Weak convergence of orthogonal polynomials

13 Pages·0.17 MB·English

by  Walter Van Assche

WEAK CONVERGENCE OF ORTHOGONAL POLYNOMIALS
4
9
9 Walter Van Assche
1
Katholieke Universiteit Leuven
n
a
J
Abstract. The weak convergence of orthogonal polynomials is given under conditions on the asymptotic
1
behaviour of the coefficients in the three-term recurrence relation. The results generalize known results and
2
are applied to several systems of orthogonal polynomials, including orthogonal polynomials on a finite set of

] points.
A
C
.
h
t 1. Introduction
a
m
Let p (x) be a system of orthonormal polynomials on the real line, with orthogonality measure µ, i.e.,
[ n
µ is a probability measure for which all the moments exist and

1
v
9 p (x)p (x)dµ(x) = δ , m,n 0.
n m m,n
0 ≥
2 Z
1 When the support of µ consists of a finite number of points x ,x ,... ,x , then we only consider the
0 1 2 N
polynomials up to degree N and p (x) has its zeros at the support x ,x ,... ,x . It is well known
4 N 1 2 N
{ }
9 that orthonormal polynomials satisfy a three-term recurrence relation
/
h
t (1.1) xp (x) = a p (x)+b p (x)+a p (x), n 0,
a n n+1 n+1 n n n n−1
≥
m
: with initial values p0(x) = 1 and p−1(x) = 0. Here
v
i
X
a = xp (x)p (x)dµ(x), b = xp2(x)dµ(x) R, n 0.
r n+1 n n+1 n n ∈ ≥
a Z Z
Usuallytheorthonormalpolynomialsarechoseninsuchawaythattheleadingcoefficientγ = (a a a
n 1 2 n
···
is positive, and then a = γ /γ > 0 for every n 1.
n n n−1
≥
Weareinterested intheweak asymptoticbehaviourofthe orthonormalpolynomialsp (x). Thismeans
n
that we want to investigate the behaviour for n of integrals of the form
→ ∞
f(x)p2(x)dµ(x), f C ,
n ∈ b
Z
where C is the linear space of bounded and continuous functions on R.
b
We will consider a one-parameter orthogonality measure µ (k N), the parameter being discrete.
k
∈
This implies that the recurrence coefficients and the orthogonal polynomials all depend on this discrete
parameter and we write
(1.2) xp (x;µ ) = a p (x;µ )+b p (x;µ )+a p (x;µ ), n 0.
n k n+1,k n+1 k n,k n k n,k n−1 k
≥
Our main result will be the limit as n of integrals of the form
→ ∞
Z
2 WALTER VAN ASSCHE
The special case k = l = 0 then gives the desired weak convergence, but the general case with k,l Z
∈
also has useful applications: these integrals are related to the transition probabilities of birth and death
processes and random walks.
Such one-parameter families do occur frequently in various applications and limiting procedures. For
example the rescaling of orthogonal polynomials
p (c x;µ) = p (x;µ )
n k n k
gives the one-parameter family of measures µ with distribution functions satisfying
k
µ (t) = µ(c t), t R.
k k
∈
Other examples include orthogonal polynomials in which some of the parameters are allowed to tend
(α,β)
to infinity together with the degree, e.g., if P (x) is the Jacobi polynomial of degree n, with weight
n
function
w(x) = (1 x)α(1+x)β, 1 < x < 1,
− −
(kα+γ,kβ+δ)
then P (x) is an orthogonal polynomial of degree n with weight function
n
w (x) = wk(x)(1 x)γ(1+x)δ, 1 < x < 1.
k
− −
The main result will be in terms of a doubly infinite Jacobi matrix
. .
. .
. .
.
.
 . b0 a0 
−2 −1
a0 b0 a0
 −1 −1 0 
=  a0 b0 a0 ,
J  0 0 1 
 a0 b0 a0 
 1 1 2 
 a0 b0 a0 
 2 2 3 
 . . . 
 .. .. ..
