Chain Sequences, Orthogonal Polynomials, and Jacobi Matrices

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Journal of Approximation Theory 92, 5973 (1998)

Chain Sequences, Orthogonal Polynomials, and Jacobi Matrices

Ryszard Szwarc*

Institute of Mathematics, Polish Academy of Sciences, ul. Kopernika 18, 51-617 Wrocfaw, Poland ; and

Institute of Mathematics, Wrocfaw University, pl. Grunwaldzki 24, 50-384 Wrocfaw, Poland

Received March 5, 1996; accepted in revised form December 3, 1996 Communicated by Walter Van Assche

Chain sequences are positive sequences [a_n] of the form an= g_n(1& g_n&1) for a nonnegative sequence [ g_n]. This concept was introduced by Wall in connection with continued fractions. In his monograph on orthogonal polynomials, Chihara conjectured that if a_n¹₄for each n then (a_n&¹₄)¹₄. We prove this conjecture and give other precise estimates for a_n. We also characterize the chain sequences [an] whose terms are greater than ¹4. We show connections to Jacobi matrices and orthogonal polynomials. In particular, we characterize the maximal chain sequences in terms of integrability properties of the spectral measure of the associated Jacobi matrix. 1998 Academic Press

Key words: orthogonal polynomials; chain sequence; recurrence relation.

1. INTRODUCTION

The concept of chain sequences was introduced by Wall [7] in his monograph on continued fractions. Chain sequences are sequences [a_n]_n=1 for which there exists a sequence [ g_n]_n=0such that 0 g_n1 and

a_n= g_n(1& g_n&1), for n1.

The sequence [ g_n] is called a parameter sequence and need not be unique.

The connection to continued fractions is that a nonnegative sequence [a_n] is a chain sequence if and only if the approximants of the continued fraction

1|

|1&a1|

|1&a2|

|1&a3|

|1& } } }

Article No. AT963109

59

0021-904598 25.00

* This work was partially supported by KBN (Poland) under Grant 2P03A 030 09.

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are positive and converge to a limit. We refer to [2, 7] for basic facts about chain sequences.

The constant sequence a_n=¹₄ is one of the simplest examples of a chain sequence. The constant ¹₄ cannot be enlarged and moreover if a_n¹₄ and [a_n] is a chain sequence then a_n¹

4. In [2, Theorem III.5.8] Chihara showed that if [a_n] is a chain sequence such that a_n¹₄ then

:

n=1

(a_n&¹₄)³₈.

In [2, Exercise III.5.6] he replaced ³₈with (1+- 2)8 and conjectured that

1

4 is sufficient. We show that this conjecture is correct. We also show that one has

:

n=N

\

^aⁿ^&¹⁴

+

^4N¹ ^,

and we determine when the equality holds.

Chain sequences have important applications to orthogonal polynomials (see [2]). Let p_n be symmetric orthogonal polynomials on the interval [&1, 1] relative to a probability measure + and satisfying the recurrence relation

xp_n(x)=*_n+1p_n+1(x)+*_np_n&1(x), n1, (1) with initial conditions p0(x)=1 and p1(x)=*1x. It can be shown that the support of + is contained in [&1, 1] if and only if [*²_n] is a chain sequence. The constant sequence *_n=¹₂ corresponds to the Chebyshev polynomials of the second kind. Their orthogonality measure d+(x)=

(2?)(1&x²)¹²dx is supported in [&1, 1]. When *_n¹₂the orthogonality interval can be larger than [&1, 1]. The question arises: by how much can

*_n exceed ¹₂ so that the orthogonality measure is still supported in the interval [&1, 1]? This question is connected with estimating the norms or spectral radii of the Jacobi matrix associated with (1), because the orthogonality measure is supported in [&1, 1] if and only if the spectral radius of the Jacobi matrix is less than or equal to 1. All this can readily be solved by means of chain sequences. We give necessary and sufficient conditions for sequences *_n¹₂ such that supp +/[&1, 1]. These conditions are useful in constructing such sequences.

