Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

Frank Filbir,²Roland Girgensohn,²Anu Saxena,³Ajit Iqbal Singh⁴ and Ryszard Szwarc⁵

For an orthogonal polynomial system p ˆ …pn†n2IN0and a sequence d ˆ …dn†n2IN0 of nonzero numbers, let Sp;dbe the linear operator de®ned on the linear space of all polynomials via Sp;dpnˆ dnpnfor all n 2 IN0.We investigate conditions on p and d under which Sp;dcan simultaneously preserve the orthogonality of di€erent polynomial systems. As an application, we get that for p ˆ … L_n†, a generalized Laguerre polynomial system, no d can simultaneously preserve the orthogonality of two additional Laguerre systems, … L^‡t_n ¹† and … L^‡t_n ²†, where t1; t26ˆ 0 and t16ˆ t2. On the other hand, for p ˆ … Tn†, the Chebyshev polynomial system and d ˆ … … 1†ⁿ†, Sp;d simultaneously preserves the orthogonality of uncountably many kernel polynomial systems associated with p. We study many other ex- amples of this type.

KEY WORDS: Orthogonal polynomials; dilation map; kernel polynomials;

Jacobi polynomials; Laguerre polynomials.

1. INTRODUCTION

We start with some notation. Let IN be the set of natural numbers and IN0ˆ IN [ f0g. For a set X; ] X will denote the cardinality of X. Let P1be the linear space of all polynomials and let P be the set of sequences p ˆ …p_n†_n2IN0 in P₁ with p₀ˆ 1 and deg p_nˆ n for all n 2 IN₀. We shall

0Dedicated to Professor P. L. Butzer on the occasion of his 70th birthday.

0GSFÐNational Research Center for Environment and Health, Institute of Biomathematics and Biometry, IngolstaÈdter Landstrae 1, D-85764 Neuherberg, Germany.

0Department of Mathematics, Jesus and Mary College, Chanakyapuri, New Delhi 110021, India.

0Department of Mathematics, University of Delhi South Campus, Benito Juarez Road, New Delhi 110021, India.

0Institute of Mathematics, WrocŒaw University, pl. Grunwaldzki 2/4, 50-384 WrocŒaw, Poland.

177

1521-1398/00/0400-0177$18.00/0 ß 2000 Plenum Publishing Corporation

often write …p_n†_n2IN0 simply as …p_n† or even p. S will usually stand for a degree preserving linear operator on P₁ to itself with S…1† ˆ 1. Ortho- gonality will always be with respect to some quasi-de®nite moment func- tional, whereas positive orthogonality will be with respect to some positive de®nite moment functional and in the context of real polynomials only (see [5], Theorem I.3.3). The word orthonormality will be used only in the context of positive orthogonality, where, in addition, the positive moment functional L is normalized so that L…1† ˆ 1. For any such pair …p; L† and any polynomial q, …L…jqj²††¹⁼²will be denoted by kqk_L or kqk_p; the sux L or p will be suppressed if no confusion arises. Orthogonality with respect to a symmetric moment functional (see [5], De®nition I.4.1) will be termed symmetric orthogonal. We refer to [2], [5], and [13] for the basics on orthogonal polynomials.

Operators preserving orthogonality of polynomials have been studied by various authors, among them [1], [8], and [10]. We start by quoting two simple recent results from [8] which, to a certain extent, provided the motivation for the present paper. For  2 IR n … IN†, let …L_n† be the generalized Laguerre polynomial system, given by

L_n…x† ˆXⁿ

k ˆ 0

… 1†^k n ‡  n k

 

x^k

k!; x 2 IR; n 2 IN0; where

t 0 !

ˆ 1; t

k !

ˆt…t 1†


…t k ‡ 1†

k!

ˆ…t k ‡ 1†_k

…1†_k ; t 2 IR; k 2 IN:

From Ref. [8]: Corollary 1. If S preserves the orthogonality of …L_n† for  ˆ ₀ 1; ₀; ₀‡ 1; ₀‡ 2 (with some ₀† and, moreover, k S L₁ k²ˆ 1 ‡  for at least two of f₀; ₀‡ 1; ₀‡ 2g, then for some a;

b 2 IR; …Sp†…x† ˆ p…ax ‡ b† for x 2 IR and p 2 P₁.

From Ref. [8]: Theorem 3. For any  2 IR, the linear operator S de®ned on P1via …SL_n†…x† ˆ L^‡_n …x ‡ †; x 2 IR; n 2 IN, is independent of . Further, S simultaneously preserves the orthogonality of … L_n† for all . Let Pm; Po; Pmo; and P‡o be the subsets of P consisting of monic, orthogonal, monic orthogonal, and positive orthogonal systems, resp- ectively. Let D be the set of nonconstant sequences d ˆ …dn†_n2IN0of non-zero numbers with d0ˆ 1.

In Section 2 we shall de®ne a dilation-type linear operator S_p;don P₁to itself associated to a p in P_oand d in D. In fact, S_p;dis the linear operator on P₁ to itself, given by S_p;dp_n ˆ d_np_n for all n 2 IN₀. We ®rst observe that if d 2 D is the sequence of eigenvalues of a linear di€erential operator F such that the corresponding sequence of eigenfunctions consists of an orthogonal polynomial system p, then S_p;d coincides with F. (For these topics, one can consult [7] and the related expository articles [4] and [11], or the books [3], Section 3.5, and [5], Section V.2 and Section V.3.) This is illustrated by the following simple examples.

(a) Let ;  > 1 and p ˆ …Pn^;††, the Jacobi polynomial system. Let dnˆ n …n ‡  ‡  ‡ 1† ‡ 1 for n 2 IN0. Then Sp;d is the di€er- ential operator F determined by

…Ff†…x† ˆ …1 x²† f⁰⁰…x† ‡ …  … ‡  ‡ 2† x†f⁰…x† ‡ f…x†:

(b) For p ˆ …H_n†, the Hermite polynomial system, and d given by d_nˆ 2n ‡ 1 for n 2 IN₀, S_p;d coincides with the di€erential operator F de®ned by

…Ff†…x† ˆ f⁰⁰…x† 2xf⁰…x† ‡ f…x†:

(c) Let  2 IR n … IN† and p ˆ …L_n†, the generalized Laguerre system.