 
 
and is the following
Theorem. Suppose that the recurrence coefficients in (1.2) satisfy
(1.3) lim a = a0 > 0, lim b = b0 R
n→∞ n+k,n k n→∞ n+k,n k ∈
for every k Z. Then
∈
lim f(x)p (x;µ )p (x;µ )dµ (x)
n+k n n+l n n
n→∞
Z
A (x)
= f(x)(A (x) B (x))dµ(x) l ,
k k B (x)
l
Z (cid:18) (cid:19)
for every polynomial f. Here µ is the spectral matrix of measures for the doubly infinite Jacobi matrix
J
with entries a0,b0 (n Z) and
n n ∈
0
a0p(1) (x)
−a01 n−1 for n 0,
≥
An(x) =  pn(x) !
B (x)  q (x)
n  |n|−1
(cid:18) (cid:19)  0 for n < 0,
 −aa0−01q|(n1|)−2(x)!


with p (x) and q (x) the orthonormal polynomials with recurrence coefficients respectively a0 ,b0 (n =
n n
ORTHOGONAL POLYNOMIALS 3
2. Spectral theory for Jacobi operators
If we put the coefficients a > 0 (n = 1,2,...) and b R (n = 0,1,2,...) of the recurrence relation
n n
∈
(1.1) in a tridiagonal matrix
b a
0 1
a b a
1 1 2
 a b a 
(2.1) J = 2 2 3
a b a
 3 3 4 
 . . . 
 .. .. ..
 
 
then J is the Jacobi matrix associated with these orthogonal polynomials. If µ is supported on N points,
then a = 0 = b for n N and J is a N N matrix with eigenvalues at the support x ,x ,... ,x .
n n 1 2 N
≥ × { }
Note that
(2.2) J = xp (x)p (x)dµ(x) ,
i j
(cid:18)Z (cid:19)i,j=0,1,...
and by induction we find
(2.3) Jk = xkp (x)p (x)dµ(x) .
i j
(cid:18)Z (cid:19)i,j=0,1,...
In fact, J : ℓ (N,C) ℓ (N,C) acts as a linear operator on the Hilbert space ℓ (N,C) = ψ : ψ
2 2 2 i
C and ∞ ψ 2 < → . The operator is symmetric on the initial domain consisting of fin{ite linea∈r
i=0| i| ∞}
combinations of the basic vectors e+ = (0,0,... ,0,1,0,...) : n 0 and by imposing appropriate
P { n ≥ }
n zeros
conditions on the recurrence coefficients a ,b (e.g., boundedness) this operator can be extended in
n+1 n
a unique way to a self-adjoint operator on |the {mzaxi}mal domain ψ ℓ (N,C) : Jψ ℓ (N,C) . This
2 2
{ ∈ ∈ }
operator has a cyclic vector, i.e., if we take e+ = (1,0,0,0,...), then the linear span of Jke+ : k =
0 { 0
0,1,... is dense in ℓ (N,C). The spectral theorem (see, e.g., Akhiezer and Glazman [2] or Stone [19])
2
}
then implies the existence of a measure µ and a linear mapping Λ : ℓ (N,C) L (µ) with Λe+ = 1 and
2 → 2 0
ΛJψ = MΛψ for every ψ ℓ (N,C), where M is the multiplication operator
2
∈
(Mf)(t) = tf(t).
The mapping Λ is unitary, meaning Λψ,Λφ = ψ,φ . The mapping Λ thus maps e+ to the constant
h i h i 0
function t 1, Je+ to the identity t t, and in general Λ maps Jne+ to the monomial t tn. Hence
7→ 0 7→ 0 7→
the fact that e+ is a cyclic vector is equivalent with the density of polynomials in L (µ). By induction
0 2
and by using the recurrence relation (1.1) we see that Λ maps the basic vector e+ to the polynomial p .
n n
The unitarity thus implies that
p (t)p (t)dµ(t) = e+,e+ = δ ,
n m h n mi m,n
Z
whichshowsthatthespectralmeasureµfortheoperatorJ istheorthogonalitymeasurefortheorthogonal
polynomials p (x) (n = 0,1,2,...). These elements from spectral theory, applied to the semi-infinite
n
Jacobi matrix J, are well-known (see e.g., Akhiezer [1], Dombrowski [6], Sarason [18], Stone [19]).
