upm etsiccp-seminar-vf

Comportamiento Asintotico de Secuencias dePolinomios Ortogonales e

Interpretacion Electrostatica sus Ceros

Edmundo J. Huertas

Universidad Politecnica de Madrid - Grupo SERPA-HGA

March 12, 2015- Seminario del Departamento de Matematicas eInformatica Aplicadas a las II. Civil y Naval

(UPM 2015) Polinomios Ortogonales UPM 2015 1 / 50

Outline

1 Electrostatic Interpretation of Zeros of Orthogonal PolynomialsIntroductionBasic background for MOPS and its zerosCanonical perturbations of a measure (Christoffel, Uvarov and Geronimus)The interacting particle model (M. Ismail)Example for the Laguerre-Geronimus measure with c =−1Another example for the Uvarov modification of the Laguerre measure

2 Asymptotic Behavior of Ratios of Laguerre Orthogonal PolynomialsAsymptotics for Classical Laguerre PolynomialsMotivation of the problemAn alternative (algorithmic) approachAn expansion of 1F1(a;c;z) by BuchholzA first strong asymptotic expansion valid in the whole C

A second strong asymptotic expansion valid in C\R+

Asymptotics of ratios of Laguerre polynomials


Outline






FIRST PART OF THE TALK:

ELECTROSTATIC INTERPRETATION OF ZEROS OF ORTHOGONALPOLYNOMIALS

References:

A. Branquinho, E.J. Huertas, and F.R. Rafaeli, Zeros of orthogonal polynomialsgenerated by the Geronimus perturbation of measures, Lecture Notes inComputer Science (LNCS), 8579 (2014), 44–59.

E.J. Huertas, F. Marcellan and H. Pijeira, An electrostatic model for zeros ofperturbed Laguerre polynomials, Proceedings of the American MathematicalSociety, 142 (5) (2014), 1733–1747.

E.J. Huertas, F. Marcellan and F.R. Rafaeli, Zeros of orthogonal polynomialsgenerated by canonical perturbations of measures, Applied Mathematics andComputation, 218 (13) (2012), 7109–7127.


The work of Stieltjes.

Theorem (Stieltjes 1885-1889): Suppose n unit charges at points x1,x2, . . . ,xn aredistributed in the interval [−1,1]. The energy of the system

E(x) = E(x1,x2, . . . ,xn) =n

∑k=1

V (xn,k)− ∑1≤ j≤k≤n

ln∣

∣xn, j − xn,k∣

∣ .

The above expression becomes a minimum when x1,x2, . . . ,xn are the zeros of theJacobi polynomials P(2p−1,2q−1)

n (x)

Similar results hold for the zeros of Laguerre and Hermite polynomials.


Motivation

1. Zeros of orthogonal polynomials are the nodes of the Gaussian quadraturerules and its extensions (Gauss–Radau, Gauss–Lobatto, Gauss–Kronrodrules,...etc)

∫

f (x)dµ(x)∼n

∑k=1

λk,n f (xk,n)

2. Zeros of classical orthogonal polynomials are the electrostatic equilibriumpoints of positive unit charges interacting according to a logarithmic potentialunder the action of an external field.

3. Zeros of orthogonal polynomials are used in collocation methods for boundaryvalue problems of 2nd order linear differential operators .

4. Global properties of zeros of orthogonal polynomials can be analyzed when theysatisfy 2nd order differential equations with polynomial coefficients, using theWKB method.

5. Zeros of orthogonal polynomials are eigenvalues of Jacobi matrices and its rolein Numerical Linear Algebra is very well known.


Basic background - MOPS

Let us consider the inner product 〈·, ·〉µ : P×P→ R

〈 f ,g〉µ =∫ b

af (x)g(x)dµ(x), n ≥ 0, f ,g ∈ P,

and supp(dµ) = (a,b)⊆ R.

Let {Pn(x)}n≥0 be a Monic Orthogonal Polynomial Sequence (MOPS) with respectto the above inner product.

Three-term recurrence relation (TTRR)

xPn(x) = Pn+1(x)+βnPn(x)+ γnPn−1(x), n ≥ 0,

with P−1(x) = 0, P0(x) = 1, and recurrence coefficients

βn =〈xPn,Pn〉µ

‖Pn‖2µ

, n ≥ 0 and γn =‖Pn‖2

µ

‖Pn−1‖2µ> 0, n ≥ 1.


