Another connection between orthogonal polynomials and L-orthogonal polynomials

www.elsevier.com/locate/jmaa

Another connection between orthogonal polynomials

and L-orthogonal polynomials

E.X.L. de Andrade, C.F. Bracciali, A. Sri Ranga

DCCE, IBILCE, UNESP – Universidade Estadual Paulista, Rua Cristóvão Colombo 2265, 15054-000 São José do Rio Preto, SP, Brazil

Received 28 March 2005 Available online 22 August 2006

Submitted by D. Waterman

Abstract

We consider a connection that exists between orthogonal polynomials associated with positive measures on the real line and orthogonal Laurent polynomials associated with strong measures of the classS3[0, β, b]. Examples are given to illustrate the main contribution in this paper.

_{2006 Elsevier Inc. All rights reserved.}

Keywords:Orthogonal polynomials; Orthogonal Laurent polynomials; Szeg˝o polynomials on the real line; Hyperbolic Szeg˝o transformation

1. Introduction

Let_−∞c < d_∞ and letφ be a bounded and nondecreasing function on_[c, d_]such that it has infinitely many points of increase in_[c, d_]and that the momentsμφn=_cdxndφ (x), n₌0,1,2, . . ., all exist. Thenφrepresents a positive measure on[c, d_]and one can uniquely define the set of polynomials{Qφn}∞_n=0by

Qφ_n is monic of degreen,

✩

This research was supported by grants from CNPq and FAPESP of Brazil. * _{Corresponding author. Fax: (55) 17 221 2203.}

E-mail address:[email protected] (A. Sri Ranga).

d

c

xsQφ_n(x)dφ (x)₌0, s₌0,1, . . . , n₋1.

These are the monic orthogonal polynomials associated with the measureφ. It is well known that these polynomials satisfy the three term recurrence relation

Qφ_n +1(z)=

z₋bφ_n

+1

Qφ_n(z)₋aφ_n +1Q

φ

n−1(z), n1, (1.1)

withQφ₀(z)=1 andQφ₁(z)=z−bφ₁. The coefficientsanφandbφn, given by

b_nφ₊₁₌ d

c x[Q φ

n(x)]2dφ (x) d

c[Q φ

n(x)]2dφ (x)

, n0, and aφ_n+1= d

c [Q φ

n(x)]2dφ (x) d

c[Q φ

n−1(x)]2dφ (x)

, n1,

satisfybφn∈R,n1, andanφ>0,n2. For further information on these polynomials we refer

to Chihara [1] and Szeg˝o [13].

Now let 0a < b_∞and letψ be a bounded and nondecreasing function on_[a, b_]with infinitely many points of increase in_[a, b_]and such that the momentsμψn =_abtndψ (t ),n=

0,_±1,_±2, . . ., all exist. We refer toψas a strong positive measure on_[a, b_]and define the set of polynomials_{Qψ_n}∞_n

=0by

Qψ_n is monic of degreen, b

a

t−n+sQψ_n(t )dψ (t )=0, s=0,1, . . . , n−1. (1.2)

Such polynomials were introduced in [7] in order to solve the strong Stieltjes moment problem. The sequence of functions _{t−⌊(n+1)/2⌋Qψ_n(t )_}is known as a sequence of orthogonal Laurent polynomials or orthogonal L-polynomials associated with the measure ψ (see [6]). Like the monic orthogonal polynomials, the polynomials Qψ_n are also unique and for convenience we refer to them as L-orthogonal polynomials.

It is known that the zeros ofQψ_n are all positive, distinct and lie within the interval(a, b). The zeros ofQψ_n also interlace with the zeros ofQψ

n−1. Moreover, these polynomials satisfy the three

term recurrence relation

Qψ

n+1(w)=

w₋β_n+1ψ

Qψ_n(w)₋α_n+1ψ wQψ

n−1(w), n1, (1.3)

withQψ

0(w)=1 andQ ψ

1(w)=w−β ψ 1,β

ψ 1 =μ

ψ 0/μ

ψ −1=ρ

ψ 0/σ

ψ

0 and, forn1, β_n+1ψ _{= −}α_n+1ψ σ

ψ n−1 σnψ

and α_n+1ψ ₌ ρ ψ n

ρ_n−1ψ ,

whereρnψ= b

a Q ψ

n(t )dψ (t )andσ ψ n =

b

a t−n−1Q ψ

n(t )dψ (t ). The coefficientsβ ψ

n,n1, and αψn,n2, are all positive. From (1.3) one can also note thatQ

ψ

n(0)=(−1)nβ ψ 1β

ψ 2 · · ·β

ψ n.

Assuming different symmetric behaviors for the measureψ, properties of the L-orthogonal polynomials{Qψ_n}have been explored in several articles.

We say that the strong positive measureψbelongs to the symmetric classS3[τ, β, b]if dψ (t )

tτ = −

dψ (β2/t )

where 0< β < b,a=β2/b and 2τ ∈Z_{. The classification of the symmetry is according to the}

value ofτ.

It seems that two of these symmetric classes, namely whenτ ₌1/2 andτ₌0, turn out to be more interesting than the others.

The L-orthogonal polynomials{Qψ_n}, when the measureψbelongs to the classS3_[1/2, β, b_], have been explored for example in [10]. In this case we can state the following results:

wnQψ_n(β2/w)

Qψ_n(0) =

Qψ_n(w) and β_nψ₌β forn1.

Moreover, with the transformationx(t )₌ 1 2√α(

√

t₋β/√t ), whereα >0, if the positive measure

φon the real interval_[−x(b), x(b)_]is such that

dφ (x)=t+β

2t dψ (t ), (1.4)

then the orthogonal polynomials_{Qφn}are connected to the L-orthogonal polynomials{Qψn}in

the following way:

Qφ_n x(t )

=

2√αt−n

Qψ_n(t ) and a_nφ₊₁₌ 1

4αα ψ

n+1 forn1.