We will also need to use doubly infinite Jacobi matrices of the form
. .
. .
. .
.
 
 
4 WALTER VAN ASSCHE
with a > 0,b R for k Z. Again these are operators, now acting on the Hilbert space ℓ (Z,C) = ψ :
k k 2
ψ C and ∞∈ ψ 2∈< . The spectral theory of such matrices is less known, but has also b{een
i ∈ i=−∞| i| ∞}
developped (see, e.g., Berezanski˘ı [4], Nikishin [17], Masson and Repka [13]). We will briefly recall some
P
of the elements, which we have taken from Nikishin [17] (see also Berezanski˘ı [4, Chapter VII, 3]). We
§
will first consider bounded matrices , i.e., we assume that a and b (n Z) are bounded sequences.
n n
J ∈
It turns out that it is convenient to introduce the 2 2 matrices
×
b a b 0
B = −1 0 , B = −n−1 n = 1,2,...
0 a b n 0 b
0 0 n
(cid:18) (cid:19) (cid:18) (cid:19)
a 0
A = −n , n = 1,2,... .
n 0 a
n
(cid:18) (cid:19)
We can now study the semi-infinite Jacobi block matrix
B A
0 1
A B A
1 1 2
(2.5) J =  A2 B2 A3 
A B A
 3 3 4 
 . . . 
 .. .. ..
 
 
which contains 2 2 matrices. As an operator it acts on ℓ (N,C2) in the sense that ψ for ψ ℓ (Z,C)
2 2
× J ∈
corresponds to JΨ with Ψ ℓ (N,C2) given by
2
∈
ψ
Ψ = −n−1 , n = 0,1,2,... .
n ψ
n
(cid:18) (cid:19)
In this way we have transformed the study of the doubly-infinite Jacobi matrix to the study of a semi-
infinite Jacobi block matrix. Such semi-infinite block matrices are closely connected to orthogonal matrix
polynomials, in the same way as ordinary Jacobi matrices are connected to scalar orthogonal polynomials
(see, e.g., Aptekarev and Nikishin [3]).
Consider the standard basis e (n Z) in ℓ (Z,C), i.e.,
n 2
∈
(e ) = δ , i,n Z,
n i i,n
∈
then the linear span of ke , le ,k,l N is dense in ℓ (Z,C). By the spectral theorem there exists
−1 0 2
{J J ∈ }
a matrix-measure
µ µ
µ = 1,1 1,2
µ µ
1,2 2,2
(cid:18) (cid:19)
and a unitary linear mapping Λ : ℓ (Z,C) L (µ) with
2 2
→
1 0
Λe = , Λe = ,
−1 0 0 1
(cid:18) (cid:19) (cid:18) (cid:19)
such that
Λ ψ = MΛψ,
J
where now M is the multiplication operator in the space L (µ) of vector valued functions. The inner
2
product in the space L (µ) is given by
2
(cid:18) (cid:19) (cid:18) (cid:19) Z (cid:18) (cid:19)(cid:18) (cid:19)
ORTHOGONAL POLYNOMIALS 5
whereas ne is mapped to
0
J
0
t .
7→ tn
(cid:18) (cid:19)
Let J+ be the semi-infinite Jacobi matrix
b a
0 1
a b a
1 1 2
J+ =  a b a ,
2 2 3
. . .
 .. .. ..
 
 
and similarly J− be the semi-infinite Jacobi matrix
b a
−1 −1
a b a
−1 −2 −2
J− =  a b a .
−2 −3 −3
. . .
 .. .. ..