If d_nˆ 2n ‡ 1 for n 2 IN₀, then S_p;d coincides with the di€erential operator F determined by

…Ff†…x† ˆ 2xf⁰⁰…x† ‡ 2… ‡ 1 x†f⁰…x† ‡ f…x†:

For a ®xed …p; d† 2 P_o D we shall look for those subsets S of P, which are mapped into Po by Sp;d, i.e., where orthogonality is simulta- neously induced by Sp;d. Also, for a class S contained in Poand a p in Po, we shall study the question of existence of a d in D such that Sp;d preserves the orthogonality of S.

To motivate, we may note the following result, which follows readily from the connection coecients of the Jacobi system (cf. [2], 7.32±7.34).

Theorem 1.1. Let ;  > 1 and p be the Jacobi polynomial system …Pn^;††. Let

dnˆ … 1†ⁿ … ‡ 1†_n

… ‡ 1†_nˆ … 1†ⁿ Q_n

j ˆ 1… ‡ j†

Q_n

j ˆ 1… ‡ j†

and d⁰_nˆ 1=d_n for n 2 IN₀.

(a) Then, for t > 0 and n 2 IN₀,

S_p;dP_n^;‡t†ˆ d_nP_n^‡t;† and

Sp;d⁰P_n^‡t;†ˆ d⁰_nP_n^;‡t†:

Therefore Sp;d simultaneously preserves the orthogonality of fPn^;‡t†; t > 0g and Sp;d⁰ that of fPn^‡t;†; t > 0 g.

(b) For  ˆ , we have dn ˆ … 1†ⁿ; n 2 IN0, and d ˆ d⁰. In this case, Sp;dPn^{; †} ˆ … 1†ⁿPn^{; †} for all > 1, and thus Sp;dˆ S_P^{; †}

n †;d

for > 1.

This is true for any sequence d satisfying d_2nˆ 1 and d_2n‡1ˆ d₁ for all n 2 IN₀.

Further, for such d's, it is enough to consider the case p ˆ …P… 1=2; 1=2†

n † ˆ …T_n† instead of the whole class fPn^{; †} : > 1g of symmetric Jacobi polynomials.

In Section 3, we shall de®ne and study the notion of a q ˆ …qn† in P to be analytic with respect to a p ˆ …pn† in P. Let Papbe the class of those q's in P, which are analytic with respect to p. It turns out that for a p in Pmoand q in Pap; q 2 Po if, and only if, q is a kernel polynomial K…p; y† associated with p where y 62 Zp, Zpbeing the set of zeros of pn's. An equivalent form for a p in Pmois obtained using the general method developed in Section 2. This enables us to have nice transparent conditions on d for Sp;d to simulta- neously preserve the orthogonality of various subsets of Pap\ Po. For some special d's we can obtain many di€erent subsets of P_ap, whose orthogonality is simultaneously preserved by S_p;d.

In the last section, we apply the results of the second and third sections to the class of generalized Laguerre polynomial systems and obtain a striking contrast to the situation for the Chebyshev polynomials!

2. DILATION-TYPE OPERATORS AND ORTHOGONALITY 2.1. Discussion and De®nition

Let p ˆ …pn† be an orthogonal polynomial system. Then the set P of all polynomial systems is in one±one correspondence with the set A of (in®nite) matrices A ˆ …aj;k†_j;k2IN0 with aj;kˆ 0 for k > j; a0;0ˆ 1 and aj;j6ˆ 0 for

all j via q ˆ A p, i.e., for all n 2 IN₀; q_nˆP_n

k ˆ 0 a_n;kp_k.

(a) Clearly, in the case p has real coecients, q has real coecients if, and only if, A is real.

(b) In the case p 2 P_m, i.e., p is monic and A 2 A, q ˆ A p 2 P_m if, and only if, aj;jˆ 1 for all j. Such matrices will be called monic.

The set of monic matrices will be denoted by Am.

(c) For a d ˆ …dn† 2 D let Sp;d be the linear operator on P1 to itself given by Sp;dpnˆ dnpnfor all n 2 IN0. The map induced by Sp;don P will be called the …p; d†-dilation and denoted by Sp;d only.

(d) Clearly for an A 2 A and q ˆ A p, Sp;dq corresponds to the matrix Adˆ …aj;kdk† ˆ A D; D being the diagonal matrix given by the diagonal d. In the case p is monic, the associated monic polynomial system Q has the matrix …a_j;k d_k†=…a_j;j d_j†

(e) For a ®xed p, the sequence d and the operator S. _p;ddetermine each other. However, for a ®xed d; S_p;d ˆ S_q;d can hold without p ˆ q, as is seen from Theorem 1.1. In fact, for q ˆ A p and d; d⁰2 D, S_p;d ˆ S_q;d⁰ if, and only if, a_j;k…d⁰_j d_k† ˆ 0 for all j; k 2 IN₀. As a consequence, if d has all terms distinct and p; q are monic, then p ˆ q()S_p;dˆ S_q;d. On the other hand, if d has at least two terms coincident then fq monic : S_p;d ˆ S_q;dg is in®nite.

(f) Monic orthogonal polynomial systems …pn† can be characterized via their recurrence relation

xpn…x† ˆ pn‡1…x† ‡ bnpn…x† ‡ cnpn 1…x†; x 2 IR; n 2 IN0

with recurrence sequences b ˆ …b_n†, c ˆ …c_n† and c_n6ˆ 0 for all n 2 IN₀ (see again [5]). For notational convenience, we set p 1ˆ q 1  0; aj;k ˆ 0 for j < 0 or k < 0, d 1ˆ 0, and we take empty sums to be zero and empty products to be 1.

