Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 77–89
Strong asymptotics for Laguerre polynomials with varying
weights
Christof Bosbach, Wolfgang Gawronski ∗
Abteilung Mathematik, Universitat Trier, D-54286 Trier, Germany
Received 30 September 1997; received in revised form 2 April 1998
Abstract
Supplementing and extending classical and recent results strong asymptotics for the Laguerre polynomials Ln( n ) are
established, as n → ∞; when the parameter n depends on the degree n suitably. A case of particular interest is the one
for which n grows faster than n: Rescaling the argument z appropriately the resulting asymptotic forms are described
by elementary functions, thereby extending the classical formulae of Plancherel–Rotach type for Laguerre polynomials
c 1998 Elsevier Science B.V. All rights reserved.
L( )
n (z); being xed.
AMS classi cation: 30E15; 33C45
Keywords: Laguerre polynomials; Strong asymptotics; Plancherel-Rotach formulae
0. Introduction and summary
In this paper we deal with some generalizations and extensions of asymptotic results for Laguerre
polynomials Ln( ) : The de nition of these classical orthogonal polynomials and some well-known
formulae as well we take from Szego’s classic memoir [24, Chapter V]. Here we only mention the
de ning orthogonality relation
Z
0
∞
Ln( ) (x)Lm( ) (x)x e−x
dx =
n+
n
( + 1)nm ;
¿ − 1:
(0.1)
The classical asymptotic theory of these polynomials primarily is concerned with large degrees n
and a xed parameter [24, Chapter VIII]. Among the numerous asymptotic forms the so-called
formulae of Plancherel–Rotach type [24, p. 200] give asymptotic descriptions of the rescaled Laguerre
polynomials Ln( ) (n z); n = n + ( + 1)=2; in terms of elementary functions, as n → ∞; where the
∗
Corresponding author. E-mail: gawron@uni-trier.de.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 7 - 2
78
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
cases z ∈ (r; s) = (0; 4); the interval of zeros, and z ∈ C[r; s] ; the cut plane, have to be distinguished.
Here and throughout for real r; s with r¡s we use the notation
C[r; s] := C\[r; s]
that is the complex plane with a cut along the interval [r; s]: The resulting asymptotic approximations,
which hold uniformly on compact subsets of (r; s) and of C[r; s] , respectively, are named strong
asymptotics in contrast to the so-called weak asymptotics that is the limiting distribution for the
zeros.
For instance questions of weighted polynomial approximation and in the nonrelativistic quantum
theory led various authors to considering orthogonal polynomials with varying weights, i.e., with
weight functions depending on the degree of the polynomials. The most prominent representatives
of such orthogonal functions with weights living on an unbounded interval are Laguerre polynomials
Ln( n ) with a parameter n now being dependent on the degree n: The relevant literature concerning
the corresponding asymptotic theory primarily deals with weak asymptotics [3, 4, 7, 9, 11, 12, 17–19,
23] and some papers treat strong asymptotics in case of very special sequences ( n ): We mention the
contributions [2, 9, 10, 14–16, 22, 25, 26] where strong asymptotics for Ln( n ) have been established
provided the parameter is a linear function of the degree, that is n = an + ; a ¿ 0; ∈ R being
independent of n:
n)
It is the main object of this paper to derive strong asymptotics for L(
when now ( n ) is a more
n
general real sequence satisfying throughout the condition
n
= a ¿ 0:
(0.2)
lim
n→∞ n
In particular we admit the value a = ∞: The resulting generalized formulae of Plancherel–Rotach
type are stated and proved in Sections 2 and 3.
The recent publications [4, 7] treat weak asymptotics for classical orthogonal polynomials including
( n )
Ln under the general assumption (0.2). In particular it is shown for Ln( n ) with a = ∞ in (0.2) that
the asymptotic zero distribution is a semicircle law which also is well-known to be the weak limit
of the contracted zero distribution for the classical Hermite polynomials Hn : This coincidence of
the weak asymptotics gives rise to the question: In how far is this true for strong asymptotics?
As consequence of the strong asymptotics o the zero interval the answer is made precise by the
relative asymptotics
!
√
−z
(−1)n n!2n=2 Ln( n ) (2 n n z + n )
√
√
lim
= exp
n→∞
Hn ( 2n + 1z)
2(z + z 2 − 1)
nn=2
holding uniformly on compact subsets of C[−1;1] provided that n3 = o( n ); n → ∞ (Theorem 2.4).
Strong asymptotics in the sense explained above and used in this paper involves an approximation
of the underlying polynomials by elementary functions on each the interval of zeros (r; s) and the
cut plane [r; s]: Asymptotic expansions for Ln( ) (n z) as both n; → ∞ suitably and as z varies in
domains including (r; s) and parts of the cut plane lead to approximants being higher transcendental
functions [5, 6, 8, 25, 26]. However, in this paper we will not consider this point of view and refer
to the short discussion given in [10]. The proofs of the various asymptotic formulae rely on the
saddle point approximation of certain contour integrals combined with Vitali’s theorem on compact
convergence for sequences of analytic functions.
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
79
1. Auxiliary results
In this section we collect technical details and give some preliminary results which are basic for
the main theorems below.
Starting point is the well-known representation formula
Ln( ) (z) =
ez
2i
Z
e−t
C
t
z
t
t−z
n
dt
;
t−z
z 6= 0;
(1.1)
by means of a contour integral [24, p. 105, (5.4.8)], ∈ C; n ∈ N0 : Here C is a simple closed
contour with positive orientation enclosing t = z but not t = 0: Further the non-integer power of t=z
is assumed to reduce to 1 when t = z: In the light of various known asymptotic results in the weak
and in the strong sense [e.g. 4, 7, 10] under the general assumption (0.2) we consider the rescaled
Laguerre polynomials
Qn (z) := Ln( n ) (n z);
if a ∈ [0; ∞);
(1.2)
with
n := n + 12 ( n + 1);
√
Pn (z) := Ln( n ) (2 n n z + n );
(1.3)
if a = ∞:
(1.4)
Obvious substitutions in (1.1) lead to the basic representation formulae
e n z
Qn (z) =
2iz
Z
e−qn (t) dt;
C
z 6= 0;
(1.5)
t
t
− (n + 1) log
;
z
t−z
(1.6)
with
qn (t) := n t − ( n − 1) log
√
with
e2 n n z
p
p
Pn (z) =
2i(2 n= n )n (1 + 2 n= n z)
Z
Cn
p
e−pn (t) dt;
z 6=
1
2
r
n
;
n
(1.7)
p
√
1 + 2 n= n t
1 + 2 n= n t
p
pn (t) := 2 n n t − ( n − 1) log
− (n + 1) log
:
t−z
1 + 2 n= n z
(1.8)
In (1.7)
Cn is a simple closed contour with positive orientation enclosing t = z but not t =
p
− 21 n =n: Further in addition to the choice in (1.1) for the logarithms involving the factor t − z
p
in (1.6) and (1.8) we choose those branches which are real if t=(t − z) and (1 + 2 n= n t)=(t − z)
are real and positive, respectively. Now our main task consists in a saddle point approximation of
the contour integrals in (1.5) and (1.7). In the special case of a linear sequence n = an + ; a ¿ 0;
the parameter integral corresponding to that in (1.5) [10, p. 36] could be evaluated asymptotically
by the saddle point method in standard form [13, 21, 27, 30]. In the sequel we employ a variant
of the saddle point approximation which essentially is based on the following lemma taken from [1,
Section 20.2].
