Upm etsiccp-seminar-vf

Comportamiento Asintótico de Secuencias de
Polinomios Ortogonales e
Interpretación Electrostática sus Ceros
Edmundo J. Huertas
Universidad Politécnica de Madrid - Grupo SERPA-HGA
March 12, 2015- Seminario del Departamento de Matemáticas e
Informática Aplicadas a las II. Civil y Naval
(UPM 2015) Polinomios Ortogonales UPM 2015 1 / 50

Outline
1 Electrostatic Interpretation of Zeros of Orthogonal Polynomials
Introduction
Basic background for MOPS and its zeros
Canonical perturbations of a measure (Christoffel, Uvarov and Geronimus)
The interacting particle model (M. Ismail)
Example for the Laguerre-Geronimus measure with c = −1
Another example for the Uvarov modiﬁcation of the Laguerre measure
2 Asymptotic Behavior of Ratios of Laguerre Orthogonal Polynomials
Asymptotics for Classical Laguerre Polynomials
Motivation of the problem
An alternative (algorithmic) approach
An expansion of 1F1(a;c;z) by Buchholz
A ﬁrst strong asymptotic expansion valid in the whole C
A second strong asymptotic expansion valid in CR+
Asymptotics of ratios of Laguerre polynomials

Outline
Introduction

FIRST PART OF THE TALK:
ELECTROSTATIC INTERPRETATION OF ZEROS OF ORTHOGONAL
POLYNOMIALS
References:
A. Branquinho, E.J. Huertas, and F.R. Rafaeli, Zeros of orthogonal polynomials
generated by the Geronimus perturbation of measures, Lecture Notes in
Computer Science (LNCS), 8579 (2014), 44–59.
E.J. Huertas, F. Marcell´an and H. Pijeira, An electrostatic model for zeros of
perturbed Laguerre polynomials, Proceedings of the American Mathematical
Society, 142 (5) (2014), 1733–1747.
E.J. Huertas, F. Marcell´an and F.R. Rafaeli, Zeros of orthogonal polynomials
generated by canonical perturbations of measures, Applied Mathematics and
Computation, 218 (13) (2012), 7109–7127.

The work of Stieltjes.
Theorem (Stieltjes 1885-1889): Suppose n unit charges at points x1,x2,...,xn are
distributed 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 the
Jacobi 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 quadrature
rules and its extensions (Gauss–Radau, Gauss–Lobatto, Gauss–Kronrod
rules,...etc)
f(x)dµ(x) ∼
n
∑
k=1
λk,n f(xk,n)
2. Zeros of classical orthogonal polynomials are the electrostatic equilibrium
points of positive unit charges interacting according to a logarithmic potential
under the action of an external ﬁeld.
3. Zeros of orthogonal polynomials are used in collocation methods for boundary
value problems of 2nd order linear differential operators.
4. Global properties of zeros of orthogonal polynomials can be analyzed when they
satisfy 2nd order differential equations with polynomial coefﬁcients, using the
WKB method.
5. Zeros of orthogonal polynomials are eigenvalues of Jacobi matrices and its role
in Numerical Linear Algebra is very well known.

Basic background - MOPS
Let us consider the inner product ·,· µ : P×P → R
f,g µ =
b
a
f(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 respect
to 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
2
µ
, n ≥ 0 and γn =
Pn
2
µ
Pn−1
2
µ
> 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 of
C0(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 are
dense in supp(dµ).

Basic background - Reproducing Kernel
nth-Kernel
Kn(x,y) =
n
∑
j=0
Pj(y)Pj(x)
Pj
2
µ
, ∀n ∈ N
Christoffel-Darboux formula
Kn(x,y) =
1
Pn
2
µ
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
2
µ
, ∀n ∈ N

Christoffel perturbation of a measure dµ
Let {P
c,[1]
n (x)}n≥0 be the MOPS associated with the measure
dµ[1]
= (x−c)dµ,
with (any complex or real number) c ∈ C0(supp(dµ)).
It is clear that Pn(c) = 0, ∀n ≥ 1.
The MOPS with respect to dµ[1]
is
P
c,[1]
n (x) = (x−c)−1
Pn+1(x)−
Pn+1(c)
Pn(c)
Pn(x) ,
and a trivial veriﬁcation shows that
P
c,[1]
n (x) =
Pn
2
µ
Pn(c)
Kn(x,c).