 
 
We denote by p (x) the orthonormal polynomials corresponding to the Jacobi-matrix J+ satisfying the
n
recurrence relation
(2.6) xp (x) = a p (x)+b p (x)+a p (x),
n n+1 n+1 n n n n−1
with initial values p (x) = 0,p (x) = 1, and by q (x) the orthonormal polynomials for J− satisfying
−1 0 n
(2.7) xq (x) = a q (x)+b q (x)+a q (x),
n −n−1 n+1 −n−1 n −n n−1
with initial values q (x) = 0,q (x) = 1. We will show, by induction, that for n N the mapping Λ
−1 0
∈
maps the basis vector e to Λe which is the vector function
n n
a0p(1) (t)
(2.8) t −a1 n−1
7→ p (t)
(cid:18) n (cid:19)
and the basisvector e to Λe given by
−n −n
q (t)
n−1
(2.9) t .
7→ a0 q(1) (t)
(cid:18)−a−1 n−2 (cid:19)
(1) (1)
Here p (x) and q (x) are the associated polynomials, i.e., the orthonormal polynomials corresponding
n n
to the Jacobi matrices J+ and J−, with the first row and column deleted. This is clear for e and e .
−1 0
Assume that this is true for 0 n k, then from
≤ ≤
e = a e +b e +a e
k k+1 k+1 k k k k−1
J
it follows that
Λ e = a Λe +b Λe +a Λe ,
k k+1 k+1 k k k k−1
J
and since Λ e = MΛe this gives
k k
J
(cid:16) (cid:17)!
6 WALTER VAN ASSCHE
and Λ e = MΛe , this gives
−k −k
J
(Λe )(t) = 1 (t−b−k)qk−1(t)−a−k+1qk−2(t) = qk(t) .
−k−1 a a0 (t b )q(1) (t) a q(1) (t) a0 q(1) (t)
−k −a−1 − −k k−2 − −k+1 k−3 ! (cid:18)−a−1 k−1 (cid:19)
(cid:16) (cid:17)
From the unitarity we find for m,n Z
∈
Λe ,Λe = δ ,
n m m,n
h i
Hence the matrix polynomials
q (t) a0 q(1) (t)
P (t) = n −a−1 n−1
n a0p(1) (t) p (t) !
−a1 n−1 n
satisfy
1 0
P (t)dµ(t)P (t)∗ = δ .
n m 0 1 m,n
Z (cid:18) (cid:19)
Therefore these matrix polynomials are orthonormal with respect to the matrix-measure µ. Note that
these matrix polynomials satisfy
tP (t) = A P (t)+B P (t)+A P (t),
n n+1 n+1 n n n n−1
1 0
P (t) = ,
0 0 1
(cid:18) (cid:19)
so that they are the orthonormal polynomials corresponding to the block Jacobi matrix J given in (2.5).
The polynomials p (x) (n = 0,1,2,...) are orthonormal with respect to some probability measure µ+,
n
and the polynomials q (x) (n = 0,1,2,...) are orthonormal with respect to some probability measure
n
µ−. In order to find a relation between the matrix-measure µ and the measures µ+ and µ−, we observe
that the unitarity of Λ implies
1
(z )−1e ,e = (Λe )∗dµ(t)Λe ,
n m n m
h −J i z t
Z −
where (z )−1 is the resolvent of , which is well-defined for every z outside the spectrum of . Since
−J J J
is symmetric and bounded, it follows that is self-adjoint so that its spectrum is a subset of the real
J J
line. The Stieltjes transform of the matrix-measure µ is determined by
1 1
(z )−1e ,e = dµ (t), (z )−1e ,e = dµ (t),
−1 −1 1,1 −1 0 1,2
h −J i z t h −J i z t
Z − Z −
1
(z )−1e ,e = dµ (t).