(g) I. M. She€er [12] de®ned polynomial systems of type zero as follows. The system q ˆ …qn† is of type zero if there are numbers n

with 16ˆ 0 such that 1q⁰_n‡ 2q_n⁰⁰‡


‡ nq^n†n ˆ qn 1 for all n 2 IN. Among other things, he connected systems of type zero to orthogonal systems by dilations. He proved ([12], Theorem 4.1):

Assume that a polynomial system q ˆ …dnpn† is given with d 2 D.

Then q is of type zero if, and only if, there exist numbers a₁; . . . ; a₄ with

b_nˆ a₁‡ a₂n and c_nˆ n …a₃‡ a₄n† for all n 2 IN₀:

Theorem 2.2. Let p ˆ …p_n† be a monic orthogonal polynomial system with recurrence sequences b ˆ …b_n† and c ˆ …c_n†. Let A ˆ …a_j;k† be a monic matrix and q ˆ …q_n† ˆ Ap.

(a) The system q is orthogonal if, and only if, (i) a_2;0 a_2;1a_1;06ˆ c₁‡ …b₀ b₁†a_1;0 …a_1;0†²; (ii) an‡1;n 1 an‡1;nan;n 1

6ˆc_n‡ …b_{n 1} b_n†a_{n;n 1} …a_{n;n 1}†²

‡an;n 2; n  2;

(iii) …an‡1;0 an‡1;nan;0† …an‡1;n 1 an‡1;nan;n 1†an 1;0

ˆ ‰…b₀ b_n† a_{n;n 1}Ša_n;0‡ c₁a_n;1

‰c_n‡ …b_{n 1} b_n†a_{n;n 1} …a_{n;n 1}†²‡ a_{n;n 2}Ša_{n 1;0}; n  2;

(iv) …aandn‡1;j an‡1;nan;j† …an‡1;n 1 an‡1;nan;n 1†an 1;j

ˆ a_{n;j 1}‡ ‰b_j b_n a_{n;n 1}Ša_n;j‡ c_j‡1a_n;j‡1

‰cn‡ …bn 1 bn†an;n 1 …an;n 1†²‡ an;n 2Šan 1;j;

n  3 ; 1  j  n 2:

(b) Suppose p is positive orthogonal and A is real. Then q is positive orthogonal if, and only if, all the conditions in (a) above are satis®ed with 6ˆ in (i) and (ii) strengthened to <.

Proof. (a) By Favard's Theorem ([5], Theorem I.4.4), combined with the fundamental recurrence relation ([5], Theorem I.4.1), q is orthogonal if, and only if, there exist sequences  ˆ …n†_n2IN0; ˆ … n†_n2IN0; 0 arbitrary (and ine€ective) such that for each n 2 IN; n6ˆ 0

xqn…x† ˆ qn‡1…x† ‡ nqn…x† ‡ nqn 1…x†; x 2 IR; n 2 IN0: Using the recurrence relation for the p_n's, this happens if, and only if, for all n 2 IN₀,

Xⁿ

j ˆ 0

a_n;j… p_j‡1‡ b_jp_j‡ c_jp_{j 1}†

ˆX^n‡1

j ˆ 0

a_n‡1;jp_j‡ _nXⁿ

j ˆ 0

a_n;jp_j‡ _n X^{n 1}

j ˆ 0

a_{n 1;j}p_j:

Since the pj's are linearly independent, we can equate their coecients on both sides, and simple computations then give that the conditions stated in

the theorem are necessary and sucient. Along the way we also have that 0ˆ b0 a1;0;

₁ˆ b₁‡ a_1;0 a_2;1; ₁ˆ c₁‡ b ₀ b₁ a_1;0

a_1;0 …a_2;0 a_2;1a_1;0†;

and for n  2 ;

nˆ bn‡ an;n 1 an‡1;n; _nˆ c_n‡ …b _{n 1} b_n† a_{n;n 1}

a_{n;n 1}

‡ an;n 2 …an‡1;n 1 an‡1;nan;n 1†:

(b) We shall again use the fundamental recurrence relation and Favard's Theorem. We have that for each n, b_n is real and c_n‡1> 0.

Further, q is positive orthogonal if, and only if, for each n, _n is real and _n‡1> 0. This gives us that 6ˆ in (i) and (ii) of (a) have to be replaced by

<. &

Remark 2.3.

(a) The above theorem tells us how to construct all in®nite matrices which preserve orthogonality or positive orthogonality.

Take any sequence s ˆ …sn†_n2IN0 of numbers and another sequence t ˆ …tn†_n2IN0 of numbers satisfying

t₀6ˆ c₁‡ s₀‰…b₀ b₁† …s₀ s₁†Š;

t_n t_{n 1}6ˆ c_n‡1‡ s_n‰…b_n b_n‡1† …s_n s_n‡1†Š; n  1:

Set a_n‡1;nˆ s_n and a_n‡2;nˆ t_n for all n. De®ne, for n  2; 0  j  n 2, the numbers a_n‡1;j recursively as in (a) (iii) and (iv). Finally, take a_j;j ˆ 1 and a_j;k ˆ 0 for k > j, and obtain A_s;tˆ …a_j;k†. Then q ˆ A_s;tp is orthogonal. Also, every monic matrix A having this preservation property has this form for a unique pair …s; t†.

(b) In case p is positive orthogonal, we have to restrict s and t to be real sequences and take

t₀< c₁‡ s₀‰…b₀ b₁† …s₀ s₁†Š;

tn tn 1< cn‡1 ‡ sn‰…bn bn‡1† …sn sn‡1†Š; n  1:

(c) On the other hand, if p is symmetric orthogonal, then b_nˆ 0 for all n. Thus q is symmetric orthogonal if, and only if, _nˆ 0 is satis®ed together with (a) of Theorem 2.2, i.e., the sequence s ˆ …s_n† is constant. This simpli®es the condition on t to t_n t_{n 1}6ˆ c_n‡1 for all n 2 IN, and it simpli®es (a) as follows:

(i) a_2;06ˆ c₁,

(ii) a_{n‡1;n 1}6ˆ a_{n;n 2}‡ c_n, n  2,

(iii) a_n‡1;0ˆ c₁a_n;1‡ …a_{n‡1;n 1} a_{n;n 2} c_n† a_{n 1;0}, n  2, and(iv) a_n‡1;jˆ a_{n;j 1}‡ c_j‡1a_n;j‡1‡ …a_{n‡1;n 1} a_{n;n 2} c_n† a_{n 1;j},

n  3, 1  j  n 2.