Sinceφis symmetric, that is dφ (₋x)_{= −}dφ (x), the orthogonal polynomials{Qφn}are such that Qφ_n(₋z)₌(₋1)nQφ_n(z) and b_nφ₌0 forn1.

Now the other interesting classS3_[0, β, b_]of strong positive measures is characterized by the symmetry

dψ (t )_{= −}dψ β2/t

, t_∈(a, b).

Here 0< β < b_∞anda₌β2/b. If dψ (t )₌h(t )dt then the symmetry is given by

t h(t )₌β2/thβ2/t. (1.5) In this paper we look at a connection that exists between orthogonal polynomials associated with positive measures on the real line and L-orthogonal polynomials associated with strong measures of the classS3_[0, β, b_]. We do this based on an idea similar to that of Szeg˝o (see [13]) to connect orthogonal polynomials on_[−1,1_]to orthogonal polynomials on the unit circle. Examples given in Section 6 to illustrate the main contribution of this paper also lead to some new results.

We mention that, in [14], Vinet and Zhedanov have considered a connection between or-thogonal polynomials associated with “symmetric” positive measures on the real line and Szeg˝o polynomials on the real line using the transformationx(t )₌λ(√t₊1/√t ), whereλ >0. As we will observe in Section 2, the monic Szeg˝o polynomials on the real line are the monic reciprocal polynomials of the L-orthogonal polynomialsQψ_n defined when the measureψ is of the class

S3_[0, β, b_].

Let{αn}and{βn}be any sequences of bounded positive numbers such that lim

n→∞αn=α >0 and nlim→∞βn=β >0.

Let the sequence of polynomials{B_n}be generated by

B_n+1(w)₌(w₋βn+1)Bn(w)−αn+1wBn−1(w), n1,

withB₁(w)=w−β1andB0=1. Then (see for example [12]),

lim

n→∞

1

n

B′ n(w)

Bn(w)=

1 2π

b

a

1

w₋t

1+β/t

√

(b−t )(t−a)dt

holds uniformly on any compact subset ofC_{\(0,∞), wherea=β}2_/bandb=(√_α+β+√_α)2_. 2. S3[0, β, b]measures and L-orthogonal polynomials

The L-orthogonal polynomials{Qψ_n}, whenψbelongs to the classS3_[0, β, b_], have first ap-peared in Sri Ranga, Andrade and McCabe [11]. Some aspects of these polynomials were also considered simultaneously in Common and McCabe [3]. We can state (see [11]) the following results whenψ_∈S3_[0, β, b_].

Iff is integrable with respect toψ, then

b

β2_/b

f (t )dψ (t )=

b

β2_/b

fβ2/tdψ (t ) (2.1)

and, in particular,μψn =β2nμψ₋n. For the associated L-orthogonal polynomials{Q ψ

n}the

coeffi-cients in the three term recurrence relation (1.3) satisfy

γ_nψ₊₁ γnψ

= β

2 βnψβ_nψ₊₁

, n0, (2.2)

whereγnψ=βnψ+α_nψ₊₁, withβ₀ψ=γ₀ψ=1. Consequently,

Q_nψ(0)=(−1)nβ₁ψβ₂ψ· · ·β_nψ=(−β)n

βnψ/γnψ, n1. (2.3)

Moreover, if

˜

Qψ_n(w)₌w n_Qψ

n(β2/w)

Qψ_n(0)

, n1,

then

˜

Qψ_n(w)=Qψ_n(w)−α_nψ₊₁Qψ

n−1(w)=

Qψ

n+1(w)+β ψ n+1Q

ψ n(w)

/w, n1. (2.4) Now, lettingw=β, and also lettingw= −β, in (2.4) we obtain

1₋ β

n

Qψ_n(0)

Qψ_n(β)₌α_nψ₊₁Qψ

n−1(β) and

1₋ (−β)

n

Qψ_n(0)

Qψ_n(₋β)₌α_nψ₊₁Qψ_n

forn1. Thus, for example,

Qψ_2n(β)_{= −}β_2nψβ_2nψ −1

γ_2nψ/β_2nψ ₊1γ_2nψ −1/β

ψ 2n−1−1

Qψ_2n

−2(β), n1,

Q_2nψ(₋β)₌β_2nψβ_2n−1ψ γ_2nψ/β_2nψ ₊1γ_2n−1ψ /β_2n−1ψ ₊1Qψ

2n−2(−β), n1. (2.5)

Note that the monic polynomials Q˜ψ_n satisfy the orthogonality (to be more precise the biorthogonality)

b

a

t−sQ˜ψ_n(t )dψ (t )₌0, s₌0,1, . . . , n₋1. (2.6)

If we define the reciprocal polynomials of Q˜ψ_n by Q˜ψ_n∗(w)₌ (w/β2)nQ˜ψ_n(β2/w), then

˜

Qψ_n∗(w)₌Qψ_n(w)/Qψ_n(0),n1, and from (2.4),

˜

Qψ_n(w)₌wQ˜ψ

n−1(w)+ ˜δnQ˜ ψ∗

n−1(w), n1, (2.7)

whereδn˜ = β2n

Qψn(0)=

(−β)n

γnψ/βnψ forn1.

Note that, if the integration in (2.6) is done over the unit circle thent−1can be replaced by the conjugate oft. Hence, based on (2.6) and (2.7), the polynomialsQ˜ψ_n can be called the monic Szeg˝o polynomials on the positive interval[a, b]. The numbersδn˜ = ˜Qψ_n(0), which play the role of reflection coefficients, satisfy

|˜δn|> βn forn1.