Properties of the zeros of the MOPS

1 For each n ≥ 1, the polynomial Pn(x) has n real and simple zeros in the interior ofC0(supp(dµ)).

2 Interlacing property: The zeros of Pn+1(x) interlace with the zeros of Pn(x).

3 Between any two zeros of Pn(x) there is at least one zero of Pm(x), for m > n ≥ 2.

4 Each point of supp(dµ) attracts zeros of the MOPS. In other words, the zeros aredense in supp(dµ).


Basic background - Reproducing Kernel

nth-Kernel

Kn(x,y) =n

∑j=0

Pj(y)Pj(x)

‖Pj‖2µ

, ∀n ∈ N

Christoffel-Darboux formula

Kn(x,y) =1

‖Pn‖2µ

Pn+1(x)Pn(y)−Pn(x)Pn+1(y)x− y

, ∀n ∈ N

Confluent form of Kn

Kn(x,x) =P′

n+1(x)Pn(x)−P′n(x)Pn+1(x)

‖Pn‖2µ

, ∀n ∈ N


Christoffel perturbation of a measure dµ

Let {Pc,[1]n (x)}n≥0 be the MOPS associated with the measure

dµ [1] = (x− c)dµ ,

with (any complex or real number) c 6∈C0(supp(dµ)).

It is clear that Pn(c) 6= 0, ∀n ≥ 1.

The MOPS with respect to dµ [1] is

Pc,[1]n (x) = (x− c)−1

[

Pn+1(x)−Pn+1(c)Pn(c)

Pn(x)

]

,

and a trivial verification shows that

Pc,[1]n (x) =

‖Pn‖2µ

Pn(c)Kn(x,c).


Uvarov perturbation of a measure dµ

Let {PNn (x)}n≥0 be the MOPS associated with the measure

dµN = dµ +N δ (x− c),

with N ∈ R+, δ (x− c) the Dirac delta function in x = c, and c 6∈C0(supp(dµ)).

Connection formula for Uvarov perturbed MOPS

The polynomials {PNn (x)}n≥0, can be represented as

PNn (x) = Pn(x)−

(

NPn(c)1+NKn−1(c,c)

)

Kn−1(c,x).


Modification of dµ by a linear divisor

Let {Qcn(x)}n≥0 be the MOPS associated with the measure

dν =1

(x− c)dµ ,

with c 6∈C0(supp (dµ)), and let ycn,k := yc

n,k(c) be the zeros of Qcn(x).

The MOPS with respect to dν can be represented as

Qcn(x) = Pn(x)−

Fn(c)Fn−1(c)

Pn−1(x), n = 0,1,2, . . . ,

where Qc0(x) = 1, F−1(c) = 1.

The functions

Fn(s) =∫

E

Pn(x)x− s

dµ(x), s ∈ C�E,

are the Cauchy integrals of {Pn(x)}n≥0, or functions of the second kind.


Geronimus perturbation of a measure dµ

In the former modification by a linear divisor, we add a mass point exactly at thepoint c. Then we obtain a Geronimus transformation of the measure dµ .

Let {Qc,Nn (x)}n≥0 be the MOPS associated with the measure

dνN =1

(x− c)dµ +Nδ (x− c),

with c 6∈C0(supp (dµ)), and let yc,Nn,k := yc,N

n,k (c) be the zeros of Qc,Nn (x).

Geronimus (1940), conclude that the sequences associated to dνN must be of theform

Pn(x)+anPn−1(x), an 6= 0,

for certain numbers an ∈ R.

Maroni (1990), stated that the sequence {Pn+1(x)}n≥0, orthogonal with respect tou = δc +λ (x− c)−1L, can be represented as

Pn+1(x) = Pn+1(x)−anPn(x), n ≥ 0,

where

an =−Pn+1(c;−λ )Pn(c;−λ )

, Pn(c;−λ ) = Pn(c)+λP(1)n−1(c).