As before, for positive orthogonality, in (i) and (ii), 6ˆ has to be replaced by <.

(d) Given a monic matrix A, the above theorem also gives us conditions on the sequences b and c for the polynomial system q ˆ Ap to be orthogonal.

Theorem 2.4. Let A be a monic in®nite matrix, d a sequence of nonzero numbers with d₀ˆ 1 and p ˆ …p_n† a monic orthogonal system with recurrence sequences b and c. Let q ˆ A p and r ˆ S_p;dq.

(a_d) The system r is orthogonal if, and only if, (i) 1

d2…a_2;0 a_2;1a_1;0† 6ˆ c₁‡ 1

d1…b₀ b₁†a_1;0 1 d1a_1;0

 ₂

;

(ii) d_{n 1}

d_n‡1…a_{n‡1;n 1} a_n‡1;na_{n;n 1}† 6ˆ c_n‡d_{n 1}

dn …b_{n 1} b_n†a_{n;n 1} d_{n 1} dn a_{n;n 1}

 ₂

‡dn 2

d_n an;n 2; n  2, (iii) 1

dn‡1…an‡1;0 an‡1;nan;0† …an‡1;n 1 an‡1;nan;n 1†an 1;0

ˆ 1 dn

"

b₀ b_n d_{n 1} dn a_{n;n 1}

!

a_n;0‡ d₁c₁a_n;1 d_n

dn 1 c_n‡d_{n 1}

dn …b_{n 1} b_n†a_{n;n 1} dn 1

d_n a_{n;n 1}

 ₂

‡dn 2

d_n a_{n;n 2}

! a_{n 1;0}

#

; n  2;

(iv) 1

dn‡1…a_n‡1;j a_n‡1;na_n;j† …a_{n‡1;n 1} a_n‡1;na_{n;n 1}†a_{n 1;j}

ˆ 1 d_n

"

dj 1

d_j a_{n;j 1}‡ b_j b_n dn 1

d_n a_{n;n 1}

 

a_n;j

‡d_j‡1

dj c_j‡1a_n;j‡1 d_n

dn 1cn‡ …bn 1 bn†an;n 1 d_{n 1}

dn …an;n 1†²

‡d_{n 2} dn 1a_{n;n 2}

 a_{n 1;j}

#

; n  3 ; 1  j  n 2:

(b_d) Suppose p is positive orthogonal. Then q is positive orthogonal if, and only if, …d_k=d_j†a_j;k

is a real matrix and all the conditions in (a_d) above are satis®ed with 6ˆ in (i) and (ii) strengthened to <.

Proof. (ad) We ®rst note that r is orthogonal if, and only if, the cor- responding monic polynomial system R ˆ …Rn† ˆ …1=d… n†rn† is orthogonal.

The matrix associated with R is eA ˆ …eaj;k†, given by eaj;k ˆ …dk=dj†aj;k for j; k 2 IN₀. Thus, by Theorem 2.2 above, R is orthogonal if, and only if, eA satis®es the Condition 2.2(a) where a is replaced by ea everywhere. We al- ready bracketed the expressions in 2.2(a) in such a way that (a) transforms to (a_d) immediately.

(b_d) By 2.1(a), eA has to be real in the ®rst place. The rest follows from (a_d) in the same manner as (b) does from (a) in Theorem 2.2. &

Remark 2.5.

(a) The above theorem is a two-edged sword, although not equally sharp! The sharper one says that if a d is given, then we can construct all those A's for which r is orthogonal (respectively, positive orthogonal in case p is so). All we have to do is to choose any sequence …sn† of numbers and then take another sequence …tn† satisfying

1

d2t₀6ˆ c₁‡ 1

d1s₀ …b₀ b₁† 1 d1s₀ d₁

d2s₁

 

 

;

and, for n  1 , dn

dn‡2tn dn 1

dn‡1tn 16ˆ cn‡1

‡ d_n

dn‡1s_n …b_n b_n‡1† d_n

dn‡1s_n d_n‡1 dn‡2s_n‡1

 

 

:

Then we can set a_n‡1;nˆ s_n and a_n‡2;nˆ t_n for all n 2 IN. Next we can de®ne, for n  2 and 0  j  n 1 , the numbers a_n‡1;j recursively using (iii) and (iv) of (a_d). Finally, take a_j;jˆ 1 for all j and a_j;kˆ 0 for all k > j and obtain A_s;tˆ …a_j;k†. Then for q ˆ A_s;tp, the system S_p;dq is orthogonal.

In addition, every monic matrix A having this preservation property has this form for a unique …s; t†.

In case p is positive orthogonal, we have to restrict s ˆ …s_n† so that …d_n=…d_n‡1††s_nis real for all n and t ˆ …t_n† so that …d_n=…d_n‡2††t_nis real for all n, and strengthen the inequalities to < in the above construction.

(b) We now come to the question of determining the sequence d when a monic matrix A is given. For j; n 2 IN, we put

u_nˆ a_{n‡1;n 1} a_n‡1;na_{n;n 1}ˆ a_{n;n 1} an‡1;n 1

1 an‡1;n

; v_n;jˆ a_n‡1;j a_n‡1;na_n;j;

and for n  2, 0  j  n 2, we put

w_n;jˆ v_n;j u_na_{n 1;j}ˆ an 1;j 1 0

an;j an;n 1 1

an‡1;j an‡1;n 1 an‡1;n

  : For each n  2, one of the following two conditions is satis®ed:

(n1) for some jn; 0  jn  n 2; wn;jn 6ˆ 0; wn;jˆ 0; 0  j < jn. (n2) wn;jˆ 0; 0  jn  n 2. This holds if, and only if, for

0  j  n; an‡1;jˆ an‡1;nan;j‡ unan 1;j if, and only if, qn‡1ˆ pn‡1‡ an‡1;nqn‡ unqn 1.