0 0 2,2
h −J i z t
Z −
On the other hand, if we write
(z )−1e = r = (... ,r ,r ,r ,r ,r ,...),
0 −2 −1 0 1 2
−J
then (z )r = e , which gives the infinite system of equations
0
−J
ORTHOGONAL POLYNOMIALS 7
combination of the orthogonal polynomials p (z) (respectively q (z)) and the functions of the second
n n
kind p˜ (z) (respectively q˜ (z)) given by
n n
p (x) q (x)
p˜ (z) = n dµ+(x), q˜ (z) = n dµ−(x),
n n
z x z x
Z − Z −
with a p˜ (z) = 1 = a q˜ (z). These functions of the second kind have the property that they are a
0 −1 0 −1
minimalsolutionoftherecurrence relation,andforz C Rtheysatisfylim p˜ (z) = lim q˜ (z) =
n→∞ n n→∞ n
∈ \
0. The fact that r ℓ (Z,C) thus implies that r and r are (up to a constant factor) given by the
2 n −n
∈
functions of the second kind p˜ (z) and q˜ (z) respectively. The constant multiple is determined by
n n−1
setting n = 0, giving
p˜ (z)
n
r = r , r = a r q˜ (z), n 0.
n 0 −n 0 0 n−1
p˜ (z) ≥
0
In particular r = r p˜ (z)/p˜ (z) and r = a r q˜ (z). Inserting in (2.10) gives
1 0 1 0 −1 0 0 0
1
r = ,
0 z a2q˜ (z) b a p˜ (z)/p˜ (z)
− 0 0 − 0 − 1 1 0
which by using a p˜ (z) = (z b )p˜ (z) 1 gives
1 1 0 0
− −
p˜ (z)
0
r = .
0 1 a2p˜ (z)q˜ (z)
− 0 0 0
In a similar way we may investigate
(z )−1e = s = (... ,s ,s ,s ,s ,s ,...),
−1 −2 −1 0 1 2
−J
which gives the infinite system of linear equations
zs = a s +b s +a s , k 0,
k k k−1 k k k+1 k+1
≥
(2.11) zs = a s +b s +a s , k 2,
−k −k −k−1 −k −k −k+1 −k+1
≥
zs 1 = a s +b s +a s .
−1 −1 −2 −1 −1 0 0
−
Now we find
q˜ (z)
n−1
s = a s p˜ (z), s = s ,
n 0 −1 n −n −1
q˜ (z)
0
and inserting this in (2.11) gives
1
s = ,
−1 z a q˜ (z)/q˜ (z) b a2p˜ (z)
− −1 1 0 − −1 − 0 0
which by using a q˜ (z) = (z b )q˜ (z) 1 becomes
−1 1 −1 0
− −
q˜ (z)
0
s = .
−1 1 a2p˜ (z)q˜ (z)
− 0 0 0
The Stieltjes transform of the matrix of measures µ is thus given by
1 q˜ (z) 1 p˜ (z)
0 0
dµ (t) = , dµ (t) = ,
z t 1,1 1 a2p˜ (z)q˜ (z) z t 2,2 1 a2p˜ (z)q˜ (z)
Z − − 0 0 0 Z − − 0 0 0
Z
8 WALTER VAN ASSCHE
3. Proof of the theorem
The first important observation is that for f(x) = xm one has
xmp (x;µ )p (x;µ )dµ (x) = Jme+ ,e+ ,
n+k n n+l n n h n n+k n+li
Z
where J is the semi-infinite Jacobi matrix with the recurrence coefficients a ,b , (k = 0,1,2,...).
n k,n k,n
Consider the semi-infinite operator J as a doubly infinite Jacobi matrix by taking (J ) = 0 whenever
n n i,j
i < 0 or j < 0. If S is the shift operator on ℓ (Z,C) acting as Se = e , then
2 k k+1
xmp (x;µ )p (x;µ )dµ (x) = (S∗)nJmSne ,e .
n+k n n+l n n h n k li
Z
One easily verifies that this expression is given by
(S∗)nJmSne ,e = (J ) (J )
h n k li n n+k,n+k+i1 n n+k+i1,n+k+i1+i2
i1,i2,...,imX−1∈{−1,0,1}
i1+i2+···+im−1+k−l∈{−1,0,1}
(J ) .