Geronimus perturbation of a measure dµ

Theorem: Connection formula 1

The monic polynomials {Qc,Nn (x)}n≥0, can be represented as

Qc,Nn (x) = Pn(x)+Λc

n(N)Pn−1(x),

with

Λcn(N) =

Pn(c)Pn−1(c)

− Fn(c)Fn−1(c)

1+NBcn

− Pn(c)Pn−1(c)

.

Theorem: Connection formula 2

The polynomials {Qc,Nn (x)}n≥0, with Qc,N

n (x) = κnQc,Nn (x), can be represented as

Qc,Nn (x) = Qc

n(x)+NBcn · (x− c)Pc,[1]

n−1 (x),

with κn = 1+NBcn, and

Bcn =

−Qcn(c)Pn−1(c)

‖Pn−1‖2µ

= Kcn−1(c,c)> 0.


Zeros of Geronimus perturbed MOPS

The mass point c attracts exactly one zero of Qc,Nn (x), when N → ∞.

When either c < a or c > b, at most one of the zeros of Qc,Nn (x) is located outside of

C0(supp (dµ)) = (a,b). In the next result, we will give explicitly the value N0 of themass such that for N > N0 one of the zeros is located outside (a,b).If C0(supp (dµ)) = (a,b) and c < a, then the largest zero yc,N

n,n satisfies

Corollary: Minimum mass, case c > b

yc,Nn,n < b, for N < N0,

yc,Nn,n = b, for N = N0,

yc,Nn,n > b, for N > N0,

with N0 = N0(n,c,b) =−Qc

n(b)

Kcn−1 (c,c)(b− c)Pc,[1]

n−1 (b)> 0.


The interacting particle model (M. Ismail)

M. E. H. Ismail, An electrostatics model for zeros of general orthogonalpolynomials, Pacific J. Math. 193, (2000), 355-369.

The model obtains the second order differential equation for Qc,Nn (x) and the total

energy at the equilibrium position of the system.

This model can be applied to MOPS which satisfies a Structure Relation as

σ(x)[Pn]′(x) = a(x,n)Pn(x)+b(x,n)Pn−1(x),

a Three Term Recurrence Relation as

xPn(x) = Pn+1+βnPn(x)+ γnPn−1(x).

Corollary: {Qc,Nn (x)}n≥0 can be also represented as

Qc,Nn (x) = Pn(x)+Λc

n(N)Pn−1(x),


Second order differential equation for Geronimus perturbed MOPS

The Geronimus perturbed MOPS {Qc,Nn (x)}n≥0 satisfies the second order linear

differential equation

[Qc,Nn (x)]′′+R(x;n)[Qc,N

n (x)]′+S (x;n)Qc,Nn (x) = 0,

(also known as the holonomic equation ), where

R(x;n) = −(

ξ c1(x;n)+ηc

2(x;n)+[ηc

1(x;n)]′

ηc1(x;n)

)

,

S (x;n) = ξ c1(x;n)ηc

2(x;n)−ηc1(x;n)ξ c

2(x;n)+ξ c

1(x;n)[ηc1(x;n)]′− [ξ c

1(x;n)]′ηc1(x;n)

ηc1(x;n)

.

In turn, for k = 1,2, the above expressions are given only in terms of Λcn(N), and the

coefficients βn, γn, σ(x), a(x;n) and b(x;n) of the three term recurrence relation and thestructure relation satisfied by {Pn(x)}n≥0:

ξ ck (x;n) =

Ck(x;n)B2(x;n)γn−1+Dk(x;n)Λcn−1(N)

∆(x;n)γn−1, ηc

k (x;n) =Dk(x;n)−Ck(x;n)Λc

n(N)

∆(x;n)γn−1,

with...


Second order differential equation for Geronimus perturbed MOPS

C1(x;n) =1

σ(x)

(

a(x;n)−Λcn(N)

b(x;n)γn−1

)

,

D1(x;n) =1

σ(x)

(

b(x;n)+Λcn(N)b(x;n−1)

(

a(x;n−1)b(x;n−1)

+(x−βn−1)

γn−1

))

,

A2(n) =−Λc

n(N)

γn−1, B2(x;n) = Λc

n−1(N)

(

1Λc

n−1(N)+

(x−βn−1)

γn−1

)

,

C2(x;n) = −Λc

n−1(N)

σ(x)

(

a(x;n)γn−1

+b(x;n−1)