We also discuss maximal sequences *_nwith the property that the Jacobi matrix associated with *_nhas a spectral radius equal to 1 and each Jacobi matrix associated with *_n*_n has a spectral radius equal to 1 if and only if *_n=*_nfor each n. We show that a sequence *_n>0 is maximal if and only if the series m2n is divergent, where m_n are the moments of the orthogonality measure associated with J.

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The origin of our interest in polynomials p_n orthogonal on the interval [&1, 1] and such that *_n¹₂ comes from the nonnegative linearization problem. If we express the product p_n(x) p_m(x) in terms of p_k(x) we get the linearization formula

p_n(x) p_m(x)= :

n+m

k= |n&m|

c(n, m, k) p_k(x).

By [6, Prop. 1] we get that c(n, m, k) are nonnegative for all n, m, k0 provided that *_n¹₂ and supp +/[&1, 1] (see also [5, Theorem 3]).

2. JACOBI MATRICES AND CHAIN SEQUENCES

A given sequence of real numbers *_n determines a Jacobi matrix J as follows:

0 *₁ 0 0 } } }

*1 0 *2 0 } } }

J=

\

⁰⁰^b ^*⁰^b² . . . .^*⁰³ ^*⁰³ ^{. . .}^{. . .}

+

^. ⁽²⁾

The connection between Jacobi matrices and chain sequences is exhibited in the next proposition.

Proposition 1. The Jacobi matrix J corresponds to a bounded linear operator on square summable real valued sequences, with operator norm less than or equal to 1 if and only if *²_nis a chain sequence.

Proof. Since J is a symmetric matrix we have

&J&l² l²=sup[x^TJx | x^Tx1].

On the other hand,

x^TJx=2 :

n=1

*_nx_nx_n+1.

Now the conclusion follows from [7, Theorem 20.1] (see also [2, Exer- cise III.5 13]). K

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3. ESTIMATES FOR CHAIN SEQUENCES

The next two lemmas are known. We prove them in order to remain self- contained. Note that the proof of Lemma 2 is entirely different from the one in [2, p. 99].

Lemma 1 (Wall [7]). Let a_n be a chain sequence with a parameter sequence g_n. If a_n¹₄the sequence g_n is increasing and it tends to ¹₂. In particular, a_n tends to ¹₄.

Proof. We have

g_n= a_n

1& g_n&1 1

4(1& g_n&1) g_n&1. Thus g_nZg and a_n g(1& g)¹₄. Hence g=¹₂.

Lemma 2 ([2], p. 99). Let a_n¹₄ be a chain sequence with a parameter sequence g_n. Then

01

2& g_n 1 2(n+1). Proof. Let

$_n=1&2 g_n.

In view of the preceding lemma we have 0$_n1 and

$_n=$_n&1&(4a_n&1)

1+$_n&1 $_n&1 1+$_n&1. Hence

$_n f ($_n&1), where f (x)= x x+1. Therefore

$_n f b f b } } } b f

n times

($0)= $₀

n$0+1 1 n+1.

This gives the conclusion. K

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Theorem 1. Let a_n14 be a chain sequence. Then :

m=n

\

^a^m^&¹⁴

+

⁴ⁿ¹ ^, ^n1. ⁽³⁾

If for some n1 equality holds in (3) then a_m=14 for m>n and a_n=14+14n.

In particular

:

n=1

\

^aⁿ^&¹⁴

+

¹⁴ ⁽⁴⁾

and the equality holds if and only if a₁=12 and a_n=14 for n2.

Proof. Let g_n be any parameter sequence for a_n. By Lemma 1 we have g_n¹₂. Therefore

a_n&¹₄= g_n(1& g_n&1)&¹₄

=¹₂( g_n& g_n&1)&(¹₂& g_n&1)(¹₂& g_n)¹₂( g_n& g_n&1).

Next, adding up the terms and using Lemma 2 gives :

m=n

\

^a^m^&¹⁴

+

²¹

\

¹²^{& g}^n&1

+

⁴ⁿ¹ ^.