Let

T₁ˆ n : n  2; …n1† holdsf g;

and

T2ˆ n : n  2; …n2† holdsf g [ f0; 1g:

The problem reduces to ®nding a suitable function  on the set T₂to the set of nonzero numbers such that, setting d_n‡1ˆ _n for n 2 T₂ and then de-

®ning the remaining d_n's recursively using (a_d)(iii) if j_{n 1}ˆ 0 and (a_d)(iv) if j_{n 1}> 0, the condition (a_d) [respectively, (b_d)] is satis®ed.

Theorem 2.6. Let p; A; d; q and r be as in Theorem 2.4 above.

(a) We may combine the conditions (a) of Theorem 2.2 and (ad) of Theorem 2.4 to obtain necessary and sucient conditions on A and d for both q and r to be orthogonal.

(b) For positive orthogonality, we have to combine (b) of Theorem 2.2 with (bd) of Theorem 2.4 which, in turn, put together are equivalent to:

(i) Both A and d are real,

(ii) a_2;0 a_2;1a_1;0< c₁‡ …b₀ b₁†a_1;0 …a_1;0†²; and1

d2 …a_2;0 a_2;1a_1;0† < c₁‡ 1

d1…b₀ b₁†a_1;0 1 d1a_1;0

 ₂

;

(iii) a_{n‡1;n 1} a_n‡1;na_{n;n 1}

< cn‡ …bn 1 bn†an;n 1 …an;n 1†²‡ an;n 2

anddn 1

d_n‡1…an‡1;n 1 an‡1;nan;n 1†

< cn‡d_{n 1}

dn …bn 1 bn†an;n 1 d_{n 1} dn an;n 1

 ₂

‡d_{n 2}

dn a_{n;n 2}; n  2 ;

(iv) …a_n‡1;j a_n‡1;na_n;j† …a_{n‡1;n 1} a_n‡1;na_{n;n 1}†a_{n 1;j}

ˆ a_{n;j 1}‡ …b_j b_n a_{n;n 1}†a_n;j‡ c_j‡1a_n;j‡1

c_n‡ …b_{n 1} b_n†a_{n;n 1} …a_{n;n 1}†²‡ a_{n;n 2}

h i

a_{n 1;j}; and

1 dn‡1dj 1

d_nd_j

 

an;j 1

‡ 1 dn‡1

dn

 

…bj bn† 1 dn‡1dn 1

d²_n

 

an;n 1

 

an;j

‡ 1 d_n‡1d_j‡1 dndj

 

c_j‡1a_n;j‡1

"

1 dn‡1

d_{n 1}

 

c_n‡ 1 dn‡1

d_n

 

…b_{n 1} b_n†a_{n;n 1}

‡ 1 dn‡1dn 2

d_nd_{n 1}

 

an;n 2 1 dn‡1dn 1

d²_n

 

…an;n 1†²

#

an 1;jˆ 0;

n  2 ; 0  j  n 2:

(c) For a set A of monic matrices, we can have a set of conditions on d in order that the members of fSp;dAp : A 2 Ag are all orthogonal or positive orthogonal systems.

In this general form these conditions look formidable. We shall discuss a few special cases here and in the next two sections.

Corollary 2.7. Let d ˆ …… 1†ⁿ† , p a monic orthogonal system with recurrence sequences b and c, and A a monic in®nite matrix. Let q ˆ A p and r ˆ S_p;dq.

(a) (i) In case …bj bn†an;jˆ 0 for 0  j < n, q is orthogonal if, and only if, r is orthogonal.

(ii) Suppose b is a constant sequence. Then q 2 Poif, and only if, Sp;dq 2 Po. Thus Sp;dPoˆ Po. In particular, it is so if p is a symmetric orthogonal system. In this case, we also have that q is symmetric if, and only if, r is.

(iii) Suppose b is eventually constant, i.e. , for a least number j12 IN0; bnˆ bj for all n  j1. Then … Sp;dPo† \ Po is in®nite.

(b) Suppose p is positive orthogonal. Then (i), (ii) and (iii) above hold with orthogonal replaced by positive orthogonal and P_o by P_‡o.

Proof. (a) (i) In this case Condition (a) of Theorem 2.2 and Condition (a_d) of Theorem 2.4 become equivalent.

(ii) Because b is a constant sequence, every monic matrix A satis®es the requirement in (i). For symmetric orthogonal polynomial systems, we have bnˆ 0 for all n.

(iii) Let Aj1ˆ f A 2 Am : an;jˆ 0 for 0  j  n ; j < j1 g. Then every A in Aj1 satis®es the requirement in (i) and, therefore, q ˆ A p is orthogonal if, and only if, Sp;d q is orthogonal if, and only if, A satis®es (a) of Theorem 2.2. There are in®nitely many matrices in Aj1 satisfying

this condition, as can be easily seen, because a_j0‡1;j0 can be chosen arbi- trarily.

(b) In the above proof of Part (a), we only have to use (b) of Theorem 2.2 and (b_d) of Theorem 2.4 instead of (a) and (a_d). &

Remark 2.8.

(a) Examples of p satisfying condition (iii) of Corollary 2.7 above are provided by Modi®ed Lommel polynomials, Tricomi±Carlitz poly- nomials and polynomials related to Bernoulli numbers (cf. [5], VI.6, VI.7 and VI.8).

(b) For any d1 with 0 6ˆ d²₁6ˆ 1, de®ne the sequence d ˆ …dn† ˆ …dⁿ₁†.

Let p be a symmetric orthogonal system and A; q; r; as in the above corollary. Then both q and r are symmetric orthogonal if, and only if, a_j;kˆ 0 for 0  k  j 3 ; j  3 ; a_k‡2;kˆ a_2;06ˆ c₁ and a_k‡1;kˆ a_1;0; k 2 IN₀. For the positive orthogonal case, the extra conditions d₁; a_1;0; a_2;02 IR and a_2;0< minf c₁; d²₁c₁g are needed.