Uvarov perturbation of a measure dµ
Let {PN
dµN
= dµ +N δ(x−c),
with N ∈ R+, δ(x−c) the Dirac delta function in x = c, and c ∈ C0(supp(dµ)).
Connection formula for Uvarov perturbed MOPS
The polynomials {PN
n (x)}n≥0, can be represented as
PN
n (x) = Pn(x)−
NPn(c)
1+NKn−1(c,c)
Kn−1(c,x).

Modiﬁcation of dµ by a linear divisor
Let {Qc
n(x)}n≥0 be the MOPS associated with the measure
dν =
1
(x−c)
dµ,
with c ∈ C0(supp (dµ)), and let yc
n,k := yc
n,k(c) be the zeros of Qc
n(x).
The MOPS with respect to dν can be represented as
Qc
n(x) = Pn(x)−
Fn(c)
Fn−1(c)
Pn−1(x), n = 0,1,2,...,
where Qc
0(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 the
point c. Then we obtain a Geronimus transformation of the measure dµ.
Let {Qc,N
dνN =
1
(x−c)
dµ +Nδ(x−c),
with c ∈ C0(supp (dµ)), and let yc,N
n,k := yc,N
n,k (c) be the zeros of Qc,N
n (x).
Geronimus (1940), conclude that the sequences associated to dνN must be of the
form
Pn(x)+anPn−1(x), an = 0,
for certain numbers an ∈ R.
Maroni (1990), stated that the sequence { ˜Pn+1(x)}n≥0, orthogonal with respect to
u = δc +λ(x−c)−1
L, 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,N
n (x)}n≥0, can be represented as
Qc,N
n (x) = Pn(x)+Λc
n(N)Pn−1(x),
with
Λc
n(N) =
Pn(c)
Pn−1(c)
−
Fn(c)
Fn−1(c)
1+NBc
n
−
Pn(c)
Pn−1(c)
.
Theorem: Connection formula 2
The polynomials { ˜Qc,N
n (x)}n≥0, with ˜Qc,N
n (x) = κnQc,N
n (x), can be represented as
˜Qc,N
n (x) = Qc
n(x)+NBc
n ·(x−c)P
c,[1]
n−1 (x),
with κn = 1+NBc
n, and
Bc
n =
−Qc
n(c)Pn−1(c)
Pn−1
2
µ
= Kc
n−1(c,c) > 0.

Zeros of Geronimus perturbed MOPS
The mass point c attracts exactly one zero of Qc,N
n (x), when N → ∞.
When either c < a or c > b, at most one of the zeros of Qc,N
n (x) is located outside of
C0(supp (dµ)) = (a,b). In the next result, we will give explicitly the value N0 of the
mass 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,N
n,n < b, for N < N0,
yc,N
n,n = b, for N = N0,
yc,N
n,n > b, for N > N0,



with N0 = N0(n,c,b) =
−Qc
n(b)
Kc
n−1 (c,c)(b−c)P
c,[1]
n−1 (b)
> 0.

M. E. H. Ismail, An electrostatics model for zeros of general orthogonal
polynomials, Paciﬁc J. Math. 193, (2000), 355-369.
The model obtains the second order differential equation for Qc,N
n (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,N
n (x)}n≥0 can be also represented as
Qc,N
n (x) = Pn(x)+Λc
n(N)Pn−1(x),

Second order differential equation for Geronimus perturbed MOPS
The Geronimus perturbed MOPS {Qc,N
n (x)}n≥0 satisfies the second order linear
differential equation
[Qc,N
n (x)]′′
+R(x;n)[Qc,N
n (x)]′
+S (x;n)Qc,N
n (x) = 0,
(also known as the holonomic equation), where
R(x;n) = − ξc
1 (x;n)+ηc
2(x;n)+
[ηc
1(x;n)]′
ηc
1(x;n)
,
S (x;n) = ξc
1 (x;n)ηc
2(x;n)−ηc
1(x;n)ξc
2 (x;n)+
ξc
1 (x;n)[ηc
1(x;n)]′ −[ξc
1 (x;n)]′ηc
1(x;n)
ηc
1(x;n)
.
In turn, for k = 1,2, the above expressions are given only in terms of Λc
n(N), and the
coefficients βn, γn, σ(x), a(x;n) and b(x;n) of the three term recurrence relation and the
structure relation satisfied by {Pn(x)}n≥0:
ξc
k (x;n) =
Ck(x;n)B2(x;n)γn−1 +Dk(x;n)Λc
n−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)−Λc
n(N)
b(x;n)
γn−1
,
D1(x;n) =
1
σ(x)
b(x;n)+Λc
n(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)
1
Λc
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
1
Λc
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
1
Λc
n−1(N)
+
(x−βn−1)
γn−1