Like the zeros ofQψ_n the zeros of Q˜ψ_n also lie within the interval(a, b), here witha₌β2/b. Hence it is important that|˜δn|> βn(see Vinet and Zhedanov [14]). Otherwise, if|˜δn|< βn, then

(just as the scaled Szeg˝o polynomials orthogonal on the circle of radiusβ) the zeros ofQ˜ψ_n would lie within the disk|w_|< β.

To be able to talk about Szeg˝o polynomials on a positive interval[a, b_]it is important that the measure is of the classS3_[0, β, b_], witha₌β2/b. Otherwise the monic polynomials{ ˜Qψ_n}

defined on [a, b_]by (2.6) do not satisfy (2.7). For example, if the measure ψ is of the class

S3_[−1/2, β, b_], then for the monic polynomials{ ˜Qψ_n}defined by (2.6) we would have, instead of (2.7),

˜

Q_nψ(w)_{= ˜}δnQ˜_nψ∗(w)₌wQ˜ψ

n−1(w)+ ˜δnQ˜ ψ∗

n−1(w)− ˜αnwQ˜ ψ n−2(w),

forn2, whereδ˜n= ˜Qnψ(0)andα˜n>0.

3. Hyperbolic Szeg˝o transformation

Withβ >0, the transformation given by

z₌T (w)₌1

2

_w

β + β w

,

semi-circular region{w=βeiθ: −πθ0}onto the interval[−1,1]). In this special case the transformation is also known as the Szeg˝o transformation.

Here, we consider another special case of the transformationT (w)which maps the interval

[β,_∞)onto the interval[1,_∞). Also the interval(0, β_]onto the interval[1,_∞). We call this the Hyperbolic Szeg˝o Transformation and write

x₌x(t )₌1

2

_t

β + β

t

,

for t _{∈ [}β2/b, b_] andx _{∈ [}1, d_], withd ₌ 1₂(b/β₊β/b). Interestingly, we can consider this transformation in two stages.

Stage 1: θ₌ln(_βt).

This is a mapping from_[β2/b, b_]onto_[−ϑ, ϑ_], whereϑ₌ln(b/β). Note thatθis a strictly increasing function oft. The inverse of this mapping ist₌βeθ.

Stage 2: x₌coshθ₌1₂(eθ_+e−θ_{), hence the use of the word Hyperbolic.}

This can be considered as a mapping from_[0, ϑ_]onto_[1, d_], whered₌coshϑ₌1₂(b/β₊ β/b). Herex is a strictly increasing function ofθ. The inverse of this mapping isθ=ln(x+ √

x2−1), with ln(b/β)=ϑ=ln(d+√d2−1).

This can also be considered as a mapping from[−ϑ,0]onto[1, d_]. Nowx is a strictly de-creasing function ofθ. The inverse of this mapping isθ₌ln(x₋√x2_{−1), with ln(β/b)=}

−ϑ₌ln(d₋√d2_−1).

Now if we consider the integralI₌d

1 f (x)dφ (x), whereφis any positive measure on[1, d],

then we can write

2I ₌ 1

d f (x)

−dφ (x)

+

d

1

f (x)dφ (x).

Here, applying Stage 2 of our transformation, we obtain

2I ₌ 0

−ϑ

f (coshθ )₋dφ (coshθ )₊ ϑ

0

f (coshθ )dφ (coshθ ). (3.1)

Applying Stage 1 of our transformation, whereeθ₌t /β, then gives

2I ₌ β

β2_/b

fx(t )₋dφx(t )₊ b

β

fx(t )dφx(t ).

Hence,

d

1

f (x)dφ (x)₌ b

β2_/b

fx(t )dψ (t ), (3.2)

where dψ (t )=1₂dφ (x(t ))for t∈(β, b)and dψ (t )= −1₂dφ (x(t ))fort∈(β2/b, β). Clearly,

μψ_{0 =}μφ₀. Moreover, one can verify thatψ is a strong positive measure on(β2/b, b)and that

In particular, if dφ (x)=k(x)dx then (3.1) and (3.2) can be given in the form

d

1

f (x)k(x)dx₌1

2

ϑ

−ϑ

f (coshθ )k(coshθ )_|sinhθ_|dθ₌ b

β2_/b

fx(t )h(t )dt,

where

h(t )₌ dψ (t )

dt =

1 4t t β −

β t

k x(t )

fort_∈

β2/b, b

. (3.3)

The following lemma gives information on some auxiliary polynomials used in the next two sections.

Lemma 3.1.Letx₌coshθ. Let

Tn(x)=coshnθ and Un(x)=

sinh(n+1)θ

sinhθ , n0.

Then the following hold:

• T0(x)=1and, for anyn1,Tnis a polynomial of degreenwith leading coefficient2n−1.

Precisely,T1(x)=x,T2(x)=2x2−1and, forn=1,2, . . . , T_2n+1(x)₌22nx2n+1₋

n

j=1 _2n

+1

j

T_2n+1−2j(x),

T2n+2(x)=22n+1x2n+2− n

j=1

2n₊2

j

T2n+2−2j(x)−

1 2

2n₊2

n₊1

.

• For anyn0,Unis a polynomial of degreenwith leading coefficient2n. Precisely,U0(x)=

1and, forn₌1,2, . . . ,

U_2n−1(x)₌2

n−1

j=0

T_2n−1−2j(x) and U_2n(x)₌1+2

n−1

j=0

T_2n−2j(x).

Proof. Using the binomial expansion

(2x)2n₌

eθ₊e−θ2n

=

2n

j=0 _2n

j

e(2n−2j )θ.

Since2n j

= 2n 2n−j

,

(2x)2n_{= 2n}

n

+

n−1

j=0 _2n

j

e(2n−2j )θ₊e−(2n−2j )θ

=

2n n

+2

n−1

j=0

2n j

T_2n−2j(x).