··· n n+k+i1+i2+···+im−1,n+l
This is a finite sum, containing the matrix entries (J ) where r and s remain bounded. The
n n+r,n+s
hypothesis (1.3) implies that (S∗)nJ Sn converges entrywise to , where is the doubly infinite Jacobi
n
J J
matrix containing the coefficients a0,b0, (n Z). The hypothesis (1.3) thus implies that
n n ∈
lim xmp (x;µ )p (x;µ )dµ (x)
n+k n n+l n n
n→∞
Z
= ,
Jk,k+i1Jk+i1,k+i1+i2 ···Jk+i1+i2+···+im−1,l
i1,i2,...,imX−1∈{−1,0,1}
i1+i2+···+im−1+k−l∈{−1,0,1}
and the latter expression is equal to
me ,e .
k l
hJ i
Therefore we find that (S∗)nJmSn converges entrywise to m. By the unitarity of the mapping Λ :
n J
ℓ (Z,C) L (µ), transforming the action of on ℓ (Z,C) to the action of the multiplication operator
2 2 2
→ J
M on L (µ), we have
2
me ,e = tm(Λe )∗dµ(t)Λe ,
k l k l
hJ i
Z
and thus the theorem for f(x) = xm (and hence for every polynomial f) follows from (2.8) and (2.9).
In order to proof the theorem for every f C , we consider the linear operator H = (S∗)nJ Sn. We
b n n
∈
would like to prove that f(H ) = (S∗)nf(J )Sn converges weakly to f( ), since then
n n
J
lim (S∗)nf(J )Sne ,e = f( )e ,e ,
n k l k l
n→∞h i h J i
and this is precisely the weak convergence stated in our theorem. Observe that H e = a e +
n k k+n,n k−1
b e +a e whenever k+n > 0, hence the condition (1.3) implies the convergence of H e
k+n,n k k+n+1,n k+1 n k
to e = a0e +b0e +a0 e for every k Z. All finite linear combinations of the basis elements
J k k k−1 k k k+1 k+1 ∈
form a core of because we assume that there is only one self-adjoint extension of to its maximal
ORTHOGONAL POLYNOMIALS 9
4. Examples
The class M(a,b). The class M(a,b) consists of all orthogonal polynomials p (x;µ) (or all probability
n
measures µ) with recurrence coefficients that satisfy
lim a = a/2, lim b = b.
n n
n→∞ n→∞
We can apply the theorem with the family of measures µ µ, i.e., with all the orthogonality measures
k
≡
the same. If a > 0 then we can, without loss of generality, only consider M(1,0). The doubly infinite
Jacobi matrix then consists of 0 on the diagonal and 1/2 on the subdiagonals. The semi-infinite Jacobi
J
matrices J+ and J− are the same and the corresponding orthogonal polynomials are the Chebyshev
polynomials of the second kind, which are orthogonal with respect to the measure (2/π)√1 x2dx on
−
the interval [ 1,1]. The Stieltjes transform of this measure is
−
p˜ (z) = q˜ (z) = 2[z z2 1].
0 0
− −
p
Therefore the Stieltjes transform of the spectral matrix of measures for the Jacobi matrix is given by
J
1 1 1
dµ (x) = = dµ (x),
1,1 2,2
z x √z2 1 z x
Z − − Z −
1 z √z2 1
dµ (x) = − − ,
1,2
z x √z2 1
Z − −
from which one easily finds that the spectrum of is [ 1,1] and
J −
1 dx 1 xdx
dµ (x) = dµ (x) = , dµ (x) = .
1,1 2,2 1,2
π √1 x2 π √1 x2
− −
From our theorem we thus find
1 1 f(x)
lim f(x)p2(x;µ)dµ(x) = f(x)dµ (x) = dx,
n→∞ n 2,2 π √1 x2
Z Z Z−1 −
and in general
1 1 U (x) xU (x)
k k−1
lim f(x)p (x;µ)p (x;µ)dµ(x) = f(x) − dx
n n+k
n→∞ π √1 x2
Z Z−1 −
1 1 T (x)
k
= f(x) dx.