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C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
Lemma 1.1. Suppose that G ⊂ C is a domain and gn are complex valued functions; n ∈ N; being
twice di erentiable on G: Further let tn ∈ G; !n be complex numbers such that the straight lines
Tn = [tn − !n ; tn + !n ] are contained in G; n ∈ N. If (¿0)
gn′ (tn ) = 0;
n ∈ N;
(1.9)
lim |!n2 gn′′ (tn )| = ∞;
(1.10)
n→∞
|arg(!n gn′′ (tn )1=2 )| 6
− ;
4
gn′′ (t) = gn′′ (tn )(1 + o(1));
n ¿ n0 ();
(1.11)
n → ∞; uniformly with respect to t ∈ Tn ;
(1.12)
then we have
Z
−gn (t)
e
Tn
dt =
s
2
gn′′ (tn )
e−gn (tn ) (1 + o(1));
n → ∞:
Condition (1.11) is to x the correct branch of the square root gn′′ (tn )1=2 :
Finally, in this section we list some special Poisson integrals the computations of which are
readily performed either by residue calculus or by mean value formulae for holomorphic functions
[24, pp. 275–277; 10, p. 40]. Therefore we omit their straightforward proofs.
Lemma 1.2. Suppose that the real numbers c; d satisfy c¿0; c ¿ d ¿ 0 and is a complex number
with ||¡1; then
(i)
1
2
Z
1
2
Z
(c + d cos )
−
1 + e−i
d = c + d;
1 − e−i
(ii)
log(c2 + d2 + 2cd cos )
−
1 + e−i
d = 2 log(c + d);
1 − e−i
log being the principal branch; that is log x is real if x is real and positive;
(iii)
1
2
1 + e−i
log(sin )
d = log
1 − e−i
−
Z
2
(
1
−
2
2 )
:
2. The case n =n → ∞
In this section we expand the contour integral in (1.7) asymptotically thereby establishing the
strong asymptotics for the polynomials Pn de ned in (1.4). Basically we follow arguments in [10].
Although here the technical details are more involved we restrict our computations and reasonings
to some essential steps.
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
81
The saddle point equation pn′ (t) = 0 for (1.7) now is equivalent to
√
2 n n −
n + n
p
2
1 + 2 n= n t
r
n
n+1
+
= 0:
n
t−z
(2.1)
Its solutions are given explicitly by
!
p
1
1
z− √
± (z − rn )(z − sn )
tn± (z) =
2
2 n n
(2.2)
with a proper determination of the square root function below and
p
1
rn = √ (1 + 2n − 2 (n + 1)(n + n ));
2 n n
(2.3)
p
1
sn = √ (1 + 2n + 2 (n + 1)(n + n )):
2 n n
(2.4)
Obviously, on account of (0.2) we have rn → −1; sn → 1 as n → ∞: Next, for z ∈ C[rn ; sn ] the variable
wn is de ned through
rn + sn sn − rn wn + wn−1 1 + 2n
z=
+
+
= √
2
2
2
2 n n
s
(n + 1)(n + n ) wn + wn−1
;
n n
2
(2.5)
|wn |¿1: This means the exterior of the unit circle in the wn -plane is mapped conformally onto the
domain C[rn ; sn ] in the z-plane such that the point wn = ∞ corresponds to z = ∞: Moreover the upper
and lower edge of the cut in the z-plane is mapped onto the upper and lower half of the unit circle
in the wn -plane, respectively. From (2.5) we have
√
√
z − r n + z − sn
2
r n + sn p
√
+ (z − rn )(z − sn )
=
z−
wn = √
z − r n − z − s n sn − rn
2
s
=
!
1 + 2n p
n n
z− √
+ (z − rn )(z − sn ) :
(n + 1)(n + n )
2 n n
(2.6)
In view of (2.2) we put
1
tn :=
2
s
(n + 1)(n + n )
wn +
n n
r
n
n
!
(2.7)
giving
!
p
1
1
tn =
z− √
+ (z − rn )(z − sn )
2
2 n n
(2.8)
which means that in (2.2) we make thep
choice of p
the branch such that C[rn ; sn ] is mapped conformally
onto the exterior of the circle |t − 12 n= n |¿ 12 (n + 1)(n + n )=n n : Further on the cut (rn ; sn )
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C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
in the z-plane and its images according to (2.5) and (2.7) we have
1 + 2n
z = √
+
2 n n
s
(n + 1)(n + n )
cos ;
n n
0¡¡:
s
wn =ei and tn = 21
(n + 1)(n + n ) i
e +
n n
r
n
n
!
(2.9)
;
Now similar to [10, pp. 38,39] straightforward calculations lead to connection formulae in
Lemma 2.1. If the complex variables z; wn , and tn are related by (2.5)–(2.8), then we have
(i)
q
1 n + 1 wn +
tn − z = − √
2 n n
wn
n+ n
n+1
;
(ii)
1+2
r
n
tn =
n
√
(n + 1)(n + n )
wn +
n
s
n + n
n+1
!
;
(iii)
q
1+2
n
t
n n
tn − z
= −2
r
n
n
s
n + n
wn ;
n+1
(iv)
p
1 + 2 (n= n )tn
wn
p
p
=
:
1 + 2 (n= n )z wn + (n + 1)=(n + n )
Next, from (1.8) we get
pn′′ (t) =
n+1
4n
n + n
p
−
n (1 + 2 (n= n )t)2 (t − z)2
(2.10)
from which the following identity is immediate by Lemma 2.1.