The polynomials{Tn}and{Un}are, respectively, the Chebyshev polynomials of the first and second kind. That is, within the interval [−1,1] these polynomials can be given by Tn(x)=

cosnθandUn(x)=sin(n+1)θ/sinθwithx=cosθ. Note thatTnandUnare such that

Tnx(t )=

1 2

(t /β)n₊(β/t )n

, n0,

Unx(t )=(t /β)n+1−(β/t )n+1(t /β)−(β/t ), n0. (3.4) 4. Orthogonal polynomials and L-orthogonal polynomials

Let 0< β < bandψ_∈S3(0, β, b). Let us consider the functions{Rn(1)}given by λ(1)_n R(1)_n (t )₌

1

t n

Qψ_2n(t )_{+ t}

β2 n

Qψ_2n

_β2 t

, n0,

whereλ(1)n =(2β)n1+

β_2nψ/γ_2nψ

. Using (2.4) one can also show that

R_n(1)(t )_{= −}

β_2nψγ_2nψ (2β)n

1

t n

Qψ

2n−1(t )+ _t

β2 n

Qψ

2n−1 _β2

t

, n1.

Theorem 4.1.Let d₌x(b)and let the positive measureφ1defined on the interval[1, d]be such

that

dφ₁ x(t )

=2 dψ (t ) fort_{∈ [}β, b_].

Let the sequence of functions{Pn(1)}be such that

P_n(1)x(t )=R_n(1)(t ) forn1.

Then for anyn1,Pn(1)(x)=Q φ1

n (x), the monic orthogonal polynomial of degreenassociated

with the positive measureφ1on[1, d]. Moreover,μφ01=μ ψ 0,b

φ1

1 =δ ψ

1 and, forn1, bφ1

n+1=

1 2δ

ψ 2n+1

δψ_2n+1

+1

2δ

ψ 2n−1

δ_2nψ ₋1

,

aφ1

n+1=

1 4

δ_2nψ₋₂+1

δψ_2n−1+1

δ_2nψ₋₁−1 δ_2nψ −1

, (4.1)

whereδnψ=

γnψ/β ψ

n. Furthermore,

δ_2nψ₋₁=S

φ1

n (−1)+Snφ1(1) Sφ1

n (−1)−Snφ1(1)

and δ_2nψ =S_nφ₊1₁(1)−S_nφ₊1₁(−1)−1 (4.2)

forn1, whereSφ1

n (z)=Qφn1(z)/Qφn1−1(z).

Proof. First we show thatPn(1)is a monic polynomial of degreen. LettingQψn(t )=nr=0qn,rtr,

λ(1)_n R_n(1)(t )₌ 2n

r=0

q2n,rβ−n+r t β

n−r

+

_β

t n−r

=2q_2n,n+ n−1

r=0

q_2n,rβ−n+r₊q_2n,2n−rβn−r t β

n−r

+

_β

t n−r

.

Hence, from (3.4),

λ(1)_n P_n(1) x(t )

=2q2n,n+2 n−1

r=0

q2n,rβ−n+r+q2n,2n−rβn−rTn−rx(t ). (4.3)

SinceTr,r1, is a polynomial of degreerinxwith leading coefficient 2r−1,Pn(1)is a

polyno-mial of degreeninx and its leading coefficient is 2n_(q

2n,0β−n+q2n,2nβn) λ(1)n

= 2

n_(Qψ

2n(0)β−n+βn) (2β)n_[1+βψ

2n/γ ψ 2n]

.

Using (2.3) one can easily verify this to be equal to 1.

Now we establish the orthogonality property of_{Pn(1)}. We will show that

I_n,k(1)₌ d

1

Tk(x)Pn(1)(x)dφ1(x)=0 fork=0,1, . . . , n−1, n1. (4.4)

From (3.2),

I_n,k(1)=

b

β2_/b

Tkx(t )P_n(1)x(t )dψ (t )=1

2

b

β2_/b

t β

k

+

_β

t k

R(1)_n (t )dψ (t ).

We write

2λ(1)_n I_n,k(1)₌ b

β2_/b

t β

k

+

_β

t k₁

t n

Qψ_2n(t )dψ (t )

+

b

β2_/b

t β

k

+

_β

t

k _t

β2 n

Qψ_2n

_β2 t

dψ (t ).

Using (2.1) on the second integral, we obtain

λ(1)_n I_n,k(1)₌ b

β2_/b

t β

k

+

_β

t k₁

t n

Qψ_2n(t )dψ (t ). (4.5)

Thus, from (1.2),I_n,k(1)=0 fork=0,1, . . . , n−1. This proves the first part of the theorem. We also have from (4.5),

I_n,n(1)₌ μ ψ 0α

ψ 2α

ψ 3 · · ·α

ψ 2n+1

2n_β2n_[1+βψ 2n/γ

ψ 2n]

On the other hand, since the leading coefficient ofTnis 2n−1, we obtain from (4.4),

I_n,n(1)₌2n−1μφ1

0 a φ1

2 a φ1

3 · · ·a φ1

n+1, n1.

Considering the ratiosIn,n(1)/I_n−1,n−1(1) we are led to

aφ1

2 =

1 1₊

β₂ψ/γ₂ψ

αψ₂αψ₃

2β2 and a φ1

n+1=

1+

β_2n−2ψ /γ_2n−2ψ

1₊

β_2nψ/γ_2nψ

αψ_2nαψ_2n +1

4β2 , n2.

Hence from (2.2), Eq. (4.1) foraφ1

n can be obtained.

Now we compare the coefficients of degreen₋1 on both sides of the expression (4.3) and obtain

−

n

r=1 bφ1

r =

2n−1_(q

2n,1β−n+1+q2n,2n−1βn−1) (2β)n_[1+βψ

2n/γ ψ 2n]

, n1.