3. KERNEL POLYNOMIALS AND DILATIONS 3.1. Discussion and De®nition

Let p ˆ …pn† be a monic orthogonal polynomial system with the asso- ciated recurrence sequences b ˆ …bn† and c ˆ …cn†. Let Zpbe the set of zeros of pn; n 2 IN0, and Ip the open interval … inf Zp; sup Zp†. For a number y not in Zp, let p^yˆ …p^y_n† be the corresponding monic kernel polynomial system. We may refer to [5], Section 7, for this and related results to be used in this section.

(i)

p^y_n ˆ cⁿ₁ p_n…y†

Xⁿ

k ˆ 0

pk…y†

c^k₁ p_k; where c^k_j ˆY^k

lˆj

c_l :

Thus, the associated monic matrix A^yˆ …a^y_n;k† satis®es a^y_n;kˆ cⁿ_k‡1pk…y†

pn…y† for 0  k < n : (ii) Also, p^yˆ …p^y_n† is orthogonal.

(iii) Further, if p is positive orthogonal, then p^yis positive orthogonal if, and only if, y 2 IR n I_p. In this case, we denote by ep ˆ …ep_n† the corresponding orthonormal system. Then ep_nˆ 1= 

cⁿ₁

p p_n for n 2 IN, and

thus, for n 2 IN ,

Xⁿ

j ˆ 0

epj…y† ep_jˆ …cⁿ₁† ¹p_n…y† p^y_n:

This motivates the de®nition and the name of the main object of study in this section.

(iv) A polynomial system q ˆ …q_n† will be called analytic withrespect to a polynomial system p ˆ …p_n†, in short, p-analytic, if there is a sequence h ˆ …h_n† of nonzero numbers with h₀ˆ 1 such that q_nˆP_n

k ˆ 0 h_kp_k; n 2 IN₀. The sequence …h_n† will be called the coecient sequence for q. If p is monic, then the monic system Q corresponding to q is given by the matrix A ˆ …a_n;k† (with respect to p) that satis®es a_n;kˆ h_k=h_n; 0  k  n. We shall say that q is monic analytic withrespect to p, in short, p-monic analytic.

Any monic matrix A ˆ …a_n;k† having a_n;kof the form h_k=h_n; 0  k  n for a sequence …hn† 2 D gives rise to a q that is p-monic analytic. We note that A and h determine each other uniquely.

(v) If q is p-analytic or p-monic analytic, then for d; d⁰2 D , we have Sp;d6ˆ Sq;d⁰.

Theorem 3.2. Let p be a monic orthogonal system with recurrence sequences b and c, and let q be monic analytic with respect to p with monic analytic sequence h.

(a) The following conditions are equivalent:

(i) q is orthogonal,

(ii) …b₀‡ c₁h₁†h_n‡1ˆ h_n‡ b_n‡1h_n‡1‡ c_n‡2h_n‡2; n 2 IN₀, (iii) h_nˆ 1=cⁿ₁ p_n… b₀‡ c₁h₁† ; n 2 IN ;

(iv) q ˆ p^y with y ˆ b₀‡ c₁h₁:

(b) Suppose p is positive orthogonal. Then q is positive ortho±

gonal if, and only if, q ˆ p^y for some y 2 IR n I_p. Further, y and …h_n† are related via

y ˆ b₀‡ c₁h₁ and h_nˆ 1

cⁿ₁ p_n…y† ; n 2 lN:

Proof. (a) We ®rst note that the monic matrix A for q satis®es an;kak;lˆ an;l, 0  l  k  n. In this case (a)(iii) and (a)(iv) of Theorem 2.2 are equivalent.

Let q be orthogonal. Then (ii) follows from (a)(iii) of Theorem 2.2.

Moreover, (ii) gives (a)(i) and (a)(ii) of Theorem 2.2 simply because cn6ˆ 0 6ˆ hn for all n. This shows the equivalence of (i) and (ii).

Let, for n 2 IN; h⁰_n ˆ cⁿ₁h_n and h⁰₀ˆ 1. We put y ˆ b₀‡ c₁h₁. Then for n 2 IN₀; y h⁰_nˆ h⁰_n‡1‡ b_nh_n⁰ ‡ c_nh⁰_{n 1}. This gives that h⁰_nˆ p_n…y† for n 2 IN₀, which readily implies (iii).

(iii) ) (iv) follows from 3.1(i), and (iv) ) (i) is immediate from 3.1(ii).

(b) We only have to combine part (a) with 3.1(iii). &

Theorem 3.3. Let p be an orthogonal polynomial system with recurrence relation

xp_n…x† ˆ _np_n‡1…x† ‡ _np_n…x† ‡ _np_{n 1}…x† ; x 2 lR; n 2 IN₀; p0…x† ˆ 1; p 1…x† ˆ 0 :

Let wnˆQ_{n 1}

j ˆ 0j=j‡1 for n 2 IN0 and w0ˆ 1. Let q be p-analytic with coecient sequence h ˆ …h_n†.

(a) The system q is orthogonal if, and only if, for some number y 62 Z_p, we have h_nˆ w_np_n…y† , n 2 IN₀.

(b) Suppose p is a positive orthogonal polynomial system.

(i) The system q is orthogonal if, and only if, for some y 2 IR n Z_p,

hn ˆp_n…y†

kp_nk²_p ; n 2 IN0:

In particular, if p is an orthonormal polynomial system, then the condition reduces to h_nˆ p_n…y† , n 2 IN₀.

(ii) The system q is positive orthogonal if, and only if, y in (i) above satis®es y  inf Z_p or y  sup Z_p. Moreover, if p is monic then, in the former case, … 1†ⁿh_n> 0 for all n, while in the latter case, hn> 0 for all n .