This equality holds if and only if

(¹₂& g_m&1)(¹₂& g_m)=0 for mn

and g_n&1=12&12n. Since by Lemma 1 the sequence g_nis nondecreasing we get g_m=12 for mn. Therefore

a_n= g_n(1& g_n&1)=1

2

\

¹²⁺²ⁿ¹

+

⁼¹⁴⁺⁴ⁿ¹ ^,

a_m= g_m(1& g_m&1)=1

4 for m>n. K Theorem 2. Let a_n be a chain sequence such that a_nz¹

4. Then 0a_n&1

41

4

\

^tan²⁽ⁿ⁺¹⁾^?

+

²^. ⁽⁵⁾

Proof. We will make use of the following lemma.

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Lemma 3. Let

f (x)=x&=²

x+1, ==tan ..

Then denoting by f^m the mth iterate of f we have f^m(x)= x&= tan m.

(=^&1tan m.) x+1. In particular

f^m(1)= tan . tan(m+1) ..

Lemma 3 can be proved by induction using the relation between tan(m+1) . and tan m.. The more demanding reader may instead consider the corresponding 2_2 matrix

F=

\

¹¹ ^&=¹ ²

+

^.

Its iterates can be computed by finding a basis of eigenvectors for F.

Assume that

a_n=¹₄+¹₄=²_n and =_nz0.

Let g_nbe a parameter sequence for a_n. Write g_n in the form g_n=¹₂(1&$_n).

By Lemma 1 we have 0$_n1. Then using a_n= g_n(1& g_n&1) and =_m=_n gives

$_m=$_m&1&=²_m

$_m&1+1 $_m&1&=²_n

$_m&1+1, mn.

Thus

$_m f ($_m&1), where f (x)=x&=²_n

x+1, for mn.

Since f (x) is an increasing function for x0, by Lemma 3 we obtain

$_m f^m($0) f^m(1)= tan .

tan(m+1) . (6)

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for mn, where =_n=tan .. This implies that tan(m+1) .0 for mn.

Hence (n+1) .?2. Thus

=_n=tan .tan ? 2(n+1). K Using tan x4x?, for 0x<?4, gives

Corollary 1. Let a_nbe a chain sequence such that a_nz¹

4. Then a_n&1

4 1 (n+1)².

The next theorem gives a characterization of the chain sequences with all terms greater than or equal to ¹₄. In view of Proposition 1 the constant chain sequence a_n=¹₄ corresponds to the Jacobi matrix with *_n=¹₂, which is in turn associated with the Chebyshev polynomials of the second kind

U_n(cos x)=sin(n+1) x sin x .

Theorem 3. Let a_n=¹₄(1+=_n), with =_n0. Then [a_n] is a chain sequence if and only if there exists a sequence [c_n] of positive numbers such that

(i) c_n+12c_n, for n1.

(ii) c_n+1&c_n :

m=n

c_m=_m, for n1.

Proof. (O) Let a_n= g_n(1& g_n&1) and g_n=¹₂(1&$_n). Then

=_n=$_n&1&$_n&$_n&1$_n. Set c₁=1 and

c_n+1

c_n =1+$_n&1. Then c_n+12c_n, because $_n1. We have

c_n+1&c_n=c_n$_n&1. (7) Moreover

=_n=$_n&1&$_n(1+$_n&1)=$_n&1&c_n+1 c_n $_n.

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Hence

c_n=_n=c_n$_n&1&c_n+1$_n. (8)

Thus the sequence c_n$_n&1 is nonincreasing; as such it has a limit c_n$_n&1 s0.

Now summing up (8) and using (7) yields

:

m=n

c_m=_m=c_n$_n&1&s=c_n+1&c_n&sc_n+1&c_n.

(o) Set

$_n=c^&1_n+1 :

m=n+1

c_m=_m.

Then

$_nc_n+2 c_n+1&1.

Thus $_n1. Let h_n=¹₂(1&$_n). Then

4h_n(1&h_n&1)=(1&$_n)(1+$_n&1)=1+=_n+c_n+1&c_n

c_n $_n&$_n&1$_n

1+=_n+1 c_n

\

m=n^:

c_m=_m

+

^$ⁿ^&$^n&1^$ⁿ

=1+=_n=4a_n.

