αn,βn

w^∗

−−→µα,β, n−→ ∞. (2.11)

Ifα= ∞andβis finite, the limit of the measuresνn,αn,βnis the point mass at−1.

Ifαis finite andβ= ∞, the limit of the measuresνn,αn,βnis the point mass at1.

Ifα=β= ∞andαn/βn→λ >0asn→ ∞, the limit measure is the point mass at(1−λ)/(1 +λ).

Proof. For fixedα >0 andβ >0, let{x⁽_i,n^α,β⁾}ⁿ_i₌1be thenth Fekete set, and letνn,α,β

denote the discrete probability measure having mass 1/nat each Fekete point x_i,n⁽^α,β⁾. ByTheorem 1.1,

νn,α,β−−→w^∗ µwα,β, n−→ ∞. (2.12)

From (2.10), it follows that α˜n:= α_n+ 1

2(n−1)^−→α, β˜n:= βn+ 1

2(n−1)^−→β, n−→ ∞. (2.13) Furthermore,

F_n,α˜n,β˜n=q_n,αn,βn. (2.14) Assume first thatαandβare both finite. Let>0 be fixed and letN() be such thatα−≤α˜_n≤α+andβ−≤β˜_n≤β+forn≥N(). We will use

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a certain monotonicity property of the zeros of the Jacobi polynomials. For 0<

α1< α2and 0< β1< β2, w_α1,β(x)

w_α2,β(x)⁼(1−x)^α¹⁻^α², w_α,β2(x)

w_a,β1(x)⁼(1 +x)^β²⁻^β¹ (2.15) are increasing functions on (−1,1). By [6, Theorem 6.12.2],

x⁽_j,n^α²^,β⁾< x⁽_j,n^α¹^,β⁾, x⁽_j,n^α,β¹⁾< x⁽_j,n^α,β²⁾, j=1,...,n. (2.16) Therefore,

x⁽_j,n^α⁺^,β⁻⁾< x^(˜_j,n^αⁿ^,^β^˜ⁿ⁾< x⁽_j,n^α⁻^,β⁺⁾, j=1,...,n. (2.17) LetA⊂Swα,βbe an interval. We have

ν_n,αñ,βñ(A)−µwα,β(A)≤ν_n,αñ,βñ(A)−νn,α,β(A)+νn,α,β(A)−µwα,β(A). (2.18) In view of (2.12), it is enough to estimate the first term in (2.18). For any mea- surable setBand fixedα0>0 andβ0>0, from (2.2) and (2.12), it follows that

νn,α,β(B)−νn,α0,β0(B)

≤νn,α,β(B)−µwα,β(B)+µwα,β(B)−µwα0,β0(B) +νn,α0,β0(B)−µ_wα0,β0(B)−→0

(2.19)

if we letn→ ∞first, and thenα→α0andβ→β0. Next, define

J_n,α,β^L (a) :=maxj:x⁽_j,n^α,β⁾< a, J_n,α,β^R (a) :=minj:x⁽_j,n^α,β⁾> a. (2.20) LetA=[c,d]. By (2.19),

n⁻¹J_n,α^L _±_,β_±(c)−J_n,α,β^L (c)=νn,α±,β±−νn,α,β

(−∞,c)−→0 (2.21) asn→ ∞first, and then→0. Similarly,

n⁻¹J_n,α^R _±_,β_±(d)−J_n,α,β^R (d)−→0, n−→ ∞,−→0. (2.22) Furthermore, (2.17) implies

J_n,α^L ₋_,β+(c)≤J_n,^L_α_˜

n,β˜n(c)≤J_n,α^L +,β−(c),

J_n,α^R ₋_,β+(d)≤J_n,^R_α_˜_n_,β˜n(d)≤J_n,α^R +,β−(d). (2.23)

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From (2.21), (2.22), and (2.23) it follows that

ν_n,α˜n,β˜n(A)−νn,α,β(A)−→0, n−→ ∞, (2.24)

and this completes the proof for finiteαandβ.

Ifαis finite andβ= ∞,βis finite andα= ∞, orαandβare both infinite, and αn/βn→λ≥0 asn→ ∞, it immediately follows from (2.3) that the supports of the extremal measuresSwαn,βnshrink to the single point 1,−1, or (1−λ)/(1 +λ), respectively, which establishes the proof in these cases.