Proof. (a) Let b and c be the recurrence sequences for the monic polynomial system P corresponding to p. Then q is P-analytic as well with the coecient sequence …h_n…Q_{n 1}

j ˆ 0_j† ¹†. We note that _n ˆ _n=_n‡1 and

_nˆ c_n…_n‡1=_{n 1}†_n, where _n is the leading coecient of the polynomial p_n. From this we can derive w_nˆ 1=cⁿ₁²_n. All that we have to do now is to apply the above theorem to P.

(b) In this case, w_nabove has the value kp_nk_p²for each n. Finally, p_n…x†

retains the same sign in … 1; inf Z_pŠ and the same sign in ‰ sup Z_p; 1†. &

Theorem 3.4. Let p ˆ …p_n† be an orthogonal polynomial system with

; ; ; w as in Theorem 3.3 above and d 2 D.

(a) S_p;dP_ap P_ap. (b) S_p;d…P_ap\ P_o†

\ P_o6ˆ ; if, and only if, there exist distinct numbers y1 and y2, both not in Zp, satisfying dnpn…y1† ˆ pn…y2† for all n.

(c) ] ‰… Sp;d Pap\ Po

† \ PoŠ  2 if, and only if, there exist distinct pairs …y1; y2† and …y3; y4† with yi62 Zp; y16ˆ y2, y16ˆ y3, y26ˆ y4, y36ˆ y4, such that

dnˆpn…y2†

p_n…y₁†ˆpn…y4†

p_n…y₃† for all n:

(d) If p is positive orthogonal, then we can replace P_oby P_‡o, provided we make the y_j's satisfy y_j2 IR n I_p instead of just y_j62 Z_p.

Proof. For an h 2 D and q_n ˆP_n

j ˆ 0 h_jp_j for n 2 IN₀, we have S_p;dq_nˆP_n

j ˆ 0 d_jh_jp_j; n 2 IN₀. This gives (a).

Further, by Theorem 3.3, both q and S_p;dq can be in P_oif, and only if, there exist distinct numbers y₁ and y₂ not in Z_p satisfying

hnˆ wnpn…y1†; hndnˆ wnpn…y2† for all n; i:e:;

h_nˆ w_np_n…y₁†; d_np_n…y₁† ˆ p_n…y₂† for all n : Thus we have (b).

(c) and (d) follow immediately from (b) in view of 3.1. &

Corollary 3.5. Let p ˆ …pn† be a symmetric orthogonal polynomial system. Then for any two distinct nonzero numbers y₁ and y₂, both not in Z_p, and d_n ˆ p_n…y₂†=p_n…y₁†; n 2 IN₀, we have ]‰S_p;d…P_ap\ P_o† \ P_oŠ  2 .

Proof. We may take y₃ˆ y₁and y₄ˆ y₂. &

Remark 3.6.

(a) If we choose y₂ˆ y₁ in the above corollary, then we always get the sequence d ˆ …… 1†ⁿ† and, thus, Sp;d Pap\ Po

ˆ Pap\ Po. This follows from Corollary 2.7, as well!

(b) We may use Mehler's formula ([13], Ex. 23, p. 377 or [2], 2.44 on p. 16) to deduce that in the case of Hermite polynomials there are no other solutions for the condition in Theorem 2.4(iii) except those given by the proof in Corollary 3.5 or (a) above. The same is true for Chebyshev polynomials of the ®rst, as well as the second, kind, as can be easily checked

by just considering the case n ˆ 1 and 2. Thus, for these polynomials the following statements hold.

(i) For d_nˆ … 1†ⁿ, we have S_p;d P_ap\ P_o

ˆ P_ap\ P_o:

(ii) For d_nˆ p_n…y₂†=p_n…y₁† with y₁; y₂ two distinct numbers, both not in Zp, and y2 6ˆ y1, we have

S_p;d P_ap\ P_o

\ P_oˆ f p^y2; p^y2† g:

(iii) For all other d's, we have S_p;d P_ap\ P_o

\ P_oˆ ; .

(c) We can use [13], Theorem 5.1, or [2], 2.43, to show that there are no solutions to condition (iii) of Theorem 3.4 when we take p to be any Laguerre polynomial system and thus we have that for distinct y1; y2 not in Zp and dnˆ pn…y2†=pn…y1† for all n, Sp;d Pap\ Po

\ Poˆ fp^y2g.

(d) As already noted in Theorem 3.3(b)(i), the coecient sequence takes a simple form for an orthonormal polynomial system p. In fact, a direct proof of this part of Theorem 3.3 can be given using the orthogonality measure .

Theorem 3.7.

(a) Let p be an orthogonal polynomial system with recurrence sequences a; b; c, i.e.,

xp_n…x† ˆ a_np_n‡1…x† ‡ b_np_n…x† ‡ c_np_{n 1}…x†; x 2 lR; n 2 IN₀: Let g be a polynomial system such that p is g-analytic with constant one as the coecient sequence. Then g is orthogonal if, and only if,

a_n‡ b_n‡ c_nˆ a₁‡ b₁‡ c₁6ˆ a₀‡ b₀; n  2:

(b) Let …g; p; q† be a triple of polynomial systems such that p is g- analytic and q is p-analytic with constant one as the coecient sequence in each case. Then g; p; q are all orthogonal if, and only if, p is orthogonal and the recurrence sequences a; b; c of p satisfy

(i) an‡ bn‡ cnˆ a1‡ b1‡ c16ˆ a0‡ b0, n  2; and (ii) a_n‡ b_n‡1‡ c_n‡2 ˆ b₀‡ c₁, n 2 IN₀. (c) The conditions in (b) are equivalent to

(i)⁰ a₁‡ b₁‡ c₁ 6ˆ a₀‡ b₀,

(ii)⁰ a_{n 1}‡ …b_n b₀† ‡ …c_n‡1 c₁† ˆ 0, n  1, and (iii)⁰ …a_n‡1 a_{n 1} a₁† ‡ ‰…b_n‡1 b_n† …b₁ b₀†Š ˆ 0, n  1:

Proof. (a) Suppose that g is orthogonal. Let ; ; be its recurrence sequences. Then _n; _n‡16ˆ 0, n 2 IN₀. Also,