Thus h_n(1&h_n&1)a_n. This implies that a_n is a chain sequence (see also [2, Theorem 5.7, p. 97]). K

The next Corollary can be found in [2, Problem 5.7, p. 100]

Corollary 2 (Chihara). Let a_n=¹₄(1+=_n), where =_n0 and :

m=1

m=_m1.

Then [a_n] is a chain sequence.

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Proof. Apply Theorem 3 with c_n=n. K

Corollary 3. Let [c_n] be a concave nondecreasing sequence of positive numbers satisfying 2c_nc_n+1 for n1. Set

=_n=2c_n+1&c_n&c_n+2

c_n .

Then the sequence a_n=¹₄(1+=_n) is a chain sequence.

Proof. We have c_n+1&c_nzs0. Thus

:

m=n

=_mc_m=c_n+1&c_n&sc_n+1&c_n. K

4. MAXIMAL CHAIN SEQUENCES

A chain sequence [a_n] is called maximal if there is no chain sequence [b_n] such that b_na_n and [a_n]{[b_n].

Maximal chain sequences exist and moreover every chain sequence is bounded from above by a maximal one (see [7]).

The next proposition follows in part from Proposition 1.

Proposition 2. Let [a_n] and [b_n] be chain sequences. Then the sequence [c_n] defined by

- c_n=* - a_n+(1&*) - b_n

is a chain sequence for any 0<*<1. Moreover if [a_n]{[b_n] then [c_n] is not a maximal chain sequence.

Proof. Let a_n= g_n(1& g_n&1) and b_n=h_n(1&h_n&1) for 0 g_n1 and 0h_n1. Set f_n=*g_n+(1&*) h_n. Then

f_n(1& f_n&1)=[*g_n+(1&*) h_n][(1&*)(1&h_n&1)+*(1& g_n&1)]

=*(1&*)[ g_n(1&h_n&1)+h_n(1& g_n&1)]+*²a_n+(1&*)²b_n

2*(1&*) - g_n(1& g_n&1) h_n(1&h_n&1)+*²a_n+(1&*)²b_n

=(* - a_n+(1&*) - b_n)²=c_n.

Hence c_n is a chain sequence as it is bounded from above by the chain sequence f_n(1& f_n&1).

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If [c_n] is a maximal chain sequence then both [a_n] and [b_n] are maximal chain sequences. In that case g0=h0=0, because otherwise setting g0=0 or h0=0 leads to chain sequences which are greater than [a_n] or [b_n], respectively.

Also, if [c_n] is a maximal chain sequence, the calculations performed above enforce

c_n= f_n(1& f_n&1),

g_n(1&h_n&1)=h_n(1& g_n&1), n1.

Since g₀=h₀=0, the last equation implies g_n=h_n for n0. Hence a_n=b_n for n1. K

We now turn to chain sequences such that a_n¹₄.

Theorem 4. Let a_n=¹₄(1+=_n), where =0. Then [a_n] is a maximal chain sequence if and only if there exists a unique sequence [c_n] of positive numbers such that c1=1 and

(i) 2c_nc_n+1, for n1.

(ii) c_n+1&c_n :

m=n

c_m=_m, for n1.

Proof. Let [a_n] be a maximal chain sequence. By Theorem 3 a sequence [c_n] exists. Let g_n be a unique parameter sequence for [a_n]. Set g_n=¹₂(1&$_n). Analyzing the second part of the proof of Theorem 3 we get

c_n+1&c_n= :

m=n

c_m=_m=c_n$_n&1, n1.

Since c1=1 and $_nis uniquely determined by g_nwe conclude that c_nis also uniquely determined.

Assume [a_n] is not maximal. By [7] it has two different parameter sequences. Hence there exist [ g_n] and [h_n] such that

a_n= g_n(1& g_n&1)=h_n(1&h_n&1), n1,

and g0<h0. Define c_n and d_nby c1=d1=1 and c_n+1=2(1& g_n&1) c_nand d_n+1=2(1&h_n&1) d_n. This leads to two different sequences satisfying Theorem 3. K

Definition 1. The Jacobi matrix associated with the sequence [*_n] will be called maximal if &J&1 and for each Jacobi matrix J$ associated with the sequence [*$_n]{[*_n] such that |*_n| |*$_n| we have &J$&>1.