Lemma 2.2. If the complex variables z; wn and tn are related by (2.5)–(2.8), then we have
pn′′ (tn ) =
1 − wn2
4n n
n + 1 1 − wn2
:
=
p
(tn − z)2 wn2
n + 1 (wn + (n + n )=(n + 1))2
After these preparations we establish the desired strong asymptotics for Pn (z) (see (1.4)) on the
cut plane C[−1;1] : Before, we mention that passing to the limit n → ∞ in (2.6) we obtain the familiar
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
83
Jukowski function
√
√
√
z+1+ z−1
√
w := lim wn = √
= z + z2 − 1
n→∞
z+1− z−1
(2.11)
mapping C[−1;1] conformally onto the exterior of the unit circle |w|¿1:
Theorem 2.3. Suppose that the real sequence ( n ) satis es
lim
n→∞
n
=∞
n
(2.12)
and the function wn is de ned by (2:6). Then for z ∈ C[−1;1] the rescaled Laguerre polynomial Pn (z)
(see (1:4)) satis es
!
√
(−1)n n+1 n + n n=2 2
(n + 1)(n + n )
−1=2
n+1
Pn (z) = √
e
wn exp
(wn − 1)
n+1
wn
2n
×
wn
wn +
q
n+1
n+ n
n
as n → ∞:
(1 + o(1))
(2.13)
Here the branches of the non-integer powers are positive when wn is real and greater than sn (see
(2:4)). Moreover the o-term holds uniformly on compact subsets of C[−1;1] :
Proof. This uses ideas from [10, pp. 46–50; 24, p.160; 28], that is rst we establish formula (2.13)
for real z = x¿2; say, and then we extend its validity to the cut plane using Vitali’s theorem.
We start from representation (1.7). According to the considerations preceeding Lemma 2.1 we
choose tn as saddle point given by (2.7). Further as contour in (1.7) we take
Cn = Cnu ∪ Tn ∪ Cnl
consisting of the straight line
Tn := {tn − i n− | − 1 6 6 1};
where 0¡¡ 21 ; and the circular arcs
Cnu := {t = x + |x − tn − !n |ei |0 6 6
Cnl : = {t = x + |x − tn − !n |ei |2 −
n
n };
6 6 2};
where !n = − in− and − n = arc cos((x − tn )=|x − tn − !n |): (0¡arc cos ¡ for −1¡¡1:) Also
observe that on account of the above de nitions of the branches we have tn ¡x; if x¿sn : In order to
determine for the integral in (1.7) the contribution along Tn we employ Lemma 1.1. Using Lemma
2.2 it is easy to verify that conditions (1.9)–(1.12) are satis ed with arg pn′′ (tn ) = . Hence we get
1
e−p n (t) dt =
i
Tn
Z
s
2
|p′′n (tn )|
e−p n (tn ) (1 + o(1));
n → ∞:
(2.14)
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C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
Next we show that the contributions along Cnu ; Cnl can be neglected. To this end for t ∈ Cnu we
estimate Re(pn (t)−pn (tn )) from below. Using (1.8) and Lemma 2.1 elementary manipulations lead to
n
Re(pn (t) − pn (tn )) =Re (n + 1)(1 + Rn ei )
s
+ (n + n )
n + 1 1 + Rn ei
wn
n + n
+(n + 1) log Rn ;
06 6
n
− log 1 +
s
n + 1 1 + Rn ei
wn
n + n
n
!!)
n;
where (see (2.8))
Rn =
1 n−2
|x − tn − !n |
=1 +
+ O(n−4 );
x − tn
2 (x − tn )2
Further the well-known inequality
1
Re( − log(1 + )) ¿ − ||2 ;
2
implies
n → ∞:
∈ \ {−1};
|1 + Rn ei |2
+ log Rn
−
2wn2
Re(pn (t) − pn (tn )) ¿ (n + 1) 1 + Rn cos
1 + R2n
¿ (n + 1) 1 −
+ Rn cos
2wn2
06 6
yields
n:
Since Rn cos
n
(2.15)
n
1
1− 2
wn
!
!
+ log Rn ;
= − 1; nally the expansion (2.15) together with (2.11) for all t ∈ Cnu
1
Re(pn (t) − pn (tn )) ¿ (n + 1) 1 − 2
wn
n−2
+ O(n1−4 ) ¿ c n1−2 ;
2(x − tn )2
for some positive constant c: Lemma 2.2 again shows that |p′′n (tn )| grows like a positive multiple of
n and hence
Z
Z
e−p n (t) dt ;
e−p n (t) dt = o
Cnu
n → ∞:
Tn
The same estimate holds for the contribution along Cnl : Thus combining (1.7), (1.8), Lemma 2.1,
and (2.14) we have established (2.13) provided that z = x¿2:
Next, let z ∈ C[rn ; sn ] and |wn |¿1 being related as above. Putting = rei = 1=wn and using (0.1)
similar to [10, pp. 48–50] we obtain
n + n
n
( n + 1) =
∞
Z
0
√
|Ln( n ) (x)|2 x n e−x dx
¿ 2 n n
Z
sn
rn
=
√
√
|Pn (t)|2 (2 n n t + n ) n e−(2 n n t+ n ) dt
(n + 1)(n + n )
Pn
−
p
Z
1 + 2n
+
√
2 n n
s
(n + 1)(n + n )
cos
n n
! 2
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
85
p
× exp(−(1 + 2n + n + 2 (n + 1)(n + n ) cos ))
p
× (1 + 2n + n + 2 (n + 1)(n + n ) cos ) n | sin | d
p
= 2 (n + 1)(n + n )
1
lim
r→1− 2
p
Pn
−
Z
1 + 2n
+
√
2 n n
s
(n + 1)(n + n ) + −1
n n
2
¿ 2 (n + 1)(n + n )(1 − ||2 )
× Pn
where
(
Dn () = exp −
1
+
4
1
4
Z
−
1 + 2n
+
√
2 n n
s
(n + 1)(n + n ) + −1
n n
2
p
1 + 2n + n + 2 (n + 1)(n + n ) cos
! 2
|n |2 |Dn ()|2 ;
1 + e−i
1 − e−i
d
1 + e−i
log((1 + 2n + n + 2 (n + 1)(n + n ) cos ) n | sin |)
d
1 − e−i
−
Z
p
p
1
1
1 + 2n + n + 2 (n + 1)(n + n )
= exp −
2
wn
n
√
√
1
;
×
n + n + n + 1
wn
! 2
n2
| | |Dn ()|2 d
(wn2 − 1)
√
2wn
)
1=2
by Lemma 1.2, when n ¿ 1: Hence for z ∈ C[rn ; sn ] we have
q
n
)
( √
√
n+1
n=2
1=2
2
w
+
2n n + 1
(n + 1)(n + n ) n
(wn − 1)
n+ n
exp
−
Pn (z) n+1
e
n + n
wnn+1
wn
wn
√
|wn |
2|w|
6p
An → p 2
2
|wn | − 1
|w| − 1
(2.16)
by (2.11) and Stirling’s formula. Thus the left hand side of (2.16) is uniformly bounded on compact
subsets of C[−1;1] : Summarizing what we proved we obtain the statement of the theorem including
the compact convergence on C[−1;1] by Vitali’s theorem.