π √1 x2
Z−1 −
Here we have used the identity
U (x) xU (x) = T (x),
k k−1 k
−
where T (x) is the Chebyshev polynomial of the first kind. This result is well-known and can already be
k
found in [15, Theorem 13 on p. 45]. See also [20, Theorem 2 on p. 438].
Unbounded recurrence coefficients. Suppose that we have a sequence of orthogonal polynomials
p (x;µ) satisfying the three-term recurrence relation (1.1). If we rescale the variable by a positive
n
and increasing sequence c and consider the one-parameter family of polynomials p (c x;µ), then these
k n k
polynomials satisfy a recurrence relation of the form (1.2) with
10 WALTER VAN ASSCHE
and
c
n+1
lim = 1,
n→∞ cn
then it is easy to show that
lim a = a/2, lim b = b.
n+k,n n+k,n
n→∞ n→∞
Therefore the conditions of our theorem are valid and the theorem is true, with the doubly infinite
J
Jacobi matrix with constant entries b on the diagonal and a/2 on the subdiagonal. The orthogonal
polynomials for the Jacobi matrices J+ and J− are U (x−b) and the matrix of measures is supported on
n a
[b a,b+a] and given by
−
1 dx 1 (x b)dx
dµ (x) = dµ (x) = , dµ (x) = − .
1,1 2,2 1,2
π a2 (x b)2 aπ a2 (x b)2
− − − −
We thus have p p
1 b+a f(x)
lim f(x/c )p2(x;µ)dµ(x) = f(x)dµ (x) = dx.
n→∞ n n 2,2 π a2 (x b)2
Z Z Zb−a − −
p
This result can already be found in [10, Lemma 1 on p. 52] and [16, Lemma 3 on p. 1188]. In general we
have
1 b+a T ((x b)/a)
k
lim f(x/c )p (x;µ)p (x;µ)dµ(x) = f(x) − dx.
n n n+k
n→∞ π a2 (x b)2
Z Zb−a − −
Wall polynomials. The orthonormal Wall polynomials w (x;b,q), wpith (0 < q < 1,0 < b < 1) are
n
orthogonal on the geometric sequence qn,n = 1,2,3,... and have recurrence coefficients
{ }
a (b,q) = qn b(1 qn)(1 bqn−1), b (b,q) = qn[b+q (1+q)bqn].
n n
− − −
p
If we consider the Wall polynomials w (x;b;c1/k), where 0 < c < 1, then we have a one-parameter family
n
of orthogonal polynomials with recurrence coefficients
a = a (b,c1/k), b = b (b,c1/k),
n,k n n,k n
and one easily verifies that
lim a = c b(1 c)(1 bc) = A/2, lim b = (b+1 2bc)c = B.
n+k,n n+k,n
n→∞ − − n→∞ −
p
Hence we can apply the theorem, where again is a doubly infinite Jacobi matrix with constant entries
J
B on the diagonal and A/2 on the subdiagonals. Therefore again we have the asymptotic behaviour in
terms of Chebyshev polynomials (first and second kind) as in the previous two examples. These three
examples are all covered by Theorem 2 in [21, p. 307] which covers the special case of our theorem with
a doubly infinite Jacobi matrix with constant entries. In particular this asymptotic behaviour for Wall
polynomials was used in [21] to show that the product formulas for Legendre polynomials are a limiting
case of the product formulas for little q-Legendre polynomials as q 1.
→
Jacobi polynomials P(an+α,bn+β)(x). The Jacobi polynomials P(α,β)(x) are orthogonal on [ 1,1] with
n n
−
the weight function (1 x)α(1+x)β. The orthonormal Jacobi polynomials
−
2n+α+β +1 n!(α+β +2)
p(α,β)(x) = n P(α,β)(x)
n s n+α+β +1 (α+1)n(β +1)n n