Electrostatic model for zeros of Laguerre and Jacobi Geronimus
perturbed MOPS
Let introduce a system of n movable unit charges in (a,b) in the presence of a
external 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,N
n,k )]′′
[Qc,N(yc,N
n,k )]′
= −R(yc,N
n,k ;n),
and after some computations we obtain
[Qc,N(yc,N
n,k )]′′
[Qc,N(yc,N
n,k )]′
= D[lnu(x)]|x=yc,N
n,k
−
ψ(yc,N
n,k )
φ(yc,N
n,k )
.
The total external potential V(x) is given by two external fields
V(x) = −
ψ(x)
φ(x)
dx + lnu(x)
Long Range Potential
Short Range Potential

Electrostatic model for zeros of Laguerre and Jacobi Geronimus
perturbed MOPS
The equilibrium position for the zeros of {Qc,N
n (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 external
potential) 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 of
Qc,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 measures
respectively. In this examples we have deg(u(x)) = 1.
VJ(x) =
1
2
lnuJ(x;n)−
1
2
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))2
x+(2n+α +β)(2n+α +β −1)Λn(N) ,
VL(x) =
1
2
lnuL(x;n)−
1
2
lnxα+1
e−x
, with
uL(x;n) = n(n+α)+Λn(N)[x−(2n+α)+Λn(N)].

Example for the Laguerre measure with c = −1
2 1 1 2 3 4 5
10
5
5
L3
0
x

Example with two point masses (Uvarov perturbation)
1 2 3 4 5
40
20
20
40
0.0000
a1
0.0000
a2
c1 1c2 2
L4
0
x
Q4
0
x
u4 x

Example with two point masses (Uvarov perturbation)
1 2 3 4 5
40
20
20
40
2.4184
a1
0.0000
a2
c1 1c2 2
L4
0
x
Q4
0
x
u4 x

Outline
Introduction

SECOND PART OF THE TALK:
ASYMPTOTIC BEHAVIOR OF RATIOS OF LAGUERRE ORTHOGONAL
POLYNOMIALS
Reference:
A. Deaño, E.J. Huertas, and F. Marcellán, Strong and ratio asymptotics for
Laguerre polynomials revisited, Journal of Mathematical Analysis and
Applications, 403 (2) (2013), 477–486.
Cited by:
R.J. Furnstahl, S.N. More, T. Papenbrock. Systematic expansion for infrared
oscillator basis extrapolations. Physical Review C 89, 044301 (2014)
K.I. Ishikawa, D. Kimura, K. Shigaki, A. Tsuji. A numerical evaluation of vacuum
polarization tensor in constant external magnetic fields. International Journal of
Modern Physics A, 28, 1350100 (2013)
S. König, S. K. Bogner, R. J. Furnstahl, S. N. More, and T. Papenbrock. Ultraviolet
extrapolations 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 Sonin
polynomials) 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 =
+∞
0
L
(α)
m L
(α)
n xα
e−x
dx = 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 hypergeometric
function
L
(α)
n (x) =
n+α
n
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 CR+.
For α > −1 we get
L
(α)
n (x) =
1
2
√
π
ex/2
(−x)−α/2−1/4
nα/2−1/4
e2(−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 plane
with a cut along the positive real semiaxis. The bound for the remainder holds
uniformly in every closed domain of the complex plane with empty intersection
with R+.
C0(α;x) = 1, but in the original paper by Perron do not appear higher order
coefﬁcients Cm(α;x), m > 1.
Mehler-Heine type formula. Fixed j, with j ∈ N∪{0} and Jα the Bessel function
of the ﬁrst kind, then
lim
n→∞
L
(α)
n (x/(n+ j))
nα
= x−α/2
Jα 2
√
x ,
uniformly over compact subsets of C.

Known asymptotics for Laguerre polynomials
“Inner” strong asymptotics: Perron generalization of Fej´er formula on R+.
Let α ∈ R. Then for x > 0 we have
L
(α)
n (x) = π−1/2
ex/2
x−α/2−1/4
nα/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−1/2
ex/2
x−α/2−1/4
nα/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 regular
for x > 0. The bound for the remainder holds uniformly in [ε,ω]. For k = 0 we have
A0(α;x) = 1 and B0(α;x) = 0.
Main reference:
G. Szeg˝o, Orthogonal Polynomials, Coll. Publ. Amer. Math. Soc. Vol. 23, (4th
ed.), Amer. Math. Soc. Providence, RI (1975).