The left-hand side follows from the three term recurrence relation (1.1). We need to find expressions for q2n,1 and q2n,2n−1. From the three term recurrence relation (1.3), qn+1,n=

−β_nψ₊₁₋α_nψ₊₁₊q_n,n−1forn1. Thus,

q2n,2n−1−α_2nψ₊₁= − 2n

r=1

γ_rψ, n1.

From (2.4),

q2n,2n−1−α2nψ+1= β2q2n,1

Qψ_2n(0), n

1.

Therefore,

n

r=1 bφ1

r =

1 2β

2n

r=1

γ_rψ₋ α ψ 2n+1

2β[1+

β_2nψ/γ_2nψ] = 1

2β 2n−1

r=1

γ_rψ₊ 1

2β

β_2nψγ_2nψ, n1.

From this and (2.2), Eq. (4.1) forbφ1

n follows.

Sincex(β)₌1 andx(₋β)_{= −}1, we obtain fromQφ1

n (x(t ))=Rn(1)(t ),

Qφ1

n (1)=

2Qψ

2n(β) λ(1)n βn

and Qφ1

n (−1)=

2Qψ

2n(−β) λ(1)n (−β)n

, n1.

Hence, withSφ1

n (z)=Qφn1(z)/Qφn−11 (z),

Sφ1

n (1)= −

1 2

δ_2nψ₋₁₋1

δψ_2n−2+1

, n1,

Sφ1

n (−1)= −

1 2

δψ_2n−1+1

δ_2n−2ψ ₊1

, n1.

Therefore we obtain (4.2) and this completes the proof of the theorem. ✷

We remind thatQψ_n(0)₌(₋β)n/δψn,n1, and

withδψ₀ =1, which follow from (2.2) and (2.3). Thus, forn1,

b

β2_/b

Qψ_n(t )dψ (t )=μψ₀ β n

δnψ n

j=1

δ_jψ−1

δ_jψ+1 ,

b

β2_/b

t−n−1Qψ_n(t )dψ (t )₌(₋1)nμψ₀ δ ψ n+1 βδnψ

n

j=1

δψ_{j −}1

δ_jψ₊1 .

5. Companion polynomials of{Qφ1

n}and further identities

We now consider the functions{Rn(2)}given by

λ(2)_n R_n(2)(t )₌ 1 t

n+1

Qψ

2n+2(t )− _t

β2 n+1

Qψ

2n+2 _β2

t

t₋β2/t

, n0,

whereλ(2)n =(2β)n[1−

β_2nψ₊₂/γ_2nψ₊₂]. Again from (2.4),

R_n(2)(t )₌

β_2nψ₊₂γ_2nψ₊₂ (2β)n

1

t n+1

Qψ

2n+1(t )− _t

β2 n+1

Qψ

2n+1 _β2

t

t₋β2/t,

forn0.

Theorem 5.1.Let d₌x(b)and let the positive measureφ₂defined on the interval_[1, d_]be such that

1

x(t )2₋₁dφ2

x(t )=dφ1x(t )=2 dψ (t ) fort∈ [β, b].

Let the sequence of functions_{Pn(2)}be such that

P_n(2) x(t )

=R(2)_n (t ) forn0.

Then forn0,Pn(2)(x)=Qφn2(x), the monic orthogonal polynomial of degree n associated

with the positive measureφ2on[1, d]. Moreover,μφ₀2=1₂(δ₁ψ−1)(δ₁ψ+1)(δψ₂ +1)μψ₀ and, for n1,

bφ2

n =

1 2δ

ψ 2n+1

δψ_2n−1

+1

2δ

ψ 2n−1

δ_2nψ ₊1

,

aφ2

n+1=

1 4

δ_2nψ ₋1

δ_2n+1ψ ₋1

δψ_2n+1+1

δ_2n+2ψ ₊1

. (5.1)

Proof. To provePn(2)(x)=Qφn2(x), apart from establishing thatPn(2)(x(t ))=Rn(2)(t )is a

poly-nomial of degreeninx, one needs to prove that

I_n,k(2)₌ d

1

Uk(x)P_n(2)(x)dφ2(x)=0 fork=0,1, . . . , n−1, n1.

Having provedPn(2)(x)=Qφn2(x), fromIn,n(2)we have d

1

Un(x)Qφ2

n (x)dφ2(x)=

1

(2β2₎n+1

1₋

β_2nψ₊₂/γ_2nψ₊₂ b

β2_/b

Qψ

2n+2(t )dψ (t ),

forn0. From this,μφ2

0 = 1

2β2_[1−βψ 2/γ

ψ 2 ]

μψ₀α₂ψα₃ψand

2nμφ2

0 a φ2

2 · · ·a φ2

n+1=

1

(2β2₎n+1

1−

β_2nψ

+2/γ ψ 2n+2

μψ₀αψ_{2 · · ·}αψ_2n +2α

ψ

2n+3, n1.

Consequently the equation foraφ2

n in (5.1) follows.

Comparing the coefficients ofxn−1 on both sides of the equationQφ2

n (x(t ))=R(2)n (t ), the

equation forbφ2

n in (5.1) also follows. ✷

Now, from (4.1) and (5.1),

bφ1

n+1−b φ2

n =δ ψ 2n+1−δ

ψ

2n−1, n1,

withbφ1

1 =δ ψ

1. Thus, we can write δ₁ψ₌bφ1

1 and δ

ψ 2n+1=

n+1

r=1 bφ1

r − n

r=1 bφ2

r , n1.

Also, from (4.1) and (5.1),

aφ1

2 μφ2

0

=δ

ψ 2 +1 δ₂ψ₋1μ

ψ 0 and

aφ1

n+1 aφ2

n

=δ

ψ 2n−2+1 δψ_2n−2−1

δψ_2n−1

δψ_2n+1, n2. Sinceμφ1

0 =μ ψ 0, δ₂ψ₋1

δ₂ψ₊1=

aφ1

2 μ φ1

0 μφ2

0

and δ

ψ 2n+2−1 δψ_2n+2+1 =

aφ1

n+2· · ·a φ1 3 a φ1 2 μ φ1 0 aφ2

n+1· · ·a φ2

2 μ φ2

0

, n1.