As a consequence of Theorem 2.3 by straightforward
computations it is veri ed that the logarithmic
√
derivative Pn′ (z)=nPn (z) converges to 2=(z + z 2 − 1) compactly on C[−1;1] : The latter function is well
known to be the Stieltjes transform of the semicircle law and thus for real n with n =n → ∞ as
n → ∞ we have another proof of Theorem 3.1, b in [4] saying that the zeros of√Ln( n ) contracted
according to (1.4) tend weakly to the probability distribution with the density (2=) 1 − x2 on (−1,
1). On the other hand it is well-known that this semicircle law is the limit distribution of the zeros
86
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
√
of the rescaled Hermite polynomials Hn ( 2n + 1z) e.g. [3, 4, 7, 9, 20, 29]. In view of the known
strong asymptotics (w is de ned in (2.11))
!
√
√
2n=2 n!
wn+1
1 w2 + 1
(1 + o(1)) ;
Hn ( 2n + 1z) =
exp n +
(2n)1=4 (w2 − 1)1=2
2
2w2
n → ∞; uniformly on compact subsets of C[−1;1] (e.g. [10, 24, 28]) the following relative asymptotics
follows from Theorem 2.3 by direct computations. We omit the details and give the result only in
Theorem 2.4. Suppose that the real sequence ( n ) satis es
n3
=0
n→∞ n
(2.17)
lim
and the function w is de ned in (2:11). Then we have
!
√
(−1)n n!2n=2 Ln( n ) (2 n n z + n )
w2 + 1
√
lim
= exp −
n→∞
4w2
Hn ( 2n + 1z)
nn=2
(2.18)
holding uniformly on compact subsets of C[−1;1] :
For a formula being similar to (2.18) compare problem 80 in [24, p. 389]. We close this section
with the ‘companion’ to Theorem 2.3 giving the oscillatory asymptotics for Pn (z); z now being in
the interval of zeros.
Theorem 2.5. Suppose that the real sequence ( n ) is chosen as in (2.12), ∈ (0; ) is xed and
the function n is de ned through
p
n () := − (n + 1)(n + n ) sin + n +
− n arctan
n + 1
+
2
4
!
√
√
n + n − n + 1
√
tan
:
√
n + n + n + 1 2
(2.19)
If
1 + 2n
+
x= √
2 n n
s
(n + 1)(n + n )
cos ;
n n
then the rescaled Laguerre polynomial Pn (x) (see (1:4)) satis es
(−1)n n+1 n + n
Pn (x) = √
e
n+1
n sin
×
as n → ∞:
1+2
s
n=2
p
exp( (n + 1)(n + n ) cos )
n+1
n+1
cos +
n + n
n + n
!− n =2
{sin n () + o(1)}
(2.20)
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
87
In (2.19) ‘arctan’ denotes the principal branch, that is −=2¡arctan ¡=2 for real : Moreover,
it can be shown that the remainder in (2.20) holds uniformly in ∈ [; − ] for 0¡¡: Since
the proof of Theorem 2.5 largely parallels that of Theorem 2.3 we do not carry out the reasonings.
We only mention that n () in (2.19) satis es n () = (n=2)(2 − sin 2)(1 + o(1)); n → ∞; thereby
exhibiting the similarity to the corresponding Plancherel-Rotach formula (8.22.12) in [24, p. 201]
for Hermite polynomials.
3. The case n =n → a
The special case n = an+ ; a ¿ 0; of a linear parameter sequence has been treated extensively in
[2, 10]. Now in the general case (0.2), that is limn→∞ ( n =n) = a ∈ [0; ∞); we evaluate the rescaled
Laguerre polynomial Qn given by the representation (1.5) asymptotically by using Lemma 1.1 again.
The calculations and reasonings are very similar to those in Section 2 and in [10]. Therefore we
omit the detailed proofs and state the results only. To this end besides (1.3) we use the following
notations
2p
2p
rn∗ = 2 −
(n + 1)(n + n );
sn∗ = 2 +
(n + 1)(n + n );
n
n
(3.1)
q
n
(z − 2 + (z − rn∗ )(z − sn∗ ));
wn∗ = √
2 (n + 1)(n + n )
where the square root is such that the cut z-plane C[rn∗; sn∗ ] corresponds to the exterior of the unit
circle |wn∗ |¿1: Obviously, we get
rn∗ → 2 −
4 √
1 + a =: r ∗ ;
2+a
sn∗ → 2 +
4 √
1 + a =: s∗ :
2+a
Now we have
Theorem 3.1. Suppose that the real sequence ( n ) satis es (0.2) with a ∈ [0; ∞) and that the
function wn∗ is de ned in (3:1). Then for z ∈ C[r∗; s∗ ] the rescaled Laguerre polynomial Qn (z) (see
(1:2)) satis es
!
√
(−1)n n+1 n + n n=2 ∗2
(n
+
1)(n
+
)
n
Qn (z) = √
(wn − 1)−1=2 wn∗n+1 exp
e
n+1
wn∗
2n
n
wn∗
(1 + o(1)); as n → ∞:
q
×
n+1
wn∗ + n+
n
Concerning the non-integer powers and the uniform valitity on C[r∗; s∗ ] similar properties hold as in
Theorem 2:3:
Theorem 3.2. Suppose that the real sequence ( n ) is chosen according to (0:2) with a ∈ [0; ∞);
∈ (0; ) is xed and the function n is de ned in (2:19). If
x=2 +
2p
(n + 1)(n + n ) cos ;
n
88
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
then the right-hand side of (2:20) also serves as an asymptotic form for the rescaled Laguerre
polynomial Qn (x) (see (1:3)).
As in Section 2 from Theorem 3.1 we obtain a proof of the known limit distributions of the
zeros of Qn which have been derived in [4, Theorem 3.1, a; 7, Theorem 3,i)] by a di erent method.
Finally we mention that on the basis of the results of this paper we could compute strong asymptotics
for the generalized Hermite polynomials Hn(
n ) with a degree dependent parameter
n which would
extend the results of Section 3 in [10]. However, we will not present the tedious but straightforward
computations.