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In view of Proposition 1 J is a maximal Jacobi matrix if and only if [*²_n] is a maximal parameter sequence. By [7] a sequence [a_n] is a maximal chain sequence if and only if the continued fraction

1|

|1&a1|

|1&a2|

|1&a3|

|1& } } } tends to 0.

Favard's Theorem states that for each Jacobi matrix of the form (2) there exists a probability measure + symmetric about the origin, supp +/[&&J&, &J&], such that it is the orthogonality measure for the polynomials p_n given recursively by (1).

The next theorem collects facts that can be deduced from [7] and [1, Theorem 1]. Our setting is a bit different, so we provide a short inde- pendent proof.

Theorem 5. Let J be a Jacobi matrix associated with the sequence [*_n].

Assume that &J&1 and that + is the associated orthogonality measure. The following conditions are equivalent.

(i) J is a maximal Jacobi matrix.

(ii) _n=0m2n=+, where m_n=_&xⁿd+(x).

(iii) The continued fraction

1&*²₁|

|1&*²₂|

|1&*²₃|

|1& } } } tends to 0.

Proof. Assume J is not maximal. In view of Proposition 1 the sequence a_n=*²_nis not a maximal chain sequence. Then there exists a chain sequence [b_n] such that a_nb_nand [a_n]{[b_n]. Let [h_n] be a parameter sequence for [b_n] and N be the smallest index such that a_N<b_N=h_N(1&h_N&1).

Set g_N=h_N and define g_n recursively by a_n= g_n(1& g_n&1). Then it is immediate that g_n>h_n for n<N and g_nh_n for n>N. In particular we have that g0>h00.

Let r_n(x) be the polynomials defined recursively by

xr_n(x)= g_n&1r_n+1(x)+(1& g_n&1) r_n&1(x), n1, (9)

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with initial conditions r0(x)=1, r1(x)=x. Thus r_n(1)=1 and r_n(&1)=

(&1)ⁿ. Hence r_n+2(x)&r_n(x) is divisible by x²&1. Consider the polynomials q_n(x) defined by

q_n(x)=r_n+2(x)&r_n(x)

x²&1 . (10)

Then by (9) we obtain

xq_n(x)= g_n+1q_n+1(x)+(1& g_n&1) q_n&1(x), n1, and q₀(x)= g^&1₀ , q₁(x)=( g₀g₁)^&1x. Set

p_n(x)= g0

^{(1& g}⁰^{)(1& g}^g¹^g²¹^{} } } g}) } } } (1& gⁿ _n&1) g_n(x). (11) Then the polynomials p_n satisfy

xp_n(x)=- g_n+1(1& g_n) p_n+1(x)+- g_n(1& g_n&1) p_n&1(x), n1, (12) with p₀(x)=1 and p₁(x)=c^&1x, where c=- g₁(1& g₀). Then taking into account *²_n=a_n= g_n(1& g_n&1) implies

xp_n(x)=*_n+1p_n+1(x)+*_np_n(x), n1, (13) with p₀(x)=1 and p₁(x)=*^&1₁ x. Let & be a probability measure associated with the polynomials r_n, which exists by Favard's Theorem. Recall that by (13) + is the orthogonality measure for the p_ns and it is supported in [&1, 1]. By (10) and (11) the measures & and + are related by

d+(x)=c^&1(1&x²) d&(x), where c=¹_&1(1&x²) d&(x). Thus

:

n=0

m_2n=

|

_&1¹ _1&x^d+(x)²^<+. ⁽¹⁴⁾

This completes the proof of (ii) O (i).

Assume _n=0m_2n<+. By (14) the measure

d&(x)=c^&1d+(x)

1&x², where c= :

n=0

m2n,

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has total mass equal to 1 and its support is contained in [&1, 1]. Hence the zeros of the corresponding orthogonal polynomials belong to (&1, 1).

Let r_nbe these polynomials normalized at x=1, i.e., r_n(1)=1. There exists a sequence [ g_n]_n=0 such that g0>0 and

xr_n(x)= g_n&1r_n+1(x)+(1& g_n&1) r_n&1(x), n1,

where r0(x)=1 and r1(x)=x. Define the polynomials q_n and p_n by (10) and (11), respectively. Then by the relation between + and & the polynomials p_n are orthonormal relative to +. Hence combining (12) and (13) gives

*²_n= g_n(1& g_n&1).