γn−1

(

1Λc

n−1(N)+

(x−βn−1)

γn−1

))

,

D2(x;n) =Λc

n−1(N)

σ(x)

[

σ(x)−b(x;n)γn−1

+b(x;n−1) ·(

a(x;n−1)b(x;n−1)

+(x−βn−1)

γn−1

)

(

1Λc

n−1(N)+

(x−βn−1)

γn−1

)]


Electrostatic model for zeros of Laguerre and Jacobi Geronimusperturbed MOPS

Let introduce a system of n movable unit charges in (a,b) in the presence of aexternal potential V (x)To find V (x) is enough to consider the polynomial coefficients of [Qc,N(x)]′′ and[Qc,N(x)]′, evaluated in the zeros of Qc,N(x), such that

[Qc,N(yc,Nn,k )]

′′

[Qc,N(yc,Nn,k )]

′=−R(yc,N

n,k ;n),

and after some computations we obtain

[Qc,N(yc,Nn,k )]

′′

[Qc,N(yc,Nn,k )]

′= D [lnu(x)] |x=yc,N

n,k−

ψ(yc,Nn,k )

φ(yc,Nn,k )

.

The total external potential V (x) is given by two external fields

V (x) = −∫ ψ(x)

φ(x)dx + lnu(x)

Long Range PotentialShort Range Potential


Electrostatic model for zeros of Laguerre and Jacobi Geronimusperturbed MOPS

The equilibrium position for the zeros of {Qc,Nn (x)}n≥0 occurs under the presence

of a total external potential V (x) = υlong(x)+υshort(x).

υshort(x) = (1/2) lnu(x;n) represents a short range potential (or varying externalpotential) corresponding to unit charges located at the zeros of u(x).

The polynomial u(x) plays a remarkable role in the behavior of the zeros ofQc,N

n (x). As an example, we show below total external potentials VJ(x) and VL(x)when the measure dµ(x) is the classical Jacobi and Laguerre measuresrespectively. In this examples we have deg(u(x)) = 1.

VJ(x) =12

lnuJ(x;n)− 12

ln(1− x)α+1(1+ x)β+1, with

uJ(x;n) = 4n(n+α)(n+β )(n+α +β )+(2n+α +β )(2n+α +β −1)Λn(N)

·[

(2n+(α +β ))2x+(2n+α +β )(2n+α +β −1)Λn(N)]

,

VL(x) =12

lnuL(x;n)− 12

lnxα+1e−x, with

uL(x;n) = n(n+α)+Λn(N) [x− (2n+α)+Λn(N)] .


Example for the Laguerre measure with c =−1


Example with two point masses (Uvarov perturbation)


Outline






SECOND PART OF THE TALK:

ASYMPTOTIC BEHAVIOR OF RATIOS OF LAGUERRE ORTHOGONALPOLYNOMIALS

Reference:

A. Deano, E.J. Huertas, and F. Marcellan, Strong and ratio asymptotics forLaguerre polynomials revisited, Journal of Mathematical Analysis andApplications, 403 (2) (2013), 477–486.

Cited by:

R.J. Furnstahl, S.N. More, T. Papenbrock. Systematic expansion for infraredoscillator basis extrapolations. Physical Review C 89, 044301 (2014)

K.I. Ishikawa, D. Kimura, K. Shigaki, A. Tsuji. A numerical evaluation of vacuumpolarization tensor in constant external magnetic fields. International Journal ofModern Physics A, 28, 1350100 (2013)

S. Konig, S. K. Bogner, R. J. Furnstahl, S. N. More, and T. Papenbrock. Ultravioletextrapolations in finite oscillator bases. Physical Review C, 2014 - APS90, 064007(2014)


The classical Laguerre polynomials

The classical Laguerre polynomials {L(α)n }∞

n=0 (sometimes called Soninpolynomials) are orthogonal with respect to the weight function w(x) = xα e−x,α >−1, on the interval (0,+∞), so they satisfy

〈L(α)m ,L(α)

n 〉=∫ +∞

0L(α)

m L(α)n xα e−xdx = ‖L(α)

n ‖2 ·δm,n, α >−1.

We consider the normalization (not monic)

L(α)n (x) =

(−1)n

n!xn + lower degree terms.