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Strong asymptotics for Laguerre polynomials with varying
weights
Christof Bosbach, Wolfgang Gawronski ∗
Abteilung Mathematik, Universitat Trier, D-54286 Trier, Germany
Received 30 September 1997; received in revised form 2 April 1998
Abstract
Supplementing and extending classical and recent results strong asymptotics for the Laguerre polynomials Ln( n ) are
established, as n → ∞; when the parameter n depends on the degree n suitably. A case of particular interest is the one
for which n grows faster than n: Rescaling the argument z appropriately the resulting asymptotic forms are described
by elementary functions, thereby extending the classical formulae of Plancherel–Rotach type for Laguerre polynomials
c 1998 Elsevier Science B.V. All rights reserved.
L( )
n (z); being xed.
AMS classi cation: 30E15; 33C45
Keywords: Laguerre polynomials; Strong asymptotics; Plancherel-Rotach formulae
0. Introduction and summary
In this paper we deal with some generalizations and extensions of asymptotic results for Laguerre
polynomials Ln( ) : The de nition of these classical orthogonal polynomials and some well-known
formulae as well we take from Szego’s classic memoir [24, Chapter V]. Here we only mention the
de ning orthogonality relation
Z
0
∞
Ln( ) (x)Lm( ) (x)x e−x
dx =
n+
n
( + 1)nm ;
¿ − 1:
(0.1)
The classical asymptotic theory of these polynomials primarily is concerned with large degrees n
and a xed parameter [24, Chapter VIII]. Among the numerous asymptotic forms the so-called
formulae of Plancherel–Rotach type [24, p. 200] give asymptotic descriptions of the rescaled Laguerre
polynomials Ln( ) (n z); n = n + ( + 1)=2; in terms of elementary functions, as n → ∞; where the
∗
Corresponding author. E-mail: gawron@uni-trier.de.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 7 - 2
78
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
cases z ∈ (r; s) = (0; 4); the interval of zeros, and z ∈ C[r; s] ; the cut plane, have to be distinguished.
Here and throughout for real r; s with r¡s we use the notation
C[r; s] := C\[r; s]
that is the complex plane with a cut along the interval [r; s]: The resulting asymptotic approximations,
which hold uniformly on compact subsets of (r; s) and of C[r; s] , respectively, are named strong
asymptotics in contrast to the so-called weak asymptotics that is the limiting distribution for the
zeros.
For instance questions of weighted polynomial approximation and in the nonrelativistic quantum
theory led various authors to considering orthogonal polynomials with varying weights, i.e., with
weight functions depending on the degree of the polynomials. The most prominent representatives
of such orthogonal functions with weights living on an unbounded interval are Laguerre polynomials
Ln( n ) with a parameter n now being dependent on the degree n: The relevant literature concerning
the corresponding asymptotic theory primarily deals with weak asymptotics [3, 4, 7, 9, 11, 12, 17–19,
23] and some papers treat strong asymptotics in case of very special sequences ( n ): We mention the
contributions [2, 9, 10, 14–16, 22, 25, 26] where strong asymptotics for Ln( n ) have been established
provided the parameter is a linear function of the degree, that is n = an + ; a ¿ 0; ∈ R being
independent of n:
n)
It is the main object of this paper to derive strong asymptotics for L(
when now ( n ) is a more
n
general real sequence satisfying throughout the condition
n
= a ¿ 0:
(0.2)
lim
n→∞ n
In particular we admit the value a = ∞: The resulting generalized formulae of Plancherel–Rotach
type are stated and proved in Sections 2 and 3.
The recent publications [4, 7] treat weak asymptotics for classical orthogonal polynomials including
( n )
Ln under the general assumption (0.2). In particular it is shown for Ln( n ) with a = ∞ in (0.2) that
the asymptotic zero distribution is a semicircle law which also is well-known to be the weak limit
of the contracted zero distribution for the classical Hermite polynomials Hn : This coincidence of
the weak asymptotics gives rise to the question: In how far is this true for strong asymptotics?
As consequence of the strong asymptotics o the zero interval the answer is made precise by the
relative asymptotics
!
√
−z
(−1)n n!2n=2 Ln( n ) (2 n n z + n )
√
√
lim
= exp
n→∞
Hn ( 2n + 1z)
2(z + z 2 − 1)
nn=2
holding uniformly on compact subsets of C[−1;1] provided that n3 = o( n ); n → ∞ (Theorem 2.4).
Strong asymptotics in the sense explained above and used in this paper involves an approximation
of the underlying polynomials by elementary functions on each the interval of zeros (r; s) and the
cut plane [r; s]: Asymptotic expansions for Ln( ) (n z) as both n; → ∞ suitably and as z varies in
domains including (r; s) and parts of the cut plane lead to approximants being higher transcendental
functions [5, 6, 8, 25, 26]. However, in this paper we will not consider this point of view and refer
to the short discussion given in [10]. The proofs of the various asymptotic formulae rely on the
saddle point approximation of certain contour integrals combined with Vitali’s theorem on compact
convergence for sequences of analytic functions.
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
79
1. Auxiliary results
In this section we collect technical details and give some preliminary results which are basic for
the main theorems below.
Starting point is the well-known representation formula
Ln( ) (z) =
ez
2i
Z
e−t
C
t
z
t
t−z
n
dt
;
t−z
z 6= 0;
(1.1)
by means of a contour integral [24, p. 105, (5.4.8)], ∈ C; n ∈ N0 : Here C is a simple closed
contour with positive orientation enclosing t = z but not t = 0: Further the non-integer power of t=z
is assumed to reduce to 1 when t = z: In the light of various known asymptotic results in the weak
and in the strong sense [e.g. 4, 7, 10] under the general assumption (0.2) we consider the rescaled
Laguerre polynomials
Qn (z) := Ln( n ) (n z);
if a ∈ [0; ∞);
(1.2)
with
n := n + 12 ( n + 1);
√
Pn (z) := Ln( n ) (2 n n z + n );
(1.3)
if a = ∞:
(1.4)
Obvious substitutions in (1.1) lead to the basic representation formulae
e n z
Qn (z) =
2iz
Z
e−qn (t) dt;
C
z 6= 0;
(1.5)
t
t
− (n + 1) log
;
z
t−z
(1.6)
with
qn (t) := n t − ( n − 1) log
√
with
e2 n n z
p
p
Pn (z) =
2i(2 n= n )n (1 + 2 n= n z)
Z
Cn
p
e−pn (t) dt;
z 6=
1
2
r
n
;
n
(1.7)
p
√
1 + 2 n= n t
1 + 2 n= n t
p
pn (t) := 2 n n t − ( n − 1) log
− (n + 1) log
:
t−z
1 + 2 n= n z
(1.8)
In (1.7)
Cn is a simple closed contour with positive orientation enclosing t = z but not t =
p
− 21 n =n: Further in addition to the choice in (1.1) for the logarithms involving the factor t − z
p
in (1.6) and (1.8) we choose those branches which are real if t=(t − z) and (1 + 2 n= n t)=(t − z)
are real and positive, respectively. Now our main task consists in a saddle point approximation of
the contour integrals in (1.5) and (1.7). In the special case of a linear sequence n = an + ; a ¿ 0;
the parameter integral corresponding to that in (1.5) [10, p. 36] could be evaluated asymptotically
by the saddle point method in standard form [13, 21, 27, 30]. In the sequel we employ a variant
of the saddle point approximation which essentially is based on the following lemma taken from [1,
Section 20.2].