3. Zero distribution of Laguerre polynomials

Let L⁽n^α⁾(x) denote the monic Laguerre polynomials that are orthogonal with respect to the Laguerre weightwα(x)=x^αe⁻^xon [0,∞), whenα >−1. Further- more,y=L⁽n^α⁾is the unique polynomial solution of degreenof the diﬀerential equation

xy+ (α+ 1−x)y+ny=0. (3.1) Whenα >0, the extremal measureµ_wαis given by (see [4, Chapter IV, Section 5])

dµwα(t)= 1 πt

t−aα

bα−t¹^/²dt, t∈Swα, (3.2) where (see [4, Chapter IV, Section 1])

Swα= aα,bα

=

α+ 1−(2α+ 1)¹^/²,α+ 1 + (2α+ 1)¹^/². (3.3) To show that the Fekete polynomials for Laguerre weightsw_αwithα >0 are Laguerre polynomials, we set w=wα in (1.1). Since w(x)/w(x)=(α/x−1), (1.6) takes the form

xiF_nxi

+ 2(n−1)α−xi F_nxi

=0, i=1,...,n, (3.4) where Fn=Fn,α is thenth Fekete polynomial for the weight wα. Since 2(n− 1)(α−x)F_n(x) +xF_n(x) is a polynomial of degree n with leading coeﬃcient

−2n(n−1), the above equations imply thatz=Fnsatisfies the diﬀerential equation

tz+ 2(n−1)(α−t)z+ 2n(n−1)z=0. (3.5) Settingz(t)=y(x) withx=λt, we getd^kz/dt^k=λ^kd^ky/dx^kfor everyk≥0, and (3.5) becomes

λxy+ 2(n−1)(λα−x)y+ 2n(n−1)y=0. (3.6)

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Choosingλ=2(n−1), we obtain

xy+2(n−1)α−xy+ny=0. (3.7) From (3.1) and (3.7) it follows thaty(x)=L⁽²⁽n ⁿ⁻¹⁾^α⁻¹⁾(x). SinceF_n(t)=z(t)= y(2(n−1)t) we obtain

F_n,α(x)=L⁽²⁽_n ⁿ⁻¹⁾^α⁻¹⁾2(n−1)x. (3.8) Equation (3.8) shows that for every n≥1 there is a unique nth Fekete set {x_i,n⁽^α⁾}ⁿ_i₌1, and if{z_i,n⁽^γ⁾}ⁿ_i₌1denotes the zero set of the Laguerre polynomialL⁽n^γ⁾

withγ >0, then

x⁽_i,n^α⁾=z⁽²⁽_i,nⁿ⁻¹⁾^α⁻¹⁾

2(n−1) , i=1,...,n, (3.9)

where both the zeros of the Laguerre polynomial and the Fekete points are ar- ranged in increasing order.

Next, we show that the Fekete sets for a weightw_γwithγ >0 are contained in a compact set. By [4, Chapter I, Theorem 1.3],

U^µ^wγ(x)−logwγ(x)=Fwγ, x∈Swγ, (3.10) whereF_wγis a constant. Furthermore, by [7, Theorem A],U^µ^wγ(x)−logw_γ(x)≥ Fwγ,x /∈Swγ. This function is then continuously diﬀerentiable on (0,∞)\ {aγ, bγ}, its first derivative vanishes on (aγ,bγ), and

d² dx²

U^µ^wγ(x)−logw_γ(x)= _b_γ

aγ

1

(x−t)²dµ_wγ(t) + γ

x² >0, x > b_γ. (3.11) Thus, the first derivative of U^µ^wγ(x)−logwγ(x) is positive for x > bγ, and so U^µ^wγ(x)−logwγ(x)> Fwγforx > bγ. Therefore,

S^∗_w_γ:=

x:U^µ^wγ(x)−logw_γ(x)≤F_wγ

⊂

0,b_γ. (3.12) By [4, Chapter III, Theorem 1.2],{x⁽_i,n^γ⁾}ⁿ_i₌1⊂S^∗_w_γ. Thus, we conclude that{x_i,n⁽^γ⁾}ⁿ_i₌1

⊂[0,b_γ] for everyn.