They are the polynomial solutions of the second order differential equation

x[L(α)n (x)]′′+(α +1− x)[L(α)

n (x)]′+nL(α)n (x) = 0.

This polynomials can be given in terms of an 1F1 confluent hypergeometricfunction

L(α)n (x) =

(

n+αn

)

1F1(−n;α +1;x).


Classical Laguerre orthogonal polynomials.

-5 5 10 15 20

-10

-5

5

10

15

20


Known asymptotics for Laguerre polynomials

Outer strong asymptotics: Perron’s asymptotic formula in C\R+.For α >−1 we get

L(α)n (x) =

12√

πex/2 (−x)−α/2−1/4 nα/2−1/4e2(−nx)1/2

·{

d−1

∑m=0

Cm(α ;x) n−m/2+O(n−d/2)

}

.

Here Cm(α ;x) is independent of n. This relation holds for x in the complex planewith a cut along the positive real semiaxis. The bound for the remainder holdsuniformly in every closed domain of the complex plane with empty intersectionwith R+.

C0(α ;x) = 1, but in the original paper by Perron do not appear higher ordercoefficients Cm(α ;x), m > 1.

Mehler-Heine type formula . Fixed j, with j ∈ N∪{0} and Jα the Bessel functionof the first kind, then

limn→∞

L(α)n (x/(n+ j))

nα = x−α/2Jα(

2√

x)

,

uniformly over compact subsets of C.


Known asymptotics for Laguerre polynomials

“Inner” strong asymptotics: Perron generalization of Fejer formula on R+.Let α ∈ R. Then for x > 0 we have

L(α)n (x) = π−1/2ex/2x−α/2−1/4nα/2−1/4 cos{2(nx)1/2−απ/2−π/4}

·{

p−1

∑k=0

Ak(α ;x)n−k/2+O(n−p/2)

}

+π−1/2ex/2x−α/2−1/4nα/2−1/4 sin{2(nx)1/2−απ/2−π/4}

·{

p−1

∑k=0

Bk(α ;x)n−k/2+O(n−p/2)

}

,

where Ak(α ;x) and Bk(α ;x) are certain functions of x independent of n and regularfor x > 0. The bound for the remainder holds uniformly in [ε,ω]. For k = 0 we haveA0(α ;x) = 1 and B0(α ;x) = 0.

Main reference:

G. Szego, Orthogonal Polynomials, Coll. Publ. Amer. Math. Soc. Vol. 23, (4thed.), Amer. Math. Soc. Providence, RI (1975).


Motivation

Higher order coefficients in the asymptotic expansions are important whenone deals with Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials.

They play a key role in the analysis of the asymptotic behavior of these newfamilies of “perturbed” orthogonal polynomials.

One needs to estimate ratios of Laguerre orthogonal polynomials like

L(α)n+ j(x)

L(β )n (x)

,

where n = 0,1,2, . . ., j ∈ Z. Additionally, we require α ,β >−1.

More precisely, we need to know exactly the coefficient of n−d/2 to estimate theabove expressions correctly.

For example, if d = 1 we need to know the coefficient of n−1/2, if d = 2 thecoefficient of n−1, and so on.

There are some expressions in the literature, but not accurate enough.


Aim of the work

Remark: For more precise asymptotic expressions of

L(α)n+ j(x)

L(β )n (x)

we need more coefficients Cm(α ;x) in the Perron’s asymptotic formulas.

The main advantage of Perron’s expansions for Laguerre polynomials is thesimplicity of the asymptotic sequence (inverse powers of n), but it has the problemthat the coefficients Cm(α ;x) soon become very cumbersome to compute .

One possibility is to use the generating function for Laguerre polynomials:

(1− z)−α−1exp(

xzz−1

)

=∞

∑m=0

L(α)m (x)zm, |z|< 1,

write the coefficients as contour integrals and apply the standard method ofsteepest descent.

However, the computations soon become complicated, since parametrizing thepath of steepest descent is not easy in explicit form.


Very cumbersome computations

To the best of our knowledge, the only sources of information for higher ordercoefficients in the Perron expansion are

W. Van Assche, Erratum to Weighted zero distribution for polynomialsorthogonal on an infinite interval. SIAM J. Math. Anal., 32 (2001), 1169–1170.