Motivation
Higher order coefficients in the asymptotic expansions are important when
one deals with Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials.
They play a key role in the analysis of the asymptotic behavior of these new
families 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 the
above expressions correctly.
For example, if d = 1 we need to know the coefficient of n−1/2
, if d = 2 the
coefficient 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 the
simplicity of the asymptotic sequence (inverse powers of n), but it has the problem
that 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−1
exp
xz
z−1
=
∞
∑
m=0
L
(α)
m (x)zm
, |z| < 1,
write the coefficients as contour integrals and apply the standard method of
steepest descent.
However, the computations soon become complicated, since parametrizing the
path 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 order
coefficients in the Perron expansion are
W. Van Assche, Erratum to Weighted zero distribution for polynomials
orthogonal on an infinite interval. SIAM J. Math. Anal., 32 (2001), 1169–1170.
D. Borwein, J. M. Borwein, and R. E. Crandall, Effective Laguerre
asymptotics. SIAM J. Numer. Anal., 46 (6) (2008), 3285–3312.
The PAMO coefficient:
C1(α;z) =
1
4
√
−z
1
4
−α2
−2(α +1)z+
z2
3
,
named after O. Perron, W. van Assche, T. Müller and F. Olver.
Borwein et al. use complex integral representations with strict error bounds. This
provides a powerful (and very technical) method to generate the coefficients
Cm(α;z).

In this paper we propose an alternative to this approach, based solely on using an
expansion of the Laguerre polynomials that involves Bessel functions of the ﬁrst
kind.
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 approximation
is convergent in the complex plane.
Using this approach it is possible to recover easily the results in the work of
Borwein et al. (and more).

An expansion by Buchholz
H. Buchholz, The conﬂuent hypergeometric function with special emphasis on
its applications, Springer-Verlag, New York, 1969.
The following expansion holds
1F1(a;c;z) = Γ(c)ez/2
∞
∑
m=0
z
2
m
Pm(c,z)Ec−1+m(κz),
where κ =
c
2
−a, Eν (z) = z−ν/2
Jν (2
√
z) and, having P0(c,z) = 1,
Pm(c,z) = z−m/2
z
0
1
4
uPm−1(c,u)+(c−2)P′
m−1(c,u)−uP′′
m−1(u) um/2−1
du.
An important advantage of this expansion is that the coefﬁcients do no longer
depend on a (which is n in our case), so they remain bounded as n → ∞.
Remember that
L
(α)
n (x) =
n+α
n
1F1(−n;α +1;x).

The former expansion yields
L
(α)
n (z) =
n+α
n
1F1(−n;α +1;z)
=
Γ(n+α +1)
n!
ez/2
(κz)−α/2
∞
∑
m=0
z
4κ
m/2
Pm(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 this
expression in a systematic way.
The coefﬁcients become fairly complicated as well, but the procedure is easily
implemented using symbolic computation.

In order to rewrite the above expression in negative powers of n, we also use the
following asymptotic approximation for the Bessel function of large argument,
Jα (z) ∼
2
πz
1
2
cosω
∞
∑
k=0
(−1)k a2k(α)
z2k
−sinω
∞
∑
k=0
(−1)k a2k+1(α)
z2k+1
,
where |arg z| < π, ω = ω(α) = z−
απ
2
−
π
4
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)cosω
nm
+
d
∑
m=0
B2m+1(α,z)sinω
nm+1/2
+O(n−d−1
)
for some coefﬁcients Bm(α,z) independent of n. The error term holds uniformly for
z in compact sets of C, and the parameter ω is given by
ω(α) = z−
απ
2
−
π
4
Notice that the negative powers of n come from the negative powers of κ in the
former expansion for Jα (z).

The sines and cosines that appear in the asymptotic expansions can be somehow
grouped together, since
cos z−
(α +m)π
2
−
π
4
= (−1)s cos z− απ
2 − π
4 , m = 2s,
sin z− απ
2 − π
4 , m = 2s+1,
sin z−
(α +m)π
2
−
π
4
= (−1)s sin z− απ
2 − π
4 , m = 2s,
−cos z− απ
2 − π
4 , m = 2s+1,
for s = 0,1,2,....
Caution 1!: one has to take into account the different ±cosω and ±sinω factors
that 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 each
level.
In the sequel, let us assume that we ﬁx an integer d ≥ 1, so we want all terms up
to order n−d/2
.