80
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
Lemma 1.1. Suppose that G ⊂ C is a domain and gn are complex valued functions; n ∈ N; being
twice di erentiable on G: Further let tn ∈ G; !n be complex numbers such that the straight lines
Tn = [tn − !n ; tn + !n ] are contained in G; n ∈ N. If (¿0)
gn′ (tn ) = 0;
n ∈ N;
(1.9)
lim |!n2 gn′′ (tn )| = ∞;
(1.10)
n→∞
|arg(!n gn′′ (tn )1=2 )| 6
− ;
4
gn′′ (t) = gn′′ (tn )(1 + o(1));
n ¿ n0 ();
(1.11)
n → ∞; uniformly with respect to t ∈ Tn ;
(1.12)
then we have
Z
−gn (t)
e
Tn
dt =
s
2
gn′′ (tn )
e−gn (tn ) (1 + o(1));
n → ∞:
Condition (1.11) is to x the correct branch of the square root gn′′ (tn )1=2 :
Finally, in this section we list some special Poisson integrals the computations of which are
readily performed either by residue calculus or by mean value formulae for holomorphic functions
[24, pp. 275–277; 10, p. 40]. Therefore we omit their straightforward proofs.
Lemma 1.2. Suppose that the real numbers c; d satisfy c¿0; c ¿ d ¿ 0 and is a complex number
with ||¡1; then
(i)
1
2
Z
1
2
Z
(c + d cos )
−
1 + e−i
d = c + d;
1 − e−i
(ii)
log(c2 + d2 + 2cd cos )
−
1 + e−i
d = 2 log(c + d);
1 − e−i
log being the principal branch; that is log x is real if x is real and positive;
(iii)
1
2
1 + e−i
log(sin )
d = log
1 − e−i
−
Z
2
(
1
−
2
2 )
:
2. The case n =n → ∞
In this section we expand the contour integral in (1.7) asymptotically thereby establishing the
strong asymptotics for the polynomials Pn de ned in (1.4). Basically we follow arguments in [10].
Although here the technical details are more involved we restrict our computations and reasonings
to some essential steps.
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
81
The saddle point equation pn′ (t) = 0 for (1.7) now is equivalent to
√
2 n n −
n + n
p
2
1 + 2 n= n t
r
n
n+1
+
= 0:
n
t−z
(2.1)
Its solutions are given explicitly by
!
p
1
1
z− √
± (z − rn )(z − sn )
tn± (z) =
2
2 n n
(2.2)
with a proper determination of the square root function below and
p
1
rn = √ (1 + 2n − 2 (n + 1)(n + n ));
2 n n
(2.3)
p
1
sn = √ (1 + 2n + 2 (n + 1)(n + n )):
2 n n
(2.4)
Obviously, on account of (0.2) we have rn → −1; sn → 1 as n → ∞: Next, for z ∈ C[rn ; sn ] the variable
wn is de ned through
rn + sn sn − rn wn + wn−1 1 + 2n
z=
+
+
= √
2
2
2
2 n n
s
(n + 1)(n + n ) wn + wn−1
;
n n
2
(2.5)
|wn |¿1: This means the exterior of the unit circle in the wn -plane is mapped conformally onto the
domain C[rn ; sn ] in the z-plane such that the point wn = ∞ corresponds to z = ∞: Moreover the upper
and lower edge of the cut in the z-plane is mapped onto the upper and lower half of the unit circle
in the wn -plane, respectively. From (2.5) we have
√
√
z − r n + z − sn
2
r n + sn p
√
+ (z − rn )(z − sn )
=
z−
wn = √
z − r n − z − s n sn − rn
2
s
=
!
1 + 2n p
n n
z− √
+ (z − rn )(z − sn ) :
(n + 1)(n + n )
2 n n
(2.6)
In view of (2.2) we put
1
tn :=
2
s
(n + 1)(n + n )
wn +
n n
r
n
n
!
(2.7)
giving
!
p
1
1
tn =
z− √
+ (z − rn )(z − sn )
2
2 n n
(2.8)
which means that in (2.2) we make thep
choice of p
the branch such that C[rn ; sn ] is mapped conformally
onto the exterior of the circle |t − 12 n= n |¿ 12 (n + 1)(n + n )=n n : Further on the cut (rn ; sn )
82
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
in the z-plane and its images according to (2.5) and (2.7) we have
1 + 2n
z = √
+
2 n n
s
(n + 1)(n + n )
cos ;
n n
0¡¡:
s
wn =ei and tn = 21
(n + 1)(n + n ) i
e +
n n
r
n
n
!
(2.9)
;
Now similar to [10, pp. 38,39] straightforward calculations lead to connection formulae in
Lemma 2.1. If the complex variables z; wn , and tn are related by (2.5)–(2.8), then we have
(i)
q
1 n + 1 wn +
tn − z = − √
2 n n
wn
n+ n
n+1
;
(ii)
1+2
r
n
tn =
n
√
(n + 1)(n + n )
wn +
n
s
n + n
n+1
!
;
(iii)
q
1+2
n
t
n n
tn − z
= −2
r
n
n
s
n + n
wn ;
n+1
(iv)
p
1 + 2 (n= n )tn
wn
p
p
=
:
1 + 2 (n= n )z wn + (n + 1)=(n + n )
Next, from (1.8) we get
pn′′ (t) =
n+1
4n
n + n
p
−
n (1 + 2 (n= n )t)2 (t − z)2
(2.10)
from which the following identity is immediate by Lemma 2.1.