D. Borwein, J. M. Borwein, and R. E. Crandall, Effective Laguerreasymptotics. SIAM J. Numer. Anal., 46 (6) (2008), 3285–3312.

The PAMO coefficient:

C1(α ;z) =1

4√−z

(

14−α2−2(α +1)z+

z2

3

)

,

named after O. Perron, W. van Assche, T. Muller and F. Olver.

Borwein et al. use complex integral representations with strict error bounds. Thisprovides a powerful (and very technical) method to generate the coefficientsCm(α ;z).


An alternative (algorithmic) approach

In this paper we propose an alternative to this approach, based solely on using anexpansion of the Laguerre polynomials that involves Bessel functions of the firstkind.

This type of expansions go back to the works of Tricomi and Buchholz.

In this way, the different behaviors of L(α)n (x) in the complex plane are better

captured, and thus the coefficients are simpler to compute.

Moreover, apart from the large n asymptotic property, the resulting approximationis convergent in the complex plane.

Using this approach it is possible to recover easily the results in the work ofBorwein et al. (and more).


An expansion by Buchholz


H. Buchholz, The confluent hypergeometric function with special emphasis onits applications, Springer-Verlag, New York, 1969.

The following expansion holds

1F1(a;c;z) = Γ(c)ez/2∞

∑m=0

( z2

)mPm(c,z)Ec−1+m(κz),

where κ =c2−a, Eν (z) = z−ν/2Jν (2

√z) and, having P0(c,z) = 1,

Pm(c,z) = z−m/2∫ z

0

(

14

uPm−1(c,u)+(c−2)P′m−1(c,u)−uP′′

m−1(u)

)

um/2−1du.

An important advantage of this expansion is that the coefficients do no longerdepend on a (which is n in our case ), so they remain bounded as n → ∞.

Remember that

L(α)n (x) =

(

n+αn

)

1F1(−n;α +1;x).



The former expansion yields

L(α)n (z) =

(

n+αn

)

1F1(−n;α +1;z)

=Γ(n+α +1)

n!ez/2(κz)−α/2

∞

∑m=0

( z4κ

)m/2Pm(α +1,z)Jm+α (2

√κz),

where (this is important ), notice that these expansions are in negative powers of

κ = n+α +1

2,

which is essentially n.

It is possible to recover the Mehler–Heine asymptotics from the above expression.

Our aim is to construct an asymptotic expansion in negative powers of n from thisexpression in a systematic way .

The coefficients become fairly complicated as well, but the procedure is easilyimplemented using symbolic computation.



In order to rewrite the above expression in negative powers of n, we also use thefollowing asymptotic approximation for the Bessel function of large argument,

Jα (z)∼(

2πz

)12

(

cosω∞

∑k=0

(−1)k a2k(α)

z2k −sinω∞

∑k=0

(−1)k a2k+1(α)

z2k+1

)

,

where |argz|< π, ω = ω(α) = z− απ2

− π4

and the coefficients are, starting with

a0(α) = 1,

ak(α) =(4α2−1)(4α2−9) . . .(4α2− (2k−1)2)

8kk!, k ≥ 1.

We need Jm+α (2√

κz) for integers m ≥ 0. Consequently, for |arg(κz)|< 2π

Jm+α (2√

κz)∼ π−1/2(κz)−1/4

(

cosω∞

∑k=0

(−1)k a2k(α)

(4κz)k −sinω∞

∑k=0

(−1)k a2k+1(α)

(4κz)k+1/2

)


A strong asymptotic expansion in C

Theorem 1: Alternative strong asymptotic formula for L(α)n (z): Let α >−1, the

Laguerre polynomial L(α)n (z) admits the following asymptotic expansion as n → ∞

L(α)n (z) =

Γ(n+α +1)n!

π−1/2 ez/2(κz)−α/2−1/4

×[

d

∑m=0

B2m(α ,z)cosωnm +

d

∑m=0

B2m+1(α ,z)sinωnm+1/2

+O(n−d−1)

]

for some coefficients Bm(α ,z) independent of n. The error term holds uniformly forz in compact sets of C, and the parameter ω is given by

ω(α) = z− απ2

− π4

Notice that the negative powers of n come from the negative powers of κ in theformer expansion for Jα (z).



The sines and cosines that appear in the asymptotic expansions can be somehowgrouped together, since

cos

(

z− (α +m)π2

− π4

)

= (−1)s

{

cos(

z− απ2 − π

4

)

, m = 2s,

sin(

z− απ2 − π

4

)

, m = 2s+1,

sin

(

z− (α +m)π2

− π4

)

= (−1)s

{

sin(

z− απ2 − π

4

)

, m = 2s,

−cos(

z− απ2 − π

4

)

, m = 2s+1,

for s = 0,1,2, . . ..

Caution 1! : one has to take into account the different ±cosω and ±sinω factorsthat multiply the asymptotic expansion of the Bessel functions.

Caution 2! : the terms a2k(α) and a2k+1(α) depend on α , so they change at eachlevel.

In the sequel, let us assume that we fix an integer d ≥ 1, so we want all terms upto order n−d/2.



Implementing this grouping carefully, what we have at the end will be twosummations, depending on d = 2M (even) or d = 2M+1 (odd) , for M = 0,1, . . ..

If d = 2M, we have an alternating sum of the form

S2M(α ,z) = (−1)M2M

∑m=0

(−1)m( z

4κ

)m/2Pm(α ,z)

a2M−m(α +m)

(2√

κz)2M−m

= (−4κz)−M2M

∑m=0

(−1)mzmPm(α ,z)a2M−m(α +m).

If d = 2M+1, we have a similar situation, and finally we obtain

S2M+1(α ,z) = (−1)M+12M+1

∑m=0

(−1)m( z

4κ

)m/2Pm(α ,z)

a2M+1−m(α +m)

(2√

κz)2M+1−m

= (−1)M+1(4κz)−M−1/22M+1

∑m=0

(−1)mzmPm(α ,z)a2M+1−m(α +m)


The algorithm approach

Algorithm:1 Fix the maximum order d.

2 Generate the polynomials Pm(α,z) using the recursion.

3 Compute the coefficients ak(α) of the asymptotic expansion of the Bessel functions upto order d.

4 Compute Sm(α,z) for m ≤ d, add all these terms, expand as n → ∞ and truncate.

All the previous work is doing by the computer, using MAPLE, Mathematica orsimilar software.

What we obtain is the expansion given by Theorem 1 , and it only remains toidentify the coefficients B(α ,z) in this expansion.


Example: First few coefficients

B0(α ,z) = 1,

B1(α ,z) =4z2−12α2+3

48√

z,

B2(α ,z) =− z3

288+

4α2+11192

z− (4α2−1)(4α2−9)512z

,

B3(α ,z) = − z9/2

10368+

20α2+18723040

z5/2− α +148

z3/2− (4α2−9)(4α2−25)6144

z1/2

− (α +1)(4α2−1)

64z1/2+

(4α2−1)(4α2−9)(4α2−25)

24576z3/2,

...


Advantages and disadvantages

Main advantages (wrt the Perron and Fejer formulas):

The coefficients are still complicated but they can be computed systematically, up to theaccuracy desired.

The expansion given in Theorem 1 is convergent on the whole complex plane.

Retaining the Bessel functions instead of expanding them in negative powers of n, itprovides a useful representation of the Laguerre polynomials for large degree.

Disadvantages:

One difficulty of the previous expansion is that it contains cosω and sinω terms.


A second strong asymptotic expansion in C\R+

These cosω and sinω terms can be grouped together away from [0,∞).

Theorem 2: Alternative strong asymptotics for L(α)n (z): Let α >−1, the Laguerre

polynomial L(α)n (z) admits the following asymptotic expansion as n → ∞:

L(α)n (z) =

12√

πΓ(n+α +1)

n!ez/2(−κz)−α/2−1/4e2

√−κz

∞

∑m=0

Bm(α ,z)n−m/2,

where the error term is uniform for z in bounded sets of C\ [0,∞), and thecoefficients Bm(α ,z) are related to the original ones Bm(α ,z) in the following way:

B2m(α ,z) = B2m(α ,z)

B2m+1(α ,z) = ±iB2m+1(α ,z), ±Imz > 0.