Thus,δ₂ψ₌μ

φ₂ 0 +a

φ₁ 2 μ

φ₁ 0

μφ₀2−aφ₂1μφ₀1 and

δ_2n+2ψ ₌a φ2

n+1· · ·a φ2

2 μ φ2

0 +a φ1

n+2a φ1

n+1· · ·a φ1

2 μ φ1

0 aφ2

n+1· · ·a φ2

2 μ φ2

0 −a φ1

n+2a φ1

n+1· · ·a φ1

2 μ φ1

0

, n1.

6. Examples

6.1. L-orthogonal polynomials from the Chebyshev polynomials

We consider the sequence of orthogonal polynomials{Qφ1

n }given by

Qφ1

n (x)=cnTn _2x

d₋1−

d₊1

d₋1

, n1,

whereTnare the Chebyshev polynomials (see, for example, [1,13]). The constantscnare chosen

so thatQφ1

n are monic polynomials. The following results are easily verified:

μφ1

0 = d

1

1

π√(d−x)(x−1)dx=1 and

d

1 Qφ1

n (x)Qφm1(x)

1

π√(d₋x)(x₋1)dx=δn,m _d

−1 4

2n

2, n1, m0.

The three term recurrence relation forQφ1

n is

Qφ1

2 (x)=

x−1

2(d+1)

Qφ1

1 (x)−

1 8(d−1)

2_Qφ1

0 (x), Qφ1

n+1(x)=

x−1

2(d+1)

Qφ1

n (x)−

1

16(d−1)

2_Qφ1

n−1(x), n2,

withQφ1

1 (x)=(x−d+21)andQ φ1

0 (x)=1. Moreover, forn1,

2nQφ1

n (x)= x− d₊1

2

+(d₋x)(1−x) n

+ x₋d+1

2

−(d₋x)(1₋x) n

. (6.1)

Hence, from Theorem 4.1, with the use of (3.3) we can consider the L-orthogonal polynomials

{Qψ_n}associated with theS3(0, β, b)measureψgiven by dψ (t )₌ 1

2π

1+β/t

√

(b₋t )(t₋a)dt, (6.2)

whereb₌β(d₊√d2_−1)anda=β2_/b.

Theorem 6.1. Let ψ be the S3(0, β, b) measure given by (6.2). Then for the associated L-orthogonal polynomials{Qψ_n}we have that the parametersδψn =(−β)n/Q

ψ

n(0)satisfy

δ_2nψ₋₁₌[1+η]

[1−η_]

[1+η2n−1_]

[1−η2n−1_], δ ψ 2n=

[1+η_]

[1−η_]

[1−η2n_]

[1+η2n_], n1, (6.3)

whereη= [√b−√β]2/[√b+√β]2. Consequently, in the three term recurrence relation(1.3)

satisfied by these polynomials, we haveβ₁ψ=β[1−η]2 [1+η]2,α

ψ 2 =β

8η[1+η2]

β_2nψ ₌β[1+η 2n−1

][1₊η2n_]

[1₋η2n−1_][1−η2n_], β ψ 2n+1=β

[1₋η2n_][1₋η2n+1_]

[1₊η2n_][1+η2n+1_], α_2nψ₊₁₌β 4η

[1−η]2

[1+η2n−1_][1−η2n+1_]

[1+η2n][1−η2n] ,

α_2nψ +2=β

4η

[1₋η_]2

[1₋η2n_][1₊η2n+2_]

[1₊η2n+1_][1−η2n+1_]. Proof. We obtain from (6.1),

Qφ1

n (1)=2 _d

−1 2

n

−1 2

n ,

Qφ1

n (−1)=

d₊1 2 −1

2n

+

_d

+1 2 +1

2n

−1 2

n ,

forn1. Sinced=1₂(b/β+β/b), these can be written as

Qφ1

n (1)=

2

(₋8bβ)n

(√b₊β)2n(√b₋β)2n

, n1,

Qφ1

n (−1)=

1

(₋8bβ)n

(√b₊β)4n₊(√b₋β)4n, n1.

Withη_{= [}√b₋√β_]2/_[√b₊√β_]2, we can then write

Qφ1

n (1)=2

(−2)nηn

(1−η)2n and Q φ1

n (−1)=

(−2)n(1+η2n)

(1−η)2n , n1.

Hence, forSφ1

n (z)=Qφn1(z)/Qφn−11 (z),

Sφ1

n (1)=

−2η

(1₋η)2 and S φ1

n (−1)=

−2

(1₋η)2

1₊η2n

1₊η2n−2, n1.

Thus, from (4.2), the results (6.3) of the theorem follow. With this, the results for_{βnψ}

immedi-ately follow from (4.6).

To obtain the results for{αnψ}, first we obtain

δ_2nψ

−1−1=

2η

[1₋η_]

[1₊η2n−2_]

[1₋η2n−1_], δ ψ

2n−1+1=

2

[1₋η_]

[1₊η2n_]

[1₋η2n−1_], δ_2nψ ₋1₌ 2η

[1−η]

[1−η2n−1] [1+η2n] , δ

ψ 2n+1=

2

[1−η]

[1−η2n+1] [1+η2n] ,

forn1. Now the results for_{αnψ}follow from (4.6). ✷

6.2. L-orthogonal polynomials from the Laguerre polynomials

The Laguerre polynomials{L(α)n }, forα >−1, are defined by ∞

0

L(α)_n (x)L(α)_m (x)xαe−xdx₌Ŵ(α₊1) _n

+α n

δnm, n, m=0,1,2, . . . ,

whereu v

=Ŵ(v+Ŵ(u1)Ŵ(u+1)−v+1). These polynomials are explicitly given by (see, for example, [1,13])

L(α)₀ (x)₌1 and L(α)_n (x)₌ n

j=0 _n

+α n₋j

₁ j_!(−x)

j_{, n1.}

We consider the polynomials

Qφ₀1(x)=1 and Qφ1

n (x)=(−2)nn!L(α)n _x

−1 2

, n1.