Lemma 2.2. If the complex variables z; wn and tn are related by (2.5)–(2.8), then we have
pn′′ (tn ) =
1 − wn2
4n n
n + 1 1 − wn2
:
=
p
(tn − z)2 wn2
n + 1 (wn + (n + n )=(n + 1))2
After these preparations we establish the desired strong asymptotics for Pn (z) (see (1.4)) on the
cut plane C[−1;1] : Before, we mention that passing to the limit n → ∞ in (2.6) we obtain the familiar
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
83
Jukowski function
√
√
√
z+1+ z−1
√
w := lim wn = √
= z + z2 − 1
n→∞
z+1− z−1
(2.11)
mapping C[−1;1] conformally onto the exterior of the unit circle |w|¿1:
Theorem 2.3. Suppose that the real sequence ( n ) satis es
lim
n→∞
n
=∞
n
(2.12)
and the function wn is de ned by (2:6). Then for z ∈ C[−1;1] the rescaled Laguerre polynomial Pn (z)
(see (1:4)) satis es
!
√
(−1)n n+1 n + n n=2 2
(n + 1)(n + n )
−1=2
n+1
Pn (z) = √
e
wn exp
(wn − 1)
n+1
wn
2n
×
wn
wn +
q
n+1
n+ n
n
as n → ∞:
(1 + o(1))
(2.13)
Here the branches of the non-integer powers are positive when wn is real and greater than sn (see
(2:4)). Moreover the o-term holds uniformly on compact subsets of C[−1;1] :
Proof. This uses ideas from [10, pp. 46–50; 24, p.160; 28], that is rst we establish formula (2.13)
for real z = x¿2; say, and then we extend its validity to the cut plane using Vitali’s theorem.
We start from representation (1.7). According to the considerations preceeding Lemma 2.1 we
choose tn as saddle point given by (2.7). Further as contour in (1.7) we take
Cn = Cnu ∪ Tn ∪ Cnl
consisting of the straight line
Tn := {tn − i n− | − 1 6 6 1};
where 0¡¡ 21 ; and the circular arcs
Cnu := {t = x + |x − tn − !n |ei |0 6 6
Cnl : = {t = x + |x − tn − !n |ei |2 −
n
n };
6 6 2};
where !n = − in− and − n = arc cos((x − tn )=|x − tn − !n |): (0¡arc cos ¡ for −1¡¡1:) Also
observe that on account of the above de nitions of the branches we have tn ¡x; if x¿sn : In order to
determine for the integral in (1.7) the contribution along Tn we employ Lemma 1.1. Using Lemma
2.2 it is easy to verify that conditions (1.9)–(1.12) are satis ed with arg pn′′ (tn ) = . Hence we get
1
e−p n (t) dt =
i
Tn
Z
s
2
|p′′n (tn )|
e−p n (tn ) (1 + o(1));
n → ∞:
(2.14)
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C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
Next we show that the contributions along Cnu ; Cnl can be neglected. To this end for t ∈ Cnu we
estimate Re(pn (t)−pn (tn )) from below. Using (1.8) and Lemma 2.1 elementary manipulations lead to
n
Re(pn (t) − pn (tn )) =Re (n + 1)(1 + Rn ei )
s
+ (n + n )
n + 1 1 + Rn ei
wn
n + n
+(n + 1) log Rn ;
06 6
n
− log 1 +
s
n + 1 1 + Rn ei
wn
n + n
n
!!)
n;
where (see (2.8))
Rn =
1 n−2
|x − tn − !n |
=1 +
+ O(n−4 );
x − tn
2 (x − tn )2
Further the well-known inequality
1
Re( − log(1 + )) ¿ − ||2 ;
2
implies
n → ∞:
∈ \ {−1};
|1 + Rn ei |2
+ log Rn
−
2wn2
Re(pn (t) − pn (tn )) ¿ (n + 1) 1 + Rn cos
1 + R2n
¿ (n + 1) 1 −
+ Rn cos
2wn2
06 6
yields
n:
Since Rn cos
n
(2.15)
n
1
1− 2
wn
!
!
+ log Rn ;
= − 1; nally the expansion (2.15) together with (2.11) for all t ∈ Cnu
1
Re(pn (t) − pn (tn )) ¿ (n + 1) 1 − 2
wn
n−2
+ O(n1−4 ) ¿ c n1−2 ;
2(x − tn )2
for some positive constant c: Lemma 2.2 again shows that |p′′n (tn )| grows like a positive multiple of
n and hence
Z
Z
e−p n (t) dt ;
e−p n (t) dt = o
Cnu
n → ∞:
Tn
The same estimate holds for the contribution along Cnl : Thus combining (1.7), (1.8), Lemma 2.1,
and (2.14) we have established (2.13) provided that z = x¿2:
Next, let z ∈ C[rn ; sn ] and |wn |¿1 being related as above. Putting = rei = 1=wn and using (0.1)
similar to [10, pp. 48–50] we obtain
n + n
n
( n + 1) =
∞
Z
0
√
|Ln( n ) (x)|2 x n e−x dx
¿ 2 n n
Z
sn
rn
=
√
√
|Pn (t)|2 (2 n n t + n ) n e−(2 n n t+ n ) dt
(n + 1)(n + n )
Pn
−
p
Z
1 + 2n
+
√
2 n n
s
(n + 1)(n + n )
cos
n n
! 2
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
85
p
× exp(−(1 + 2n + n + 2 (n + 1)(n + n ) cos ))
p
× (1 + 2n + n + 2 (n + 1)(n + n ) cos ) n | sin | d
p
= 2 (n + 1)(n + n )
1
lim
r→1− 2
p
Pn
−
Z
1 + 2n
+
√
2 n n
s
(n + 1)(n + n ) + −1
n n
2
¿ 2 (n + 1)(n + n )(1 − ||2 )
× Pn
where
(
Dn () = exp −
1
+
4
1
4
Z
−
1 + 2n
+
√
2 n n
s
(n + 1)(n + n ) + −1
n n
2
p
1 + 2n + n + 2 (n + 1)(n + n ) cos
! 2
|n |2 |Dn ()|2 ;
1 + e−i
1 − e−i
d
1 + e−i
log((1 + 2n + n + 2 (n + 1)(n + n ) cos ) n | sin |)
d
1 − e−i
−
Z
p
p
1
1
1 + 2n + n + 2 (n + 1)(n + n )
= exp −
2
wn
n
√
√
1
;
×
n + n + n + 1
wn
! 2
n2
| | |Dn ()|2 d
(wn2 − 1)
√
2wn
)
1=2
by Lemma 1.2, when n ¿ 1: Hence for z ∈ C[rn ; sn ] we have
q
n
)
( √
√
n+1
n=2
1=2
2
w
+
2n n + 1
(n + 1)(n + n ) n
(wn − 1)
n+ n
exp
−
Pn (z) n+1
e
n + n
wnn+1
wn
wn
√
|wn |
2|w|
6p
An → p 2
2
|wn | − 1
|w| − 1
(2.16)
by (2.11) and Stirling’s formula. Thus the left hand side of (2.16) is uniformly bounded on compact
subsets of C[−1;1] : Summarizing what we proved we obtain the statement of the theorem including
the compact convergence on C[−1;1] by Vitali’s theorem.