Implementing this grouping carefully, what we have at the end will be two
summations, 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)M
2M
∑
m=0
(−1)m z
4κ
m/2
Pm(α,z)
a2M−m(α +m)
(2
√
κz)2M−m
= (−4κz)−M
2M
∑
m=0
(−1)m
zm
Pm(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+1
2M+1
∑
m=0
(−1)m z
4κ
m/2
Pm(α,z)
a2M+1−m(α +m)
(2
√
κz)2M+1−m
= (−1)M+1
(4κz)−M−1/2
2M+1
∑
m=0
(−1)m
zm
Pm(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 up
to 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 or
similar software.
What we obtain is the expansion given by Theorem 1, and it only remains to
identify 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 +11
192
z−
(4α2 −1)(4α2 −9)
512z
,
B3(α,z) = −
z9/2
10368
+
20α2 +187
23040
z5/2
−
α +1
48
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 the
accuracy 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, it
provides 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 CR+
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) =
1
2
√
π
Γ(n+α +1)
n!
ez/2
(−κz)−α/2−1/4
e2
√
−κz
∞
∑
m=0
ˆBm(α,z)n−m/2
,
where the error term is uniform for z in bounded sets of C[0,∞), and the
coefﬁcients ˆBm(α,z) are related to the original ones Bm(α,z) in the following way:
ˆB2m(α,z) = B2m(α,z)
ˆB2m+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) = ˆB0(α,z),
C1(α,z) =
√
−z(α +1)
2
ˆB0(α,z)+ ˆB1(α,z)
=
4z2 −24(α +1)z−12α2 +3
48
√
−z
,
C2(α,z) = −
(α +1)(1−2α +z(α +1))
8
ˆB0(α,z)+
√
−z(α +1)
2
ˆB1(α,z)+ ˆB2(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 → ∞ of
arbitrary ratios of Laguerre polynomials with greater accuracy than formulas
available in the literature.
We begin rewriting Theorem 2 as
L
(α)
n (z) = f
(α)
n (z)
d−1
∑
m=0
ˆBm(α,z)n−m/2
+O(n−d/2
) ,
with the prefactor
f
(α)
n (z) =
1
2
√
π
Γ(n+α +1)
n!
ez/2
(−κ(n,α)z)−α/2−1/4
e2
√
−nz
.
Note that we emphasize that κ depends both on n and on α, since we want to
consider different degree and different parameter of the Laguerre polynomials.

Thus
L
(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. For
instance, the ﬁrst ones are
D0(α,β,z) = 1,
D1(α,β,z) =
β2 −α2
4
√
−z
,
D2(α,β,z) =
(β2 −α2)(3α2 +9β2 −4z2 −9)
96z
.
The ratio of two prefactors is
f
(α)
n+ j(z)
f
(β)
n (z)
= (−z)
β−α
2 ×
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+1/4
e2
√
−(κ+ j)z−2
√
−κz
.

The prefactor can be expanded in negative powers of n, but we need a couple of
previous computations
Lemma 1: We have
κ(n,β)β/2+1/4
κ(n+ j,α)α/2+1/4
= n
β−α
2
∞
∑
m=0
Am(j,α,β)n−m
, n → ∞,
where for m ≥ 0 we have
Am(j,α,β) =
2j +α +1
2
m
×
m
∑
k=0
(−1)m−k
β
2 + 1
4
k
m−k + α
2 − 3
4
m−k
β +1
2j +α +1
k
.

Lemma 2: We have
Γ(n+ j +α +1)
Γ(n+β +1)
∼ nj+α−β
∞
∑
m=0
Gm(α,β, j)
nm
, n → ∞,
where the coefﬁcients Gm(α,β, j) can be expressed in terms of generalized
Bernoulli polynomials:
Gm(α,β, j) =
j +α −β
m
B
( j+α−β+1)
m (j +α +1).
The generalized Bernoulli polynomials are not directly implemented in MAPLE, but
they can be computed using the generating function
t
et −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 Laguerre
polynomials has an asymptotic expansion
L
(α)
n+ j(z)
L
(β)
n (z)
∼ −
z
n
β−α
2
∞
∑
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) =
β2 −α2 +2z(β −α −2j)
4
√
−z
,
U2(α,β, j,z) = −
(6j2 +6(α −β)j +α2 +2β2 −3αβ)z
12
+
(β2 −α2 +2α −1)j
4
+
(α2 −β2 −2α −2β −1)(β −α)
8
−
(α2 +3β2 −3)(α2 −β2)
32z
.
The error term holds uniformly for z in compact sets of C[0,∞).

Thank you!

Upm etsiccp-seminar-vf

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