Since g0>0 the sequence [*²_n] is not a maximal chain sequence, which in turn implies that J is not a maximal Jacobi matrix. This shows (i) O (ii).

The equivalence (ii) (iii) follows from the formula

|

_&1¹ ^d+(x)_y&x⁼^1|_{| y}^&^*

2 1|

| y&*²₂|

| y& } } } , (15) (see [3, p. 46]) and the fact that since the measure + is symmetric about x=0

|

&1¹

d+(x) 1&x²=

|

&1¹

d+(x) 1&x.

The formula (15) holds for y [&1, 1]. We get the desired result by taking the limit when y 1⁺. K

5. MAXIMAL PARAMETER SEQUENCES

Wall [7] observed that a chain sequence [a_n] is maximal if and only if it admits a unique parameter sequence. Other chain sequences admit more parameter sequences. Among them there exists a maximal parameter sequence (see [7, Theorem 19.2; 2, Theorem III.5.3]). Wall proved that maximal parameter sequences are exactly those sequences [ g_n] for which

:

n=1

g₁g₂} } } g_n

(1& g₁)(1& g₂) } } } (1& g_n)=+. (16) (see [7, (19.10), p. 82; 2, Theorem III.6.1]). For chain sequences [a_n], with terms greater than ¹₄we have the following.

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Proposition 3. Let [a_n] be a chain sequence such that a_n¹₄ for n1.

Let [a_n] be a parameter sequence and set g_n=¹₂(1&$_n). Then [ g_n] is the maximal parameter sequence if and only if

:

n=1

exp

\

^{&2 :}k=1ⁿ

$_k

+

^=.

Proof. Observe that g1g2} } } g_n

(1& g1)(1& g2) } } } (1& g_n)= (1&$²₁)(1&$²₂) } } } (1&$²_n)

(1+$1)²(1+$2)²} } } (1+$_n)². (17) By Lemma 2 we have $_n1(n+1). Thus

n+1

2n (1&$²₁)(1&$²₂) } } } (1&$²_n)1. (18) Again by using Lemma 2 and the fact that the function x [ e^x(1+x) is increasing we obtain

1 e^$¹ 1+$1

e^$² 1+$2

} } } e^$ⁿ 1+$_n

exp(1+¹₂+ } } } +¹_n) (1+¹₂)(1+¹₃) } } } (1+¹_n)

= 2

n+1exp

\

¹⁺¹²^{+ } } } +}¹ⁿ

+

^2. ⁽¹⁹⁾

Combining (17), (18), and (19) gives the conclusion. K Corollary 4. Using the notation of Proposition 3, if

:

n=1

$_n<+

then the sequence g_n=¹₂(1&$_n) is the maximal parameter sequence.

REFERENCES

1. T. Chihara, Chain sequences and orthogonal polynomials, Trans. Amer. Math. Soc. 104 (1962 ), 116.

2. T. Chihara, ``An Introduction to Orthogonal Polynomials, Mathematics and Its Applica- tions,'' Vol. 13, Gordon 6 Breach, New York, 1978.

3. J. A. Shohat and J. D. Tamarkin, ``The Problem of Moments, Mathematical Surveys,'' Vol. 1, Amer. Math. Soc., Providence, RI, 1943.

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4. R. Szwarc, Chain sequences and compact perturbations of orthogonal polynomials, Math.

Z. 217 (1994), 5771.

5. R. Szwarc, Nonnegative linearization of orthogonal polynomials, Colloq. Math. 69 (1995), 309316.

6. R. Szwarc, Nonnegative linearization of associated q-ultraspherical polynomials, Methods Appl. Anal. 2 (1996), 399407.

7. H. S. Wall, ``Analytic Theory of Continued Fractions,'' Nostrand, New York, 1948.

Chain Sequences, Orthogonal Polynomials, and Jacobi Matrices

Chain Sequences, Orthogonal Polynomials, and Jacobi Matrices

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