As a consequence of Theorem 2.3 by straightforward
computations it is veri ed that the logarithmic
√
derivative Pn′ (z)=nPn (z) converges to 2=(z + z 2 − 1) compactly on C[−1;1] : The latter function is well
known to be the Stieltjes transform of the semicircle law and thus for real n with n =n → ∞ as
n → ∞ we have another proof of Theorem 3.1, b in [4] saying that the zeros of√Ln( n ) contracted
according to (1.4) tend weakly to the probability distribution with the density (2=) 1 − x2 on (−1,
1). On the other hand it is well-known that this semicircle law is the limit distribution of the zeros
86
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
√
of the rescaled Hermite polynomials Hn ( 2n + 1z) e.g. [3, 4, 7, 9, 20, 29]. In view of the known
strong asymptotics (w is de ned in (2.11))
!
√
√
2n=2 n!
wn+1
1 w2 + 1
(1 + o(1)) ;
Hn ( 2n + 1z) =
exp n +
(2n)1=4 (w2 − 1)1=2
2
2w2
n → ∞; uniformly on compact subsets of C[−1;1] (e.g. [10, 24, 28]) the following relative asymptotics
follows from Theorem 2.3 by direct computations. We omit the details and give the result only in
Theorem 2.4. Suppose that the real sequence ( n ) satis es
n3
=0
n→∞ n
(2.17)
lim
and the function w is de ned in (2:11). Then we have
!
√
(−1)n n!2n=2 Ln( n ) (2 n n z + n )
w2 + 1
√
lim
= exp −
n→∞
4w2
Hn ( 2n + 1z)
nn=2
(2.18)
holding uniformly on compact subsets of C[−1;1] :
For a formula being similar to (2.18) compare problem 80 in [24, p. 389]. We close this section
with the ‘companion’ to Theorem 2.3 giving the oscillatory asymptotics for Pn (z); z now being in
the interval of zeros.
Theorem 2.5. Suppose that the real sequence ( n ) is chosen as in (2.12), ∈ (0; ) is xed and
the function n is de ned through
p
n () := − (n + 1)(n + n ) sin + n +
− n arctan
n + 1
+
2
4
!
√
√
n + n − n + 1
√
tan
:
√
n + n + n + 1 2
(2.19)
If
1 + 2n
+
x= √
2 n n
s
(n + 1)(n + n )
cos ;
n n
then the rescaled Laguerre polynomial Pn (x) (see (1:4)) satis es
(−1)n n+1 n + n
Pn (x) = √
e
n+1
n sin
×
as n → ∞:
1+2
s
n=2
p
exp( (n + 1)(n + n ) cos )
n+1
n+1
cos +
n + n
n + n
!− n =2
{sin n () + o(1)}
(2.20)
C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
87
In (2.19) ‘arctan’ denotes the principal branch, that is −=2¡arctan ¡=2 for real : Moreover,
it can be shown that the remainder in (2.20) holds uniformly in ∈ [; − ] for 0¡¡: Since
the proof of Theorem 2.5 largely parallels that of Theorem 2.3 we do not carry out the reasonings.
We only mention that n () in (2.19) satis es n () = (n=2)(2 − sin 2)(1 + o(1)); n → ∞; thereby
exhibiting the similarity to the corresponding Plancherel-Rotach formula (8.22.12) in [24, p. 201]
for Hermite polynomials.
3. The case n =n → a
The special case n = an+ ; a ¿ 0; of a linear parameter sequence has been treated extensively in
[2, 10]. Now in the general case (0.2), that is limn→∞ ( n =n) = a ∈ [0; ∞); we evaluate the rescaled
Laguerre polynomial Qn given by the representation (1.5) asymptotically by using Lemma 1.1 again.
The calculations and reasonings are very similar to those in Section 2 and in [10]. Therefore we
omit the detailed proofs and state the results only. To this end besides (1.3) we use the following
notations
2p
2p
rn∗ = 2 −
(n + 1)(n + n );
sn∗ = 2 +
(n + 1)(n + n );
n
n
(3.1)
q
n
(z − 2 + (z − rn∗ )(z − sn∗ ));
wn∗ = √
2 (n + 1)(n + n )
where the square root is such that the cut z-plane C[rn∗; sn∗ ] corresponds to the exterior of the unit
circle |wn∗ |¿1: Obviously, we get
rn∗ → 2 −
4 √
1 + a =: r ∗ ;
2+a
sn∗ → 2 +
4 √
1 + a =: s∗ :
2+a
Now we have
Theorem 3.1. Suppose that the real sequence ( n ) satis es (0.2) with a ∈ [0; ∞) and that the
function wn∗ is de ned in (3:1). Then for z ∈ C[r∗; s∗ ] the rescaled Laguerre polynomial Qn (z) (see
(1:2)) satis es
!
√
(−1)n n+1 n + n n=2 ∗2
(n
+
1)(n
+
)
n
Qn (z) = √
(wn − 1)−1=2 wn∗n+1 exp
e
n+1
wn∗
2n
n
wn∗
(1 + o(1)); as n → ∞:
q
×
n+1
wn∗ + n+
n
Concerning the non-integer powers and the uniform valitity on C[r∗; s∗ ] similar properties hold as in
Theorem 2:3:
Theorem 3.2. Suppose that the real sequence ( n ) is chosen according to (0:2) with a ∈ [0; ∞);
∈ (0; ) is xed and the function n is de ned in (2:19). If
x=2 +
2p
(n + 1)(n + n ) cos ;
n
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C. Bosbach, W. Gawronski / Journal of Computational and Applied Mathematics 99 (1998) 77–89
then the right-hand side of (2:20) also serves as an asymptotic form for the rescaled Laguerre
polynomial Qn (x) (see (1:3)).
As in Section 2 from Theorem 3.1 we obtain a proof of the known limit distributions of the
zeros of Qn which have been derived in [4, Theorem 3.1, a; 7, Theorem 3,i)] by a di erent method.
Finally we mention that on the basis of the results of this paper we could compute strong asymptotics
for the generalized Hermite polynomials Hn(
n ) with a degree dependent parameter
n which would
extend the results of Section 3 in [10]. However, we will not present the tedious but straightforward
computations.
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