Then{Qφ1

n (x)}are the monic orthogonal polynomials given by ∞

1 Qφ1

n (x)Qφm1(x)dφ1(x)=22nn!(α+1)nδnm, n, m=0,1,2, . . . ,

where dφ1(x)=_2Ŵ(α+1)1 [₂1(x−1)]αe−

1

2(x−1)d_x. Here,_(u)nis the Pochhammer-symbol defined

by(u)0=1 and(u)n=u(u+1)· · ·(u+n−1)forn1. One can easily verify thatμφ₀1=1,

Qφ1

0 (x)=1 and Q φ1

n (x)=(−2)nn! n

j=0 _n

+α n−j

1

j!

1−x

2

j

, n1.

Moreover,

Qφ1

n+1(x)=

x₋(4n₊3+2α) Qφ1

n (x)−4n(n+α)Q φ1

n−1(x), n1,

withQφ₀1 =1 andQ₁φ1(x)=x−(3+2α).

Theorem 6.2.Letψbe theS3(0, β,_∞)measure given by

dψ (t )₌t −1

[√t /β₊√β/t_]

4Ŵ(α₊1)

1 2

t /β₋β/t 2α+1

e−[12(

√

t /β−√β/t )]2_dt.

Then for the associated L-orthogonal polynomials_{Qψ_n}, the parameters δnψ=(−β)n/Qψn(0)

satisfy

δ_2n−1ψ _{= −}1+2n ν (α) n

ν_n−1(α+1) =1+2

(n₊α) ν (α) n−1 ν(α_n−1+1)

, n1,

δ_2nψ _{= −}1+2ν

(α+1) n

νn(α)

=1+2ν

(α+1) n−1 νn(α)

, n1,

whereνn(α)=L(α)n (−1)=nj=0 n+α

n−j 1

Proof. These results follow from the application of Theorem 4.1 together with (3.3). To complete the proof of this theorem we observe that

Qφ1

n (1)=(−2)nn! _n

+α n

=(₋2)nn_!L(α)_n (0), n1,

Qφ1

n (−1)=(−2)nn! n

j=0 _n

+α n₋j

1

j_!=(−2) n_n

!L(α)_n (₋1), n1.

Hence, forn1,

Qφ1

n (−1)Q φ1

n (1)=22n(n!)2 _n

+α n

n

j=0 _n

+α n−j

₁ j!,

Qφ1

n (−1)Q φ1

n−1(1)+Q φ1

n (1)Q φ1

n−1(−1)

= −22n−1(n−1)!n!

n₊α _n

+α n

n

j=0

(2n₋j ) _n

+α n₋j

₁ j_!,

Qφ1

n (−1)Q φ1

n−1(1)−Q φ1

n (1)Q φ1

n−1(−1)

= −22n−1(n−1)!n!

n₊α _n

+α n

n

j=0 j

_n

+α n₋j

₁

j_!. (6.4)

To obtain the last two expressions we use

_n

−1₊α n₋1₋j

=n−j

n₊α _n

+α n₋j

, j₌0,1, . . . , n₋1.

Using the results of (6.4) in (4.2) we conclude the proof of the theorem. ✷

From (4.6), we obtain that the polynomials{Qψ_n}satisfy the three term recurrence relation

(1.3)withβ₁ψ=β(3+2α)−1,α₂ψ=4(α+1)(α+2)β₁ψ and, forn1,

β_2nψ ₌β1+2(n+α)ν (α) n−1/ν

(α+1) n−1

1+2ν_n(α+1) −1 /ν

(α) n

, β_2nψ +1=β

1+2ν_n(α+1) −1 /ν

(α) n

1+2(n₊α₊1)νn(α)/νn(α+1) ,

α_2nψ₊₁=4ν

(α+1) n−1 νn(α+1)

[νn(α)]2

β_2nψ, α_2nψ₊₂=4(n+1)(n+α+1)ν (α) n ν_n(α)₊₁

[νn(α+1)]2 β_2nψ₊₁.

6.3. L-orthogonal polynomials from the q-Laguerre polynomials

The q-Laguerre polynomials_{L(α)n (x;q)}, forα >−1, can be defined by ∞

0

L(α)_n (x_;q)L(α)_m (x_;q) x α

(−x;q)_∞dx=

(q−α_;q)_∞ (q;q)_∞

(qα+1_;q)n

(q;q)nqn Ŵ(−α)Ŵ(α+1) δnm,

where(u_;q)0=1,(u;q)n=(1−u)(1−uq)· · ·(1−uqn−1),n1. These polynomials, which

are also called the Generalized Stieltjes–Wigert polynomials [2], have the explicit representation

L(α)_n (x_;q)₌(q α+1_;q)n

(q;q)n 1Φ1

q−n qα+1

q; −q

where

1Φ1

u v q;z

=

∞

j=0 (u_;q)j (v_;q)j

(₋1)jqj (j−1)/2 z j

(q_;q)j .

The moment problem associated with the q-Laguerre polynomials is indeterminate and therefore there are many orthogonality representations. For example, there is the following representation in terms of a discrete measure.

∞

j=−∞

qj (α+1) (₋cqj_;q)

∞

L(α)_n (cqj_;q)L(α)_m (cqj_;q)

=(q;q)∞(−cq

α+1_;q)

∞(−c−1q−α;q)∞ (qα+1_;q)

∞(−c;q)∞(−c−1q;q)∞

(qα+1_;q)n (q_;q)nqn

δnm,

wherecis any positive constant.