Recovering the standard Perron expansion

It is not complicated to recover the standard Perron expansion from Theorem 2

Doing some extra work to remove the ± in the complex plane, we have:

C0(α ,z) = B0(α ,z),

C1(α ,z) =

√−z(α +1)

2B0(α ,z)+ B1(α ,z)

=4z2−24(α +1)z−12α2+3

48√−z

,

C2(α ,z) = − (α +1)(1−2α + z(α +1))8

B0(α ,z)+

√−z(α +1)

2B1(α ,z)+ B2(α ,z)

= − z3

288+

(α +1)z2

24− (20α2+48α +13)z

192

− (2α −1)(2α −3)(α +1)32

− (4α2−1)(4α2−9)512z

...


Arbitrary ratios of Laguerre polynomials

Next, we use the former results to obtain the asymptotic behavior as n → ∞ ofarbitrary ratios of Laguerre polynomials with greater accuracy than formulasavailable in the literature.

We begin rewriting Theorem 2 as

L(α)n (z) = f (α)

n (z)

(

d−1

∑m=0

Bm(α ,z)n−m/2+O(n−d/2)

)

,

with the prefactor

f (α)n (z) =

12√

πΓ(n+α +1)

n!ez/2(−κ(n,α)z)−α/2−1/4e2

√−nz.

Note that we emphasize that κ depends both on n and on α , since we want toconsider different degree and different parameter of the Laguerre polynomials.



ThusL(α)

n+ j(z)

L(β )n (z)

=f (α)n+ j(z)

f (β )n (z)

d−1

∑k=0

Dk(α ,β ,z)n−k/2+O(n−d/2),

where the coefficients Dk(α ,β ,z) can be computed in a quite automatic way. Forinstance, the first ones are

D0(α ,β ,z) = 1,

D1(α ,β ,z) =β 2−α2

4√−z

,

D2(α ,β ,z) =(β 2−α2)(3α2+9β 2−4z2−9)

96z.

The ratio of two prefactors is

f (α)n+ j(z)

f (β )n (z)= (−z)

β−α2 ×

Γ(n+ j+α +1)Γ(n+ j+1)

Γ(n+1)Γ(n+β +1)

κ(n,β )β/2+1/4

κ(n+ j,α)α/2+1/4e2√

−(κ+ j)z−2√−κz.



The prefactor can be expanded in negative powers of n, but we need a couple ofprevious computations

Lemma 1 : We have

κ(n,β )β/2+1/4

κ(n+ j,α)α/2+1/4= n

β−α2

∞

∑m=0

Am( j,α ,β )n−m, n → ∞,

where for m ≥ 0 we have

Am( j,α ,β ) =(

2 j+α +12

)m

×m

∑k=0

(−1)m−k( β

2 + 14

k

)(

m− k+ α2 − 3

4m− k

)(

β +12 j+α +1

)k

.



Lemma 2 : We have

Γ(n+ j+α +1)Γ(n+β +1)

∼ n j+α−β∞

∑m=0

Gm(α ,β , j)nm , n → ∞,

where the coefficients Gm(α ,β , j) can be expressed in terms of generalizedBernoulli polynomials:

Gm(α ,β , j) =

(

j+α −βm

)

B( j+α−β+1)m ( j+α +1).

The generalized Bernoulli polynomials are not directly implemented in MAPLE, butthey can be computed using the generating function

(

tet −1

)ℓ

ext =∞

∑n=0

B(ℓ)n (x)

tn

n!, |t|< 2π.


High-order coefficients in the ratio asymptotics

Theorem 3: Let α ,β >−1, and z ∈ C\ [0,∞), the ratio of arbitrary Laguerrepolynomials has an asymptotic expansion

L(α)n+ j(z)

L(β )n (z)

∼(

− zn

)

β−α2

∞

∑m=0

Um(α ,β , j,z)n−m/2,

where the first coefficients are

U0(α ,β , j,z) = 1,

U1(α ,β , j,z) =β 2−α2+2z(β −α −2 j)

4√−z

,

U2(α ,β , j,z) = − (6 j2+6(α −β ) j+α2+2β 2−3αβ )z12

+(β 2−α2+2α −1) j

4

+(α2−β 2−2α −2β −1)(β −α)

8− (α2+3β 2−3)(α2−β 2)

32z.

The error term holds uniformly for z in compact sets of C\ [0,∞).


Thank you!


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