Theorem3.1. Let{α_n} ⊂(0,∞)be a sequence satisfyingα_n/n→2α >0asn→ ∞. Then,

νn,αn:=1 n

n i=1

δ z_i,n⁽^αⁿ⁾

2(n−1)

w^∗

−−→µwα, n−→ ∞. (3.13)

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Proof. We have ˜α_n:=(α_n+ 1)/(2(n−1))→α >0 asn→ ∞. By (3.9),z⁽_i,n^αⁿ⁾/(2(n− 1))=x^(˜_i,n^αⁿ⁾,i=1,...,n, and by [4, Chapter III, Theorem 1.3],

1 n

n i=1

δx_i,n⁽^α⁾−−→^w^∗ µ_wα, n−→ ∞. (3.14) The rest of the proof follows the argument used in the proof ofTheorem 2.1. In this case, the zeros of the Laguerre polynomialsL^(α)n are monotone in the sense that ifα1> α2>−1, thenz⁽_i,n^α²⁾< z⁽_i,n^α¹⁾,i=1,...,n. This follows from the fact that wα1(x)/wα2(x)=x^α¹⁻^α²is an increasing function on [0,∞), and a variation of [6,

Theorem 6.12.2] for unbounded intervals.

4. Zero distribution of the Hermite polynomials

The monic Hermite polynomialsHnare orthogonal with respect to the weight w(x)=e⁻^x²,x∈R. Furthermore,y=Hnsatisfies the diﬀerential equation

y−2xy+ 2ny=0, n≥0. (4.1)

The corresponding extremal measureµw is given by (see [4, Chapter IV, Theo- rem 5.1]),

dµw(t)= 2 π

1−t²dt, t∈[−1,1]. (4.2)

To determine the relationship between the zeros of the Hermite polynomials and the Fekete sets for the weightw(x)=e⁻^x², we setw(x)=e⁻^x²in (1.1). Since w(x)/w(x)= −2x, (1.6) yields

4(n−1)xiF_nxi

−F_nxi

=0, i=1,...,n. (4.3) These equations imply that thenth degree polynomial 4(n−1)xF_n(x)−F_n(x) with leading coeﬃcient 4n(n−1) has the same zero set asFn(x). Therefore,Fn(x) is the polynomial solution of the diﬀerential equation

z−4(n−1)xz+ 4n(n−1)z=0. (4.4) Forλ >0, we sety(x)=z(λx). From (4.4), it follows that

y−4(n−1)λ²xy+ 4λ²n(n−1)y=0, (4.5) and in particulary(x)=F_n(x/2(n−1)) satisfies (4.1). Since (4.1) has a unique polynomial solution of degreen, we obtainFn(x/2(n−1))=Hn(x). Then, if {xi,n}ⁿ_i₌1and{zi,n}ⁿ_i₌1denote the zeros ofFnandHn, respectively, we have

xi,n= z_i,n

2(n−1), i=1,...,n. (4.6)

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From [4, Chapter IV, equation (5.5)], it follows thatS^∗_w=[−1,1], and then by [4, Chapter III, Theorem 1.2],{xi,n}ⁿ_i₌1⊂[−1,1] for everyn≥1. Using the argument employed in the previous sections, we establish the following theorem.

Theorem4.1. For everyn≥1, letνndenote the discrete probability measure hav- ing mass1/nat each zeroz_i,nof the Hermite polynomialH_n. Then,

νn:=1 n

n i=1

δ

z_i,n 2(n−1)

w^∗

−−→µw, n−→ ∞. (4.7) References

[1] L.-C. Chen and M. E. H. Ismail,On asymptotics of Jacobi polynomials, SIAM J. Math.

Anal.22(1991), no. 5, 1442–1449.

[2] M. Lachance, E. B. Saﬀ, and R. S. Varga,Bounds for incomplete polynomials vanishing at both endpoints of an interval, Constructive Approaches to Mathematical Models (Proc. Conf. in honor of R. J. Duﬃn, Pittsburgh, Pa, 1978), Academic Press, New York, 1979, pp. 421–437.

[3] D. S. Moak, E. B. Saﬀ, and R. S. Varga,On the zeros of Jacobi polynomialsPn^(αⁿ^,βⁿ⁾(x), Trans. Amer. Math. Soc.249(1979), no. 1, 159–162.

[4] E. B. Saﬀand V. Totik,Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften, vol. 316, Springer-Verlag, Berlin, 1997.

[5] E. B. Saﬀ, J. L. Ullman, and R. S. Varga,Incomplete polynomials: an electrostatics ap- proach, Approximation Theory, III (Proc. Conf., Univ. Texas, Austin, Tex, 1980), Academic Press, New York, 1980, pp. 769–782.

[6] G. Szeg˝o,Orthogonal Polynomials, 4th ed., Colloquium Publications, vol. 23, Ameri- can Mathematical Society, Rhode Island, 1975.

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Plamen Simeonov: Department of Computer and Mathematical Sciences, University of Houston-Downtown, Houston, TX 77002, USA

E-mail address:[email protected]

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Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

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