Here we consider the sequence of monic polynomials

Qφ1

0 (x)=1 and Qφn1(x)=(−2)n (q_;q)n qn(n+α)L

(α) n

(x₋1)/2;q

, n1.

From known results on{L(α)n (q;x)}, the following results for the polynomials{Qφn1}can be

easily written down:

∞

1 Qφ1

n (x)Qφm1(x)dφ1(x)=

22n(q_;q)n(qα+1;q)n

qn(2n+2α+1) δnm, n1,

where dφ1(x)=1_2(q−α_;q)(q;q)∞

∞Ŵ(−α)Ŵ(α+1)

((x−1)/2)α

(−(x−1)/2;q)_∞dx.

Moreover,

Qφ1

n (1)=(−2)n (q_;q)n qn(n+α)L

(α)

n (0;q)=(−2)n

(qα+1_;q)n qn(n+α) ,

Qφ1

n (−1)=(−2)n (q_;q)n qn(n+α)L

(α)

n (−1;q)=(−2)n

1

qn(n+α),

forn1. The last expression follows, for example, from the summation formula (see [8, p. 15])

1Φ1

u v q;

v u

=(u

−1_v

;q)_∞ (v_;q)_∞ .

Hence, from Theorem 4.1,

Theorem 6.3.Letψbe theS3(0, β,∞)measure given by

dψ (t )=(q;q)∞t

−1_[√_{t /β+}√_β/t]

4(q−α_;q)

∞Ŵ(−α)Ŵ(α+1)

[1₂|√t /β₋√β/t_|]2α+1 (_−[1₂(√t /β₋√β/t )_]2_;q)

∞

dt.

Then for the associated L-orthogonal polynomials{Qψ_n}, the parameters δnψ=(−β)n/Qψn(0)

satisfy

6.4. Orthogonal polynomials from the L-orthogonal polynomials associated with the log-normal distribution

In [9], Pastro gave the orthogonal Laurent polynomials associated with the log-normal distri-bution. Here we consider the L-orthogonal polynomials{Qψ_n}defined by (1.2) in relation to the strong measureψgiven by the shifted log-normal distribution

dψ (t )₌ 1

2κ√π t

−1_e−[ln(t )/(2κ)]2

dt.

It is easily verified from (1.5) that this measure belongs to the classS3_[0, β, b_], withβ₌1 and

b_{= ∞}.

From Pastro’s results, and also results given in [3,4], it follows that

Qψ_n(w)₌ n

r=0

(₋1)rq−r(n−r) _n

r

q

qr/2wn−r, n1,

and

Qψ

n+1(w)=

w₋q1/2

Qψ_n(w)₋q1/2

q−n₋1 wQψ

n−1(w), n1,

withQψ₁(w)₌w₋q1/2. Here,q₌e−2κ2 andn r

qare the q-binomial coefficients given by _n

r

q

= (q;q)n

(q;q)r(q;q)n₋r

, r₌0,1, . . . , n.

The above results can also be found in [11].

Theorem 6.4.Let the positive measuresφA andφB, both having their common support on the

entire positive real axis, be given by

dφB(x)=x(x+2)dφA(x)=

1

κ√π

x(x₊2)e−[ln(x+1+ √

x(x+2))1/(2κ)]2_dx.

ThenμφA

0 =1,μ φB

0 =q−2−1 and the associated monic orthogonal polynomials{Q φA

n }and

{QφB

n }satisfyQφ₁A(z)=z+1−q−1/2,Q₁φB(z)=z+1−1₂q−1/2(q−2+1)and QφA

n+1(z)=

z−bφA

n+1

QφA

n (z)−a φA

n+1Q φA

n−1(z), n1, QφB

n+1(z)=

z₋bφB

n+1

QφB

n (z)−a φB

n+1Q φB

n−1(z), n1,

where

bφA

n+1=

1 2

q−(2n+1)/2q−n₊1

+q−(2n−1)/2q−n₋1

−1,

aφA

n+1=

1 4

q−n+1₊1

q−n₋1

q−2n+1₋1 ,

bφB

n+1=

1 2

q−(2n+3)/2

q−n−1₋1

+q−(2n+1)/2

q−n−1₊1

−1,

aφB

n+1=

1 4

q−n−1

q−n−1+1

q−2n−1−1 ,

Proof. We haveδnψ=q−n/2,n1. Hence, applying Theorems 4.1 and 5.1, we obtain for the

measuresφ₁andφ₂, both defined on[1,_∞), dφ2(x)=x2−1dφ1(x)=

1

κ√π

x2₋1e−[ln(x+

√

x2₋₁₎1/(2κ)

]2_dx,

μφ1

0 =1,μ φ2

0 =q−2−1. Moreover, the associated sequences of monic orthogonal polynomials

{Qφ1

n }and{Qφn2}satisfyQφ₁1(x)=x−q−1/2,Qφ₁2(x)=x−1₂q−1/2(q−2+1)and Qφ1

n+1(x)=

x−bφ1

n+1

Qφ1

n (x)−a φ1

n+1Q φ1

n−1(x), n1, Qφ2

n+1(x)=

x−bφ2

n+1

Qφ2

n (x)−a φ2

n+1Q φ2

n−1(x), n1,

where

bφ1

n+1=

1 2

q−(2n+1)/2

q−n₊1

+q−(2n−1)/2

q−n₋1 ,

aφ1

n+1=

1 4

q−n+1₊1

q−n₋1

q−2n+1₋1 ,

bφ2

n+1=

1 2

q−(2n+3)/2

q−n−1₋1

+q−(2n+1)/2

q−n−1₊1 ,

aφ2

n+1=

1 4

q−n−1

q−n−1+1

q−2n−1−1 ,

forn1. Hence the substitutionx₌y₊1 gives the results of the theorem. ✷

References

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