Xⁿ

j ˆ 0

…jgj‡1…x† ‡ jgj…x† ‡ jgj 1…x††

ˆXⁿ

j ˆ 0

xgj…x† ˆ x pn…x†

ˆ a_nX^n‡1

j ˆ 0

g_j…x† ‡ b_nXⁿ

j ˆ 0

g_j…x†

‡ c_nX^{n 1}

j ˆ 0

g_j…x†; x 2 lR; n 2 lN₀:

This implies Xⁿ

j ˆ 0

…jgj‡1‡ jgj‡ jgj 1†

ˆ an

X^n‡1

j ˆ 0

gj‡ bn

Xⁿ

j ˆ 0

gj‡ cn

X^n‡1

j ˆ 0

gj; n 2 lN0:

Equating coecients of gj on both sides gives ₀ˆ a₀; ₀ˆ a₀‡ b₀;

₁ˆ a₁; ₀‡ ₁ˆ a₁‡ b₁; ₀‡ ₁ˆ a₁‡ b₁‡ c₁; nˆ an; n 1‡ nˆ an‡ bn;

_{j 1}‡ _j‡ _j‡1ˆ a_n‡ b_n‡ c_n and

0‡ 1ˆ an‡ bn‡ cn

for n  2 and 1  j  n 1. This implies an‡ bn‡ cnˆ a1‡ b1‡ c1 for n  2 and, because 16ˆ 0, it also implies a1‡ b1‡ c16ˆ a0‡ b0.

We also note that nˆ an; n 2 lN0;

₀ˆ a₀‡ b₀; _nˆ …a_n a_{n 1}† ‡ b_n; n 2 lN;

₁ˆ …a₁‡ b₁‡ c₁† …a₀‡ b₀†; _nˆ c_{n 1}; n  2:

Now, on the other hand, if a; b; c satisfy

a_n‡ b_n‡ c_n ˆ a₁‡ b₁‡ c₁6ˆ a₀‡ b₀; n  2;

then ; ; , de®ned by the expressions given above, work as recurrence sequences for g and this shows orthogonality of g.

(b) As a byproduct of the computations in the proof of (a), we note that if ; ; are sequences satisfying

_n; _n‡16ˆ 0; _n‡ _n‡1‡ _n‡2ˆ ₀‡ ₁; n 2 lN₀; then the sequences a; b; c, de®ned by

a_n ˆ _n; n 2 lN₀;

b₀ ˆ _{0 0}; b_nˆ _{n …n n 1}†; n 2 lN;

cn ˆ n‡1; n 2 IN;

satisfy an‡ bn‡ cnˆ a1‡ b1‡ c16ˆ a0‡ b0, n  2. Thus, (b) follows from (a) and its proof both applied to the pairs …g; p† and …p; q†.

(c) is trivial. &

De®nition 3.8. Let p ˆ …p_n† be an orthonormal polynomial system with the orthogonality measure . Let y 2 IR n I_p. Let q ˆ …q_n† ˆ …P_n

k ˆ 0p_k…y†

p_k† and Q be the corresponding orthonormal system, i.e., Q ˆ …Q_n† ˆ …q_n=kq_nk_q†, and let r ˆ …r_n† ˆ …P_n

k ˆ 0 Q_k…y†Q_k†. Then …p; q; r†

will be called an analytic triple through y.

Theorem 3.9. Let …p ; q ; r† be an analytic triple through y.

(a) The following are equivalent:

(i) There exists a d 2 D such that S_p;d simultaneously preserves the positive orthogonality of f q; r g.

(ii) There exist y⁰; y⁰⁰2 IR n Ip; y 6ˆ y⁰, satisfying p1…y⁰†

pn…y⁰† pn‡1…y⁰†

"

Xⁿ

j ˆ 0

pj…y⁰⁰† pj…y⁰†

#

ˆ p₁…y†

p_n…y† p_n‡1…y†

"

Xⁿ

j ˆ 0

…p_j…y††²

#

for all n 2 lN₀:

(iii) There exist y⁰; y⁰⁰2 IR n Ip; y 6ˆ y⁰, such that either …† y⁰ˆ y}and p₁…y⁰†p⁰_n‡1…y⁰†

p_n‡1…y⁰†

ˆ p₁…y†p⁰_n‡1…y†

p_n‡1…y†; n 2 IN;

or …† y⁰6ˆ y⁰⁰ and p₁…y⁰† y⁰⁰ y⁰

p_n‡1…y⁰⁰† p_n‡1…y⁰† 1



ˆp₁…y† p⁰_n‡1…y†

p_n‡1…y† ; n 2 IN:

The point y⁰(respectively y⁰⁰) chosen in (ii) can be taken to be the same as in (iii), and vice versa, and in each case d ˆ p… n…y⁰†=pn…y††.

(b) For given y⁰; y⁰⁰2 IR n Ip; y 6ˆ y⁰, Part (iii)() above is equivalent to each of the following, where _nis the leading coecient of p_n, as before.

(i) y⁰6ˆ y⁰⁰ˆ y⁰‡ 1

2

p₁…y† p₂…y⁰†

p1…y⁰† p2…y†p⁰₂…y† p⁰₂…y⁰†

 

2 IR n I_p and

p1…y⁰† y⁰⁰ y⁰

pn‡1…y⁰⁰† p_n‡1…y⁰† 1

 

ˆp₁…y†p⁰_n‡1…y⁰†

p_n‡1…y† ; n  2:

(ii) y⁰6ˆ y⁰‡ 1

₂

p1…y† p2…y⁰†

p₁…y⁰† p₂…y†p⁰₂…y† p⁰₂…y⁰†

 

2 IR n Ip

and p₁…y⁰† pn‡1…y⁰†

X^n‡1

j ˆ 1

p^j†_n‡1 j!

1

2

p₁…y† p₂…y⁰†

p1…y⁰† p2…y†p⁰₂…y† p⁰₂…y⁰†

 

 _{j 1}

" #

ˆp1…y†p⁰_n‡1…y†

pn‡1…y† ; n  2: