Ito type

Linear Algebra and its Applications 519 (2017) 111–135

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Leonard triples extended from a given totally almost bipartite Leonard pair of Bannai/Ito type Yan Wang, Bo Hou, Suogang Gao ∗ College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, PR China

a r t i c l e

i n f o

Article history: Received 7 November 2016 Accepted 29 December 2016 Available online 4 January 2017 Submitted by R. Brualdi MSC: 15A04 33D45 Keywords: Leonard pair Leonard system Leonard triple Bannai/Ito type

a b s t r a c t Let K denote a field of characteristic zero and let d denote an integer at least 3. Let ⎛

⎞ 0 2d + 1 0 0 2d ⎜1 ⎟ ⎜ ⎟ 2 0 2d − 1 ⎜ ⎟ ⎜ ⎟ 3 . . ⎟ d⎜ A = (−1) ⎜ ⎟ . . . ⎜ ⎟ ⎜ ⎟ . . d + 3 ⎜ ⎟ ⎝ d−1 0 d+2⎠ 0 d d+1 and A∗ = diag((−1)d (2d + 1), . . . , −7, 5, −3, 1) be two matrices in Matd+1 (K). Then A, A∗ is a totally almost bipartite Leonard pair on Kd+1 of Bannai/Ito type. In this paper, we determine all the matrices Aε ∈ Matd+1 (K) such that A, A∗ , Aε form a Leonard triple on Kd+1 . © 2017 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected], [email protected] (S. Gao). http://dx.doi.org/10.1016/j.laa.2016.12.039 0024-3795/© 2017 Elsevier Inc. All rights reserved.

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1. Introduction Throughout this paper we adopt the following notation and terminology. Let K denote a field of characteristic zero. Let d denote a nonnegative integer. Let Matd+1 (K) denote the K-algebra consisting of all d + 1 by d + 1 matrices that have entries in K. We index the rows and columns by 0, 1, . . . , d. Let Kd+1 denote the K-vector space consisting of all d + 1 by 1 matrices that have entries in K. Its rows are indexed by 0, 1, . . . , d. We view Kd+1 as a left module for Matd+1 (K). A square matrix X is said to be tridiagonal whenever every nonzero entry appears on, immediately above, or immediately below the main diagonal. Assume X is tridiagonal. Then X is said to be irreducible whenever all entries immediately above and below the main diagonal are nonzero. The notion of a Leonard pair was introduced by Terwilliger in [10]. Definition 1.1. ([10, Definition 1.1]) Let V denote a vector space over K with finite positive dimension. By a Leonard pair on V , we mean an ordered pair of linear transformations A : V → V and A∗ : V → V that satisfy both (i) and (ii) below. (i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A∗ is irreducible tridiagonal. (ii) There exists a basis for V with respect to which the matrix representing A∗ is diagonal and the matrix representing A is irreducible tridiagonal. Terwilliger classified the Leonard pairs up to isomorphism in [13]. By that classification, the isomorphism classes of Leonard pairs fall naturally into thirteen families: q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, affine q-Krawtchouk, quantum q-Krawtchouk, Racah, Hahn, dual Hahn, Krawtchouk, Bannai/Ito and orphan. Lemma 1.2. ([10, Theorem 7.3]) Let A, A∗ denote a Leonard pair on V . Let W denote a nonzero subspace of V with AW ⊆ W and A∗ W ⊆ W . Then W = V . Let B ∈ Matd+1 (K) be tridiagonal. We say that B is bipartite whenever Bii = 0 for 0 ≤ i ≤ d. We say that B is almost bipartite whenever exactly one of B0,0 , Bd,d is nonzero and Bii = 0 for 1 ≤ i ≤ d − 1. Definition 1.3. ([2, Definition 1.2, Definition 1.3]) A Leonard pair A, A∗ is said to be bipartite (resp. almost bipartite) whenever the matrix representing A from Definition 1.1(ii) is bipartite (resp. almost bipartite). The Leonard pair A, A∗ is said to be dual bipartite (resp. dual almost bipartite) whenever the Leonard pair A∗ , A is bipartite (resp. almost bipartite). The Leonard pair A, A∗ is said to be totally bipartite (resp. totally almost bipartite) whenever it is bipartite (resp. almost bipartite) and dual bipartite (resp. dual almost bipartite).

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To state our main result of this paper, we recall the notion of a Leonard triple, which was introduced by Curtin in [3]. Definition 1.4. ([3, Definition 1.2]) Let V denote a vector space over K with finite positive dimension. By a Leonard triple on V , we mean an ordered triple of linear transformations A : V → V , A∗ : V → V and Aε : V → V that satisfy (i)–(iii) below. (i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrices representing A∗ and Aε are irreducible tridiagonal. (ii) There exists a basis for V with respect to which the matrix representing A∗ is diagonal and the matrices representing A and Aε are irreducible tridiagonal. (iii) There exists a basis for V with respect to which the matrix representing Aε is diagonal and the matrices representing A and A∗ are irreducible tridiagonal. Note that for any Leonard triple, any two of the three form a Leonard pair. We say these Leonard pairs are associated with the Leonard triple. From [5], there are only three families of totally bipartite Leonard pairs: Krawtchouk, Bannai/Ito and q-Racah types. Now we recall the results of Leonard triples extended from given totally bipartite Leonard pairs. Let ⎛

0 d ⎜ ⎜1 0 ⎜ ⎜ 2 ⎜ ⎜ B=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0

0

d−1 0 3

d−2 . . . . .

. . d−1

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 2 ⎟ ⎟ 0 1⎠ d 0

and

B ∗ = diag(d, d − 2, d − 4, . . . , 4 − d, 2 − d, −d) be two matrices in Matd+1 (K). Terwilliger showed that B, B ∗ is a totally bipartite Leonard pair on Kd+1 of Krawtchouk type in [11]. Assume d ≥ 3, Balmaceda and Maralit determined all the matrices B ε such that B, B ∗ , B ε form a Leonard triple on Kd+1 in [1].

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When d is even, let ⎛

0 d ⎜ ⎜1 0 1−d ⎜ ⎜ −2 0 ⎜ ⎜ 3 C=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0

0

d−2 . .

. . .

. . d−1

2 0 −d

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −1 ⎠ 0

and C ∗ = diag(d, 2 − d, d − 4, . . . , 4 − d, d − 2, −d) be two matrices in Matd+1 (K). Brown showed that C, C ∗ is a totally bipartite Leonard pair on Kd+1 of Bannai/Ito type in [2]. Assume d ≥ 3, Hou, Zhang and Gao determined all the matrices C ε such that C, C ∗ , C ε form a Leonard triple on Kd+1 in [6]. For any nonzero scalar q ∈ K which satisfies q i = 1 (1 ≤ i ≤ d) and q i = −1 (1 ≤ i ≤ d − 1), let ⎛

0 ⎜c ⎜ 1 ⎜ ⎜ D=⎜ ⎜ ⎜ ⎝ 0

b0 0 c2

0

b1 0 .

. . .

. . cd

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ bd−1 ⎠ 0

and D∗ = diag(θ0∗ , θ1∗ , . . . , θd∗ ) be two matrices in Matd+1 (K), where θi∗ = q i − q d−i

(0 ≤ i ≤ d),

bi =

q (q − 1) d 2i q +q

ci =

q d (1 − q 2i ) q d + q 2i

2d

2i−2d

(1 ≤ i ≤ d − 1),

(1 ≤ i ≤ d − 1),

b0 = cd = 1 − q d . Hou, Wang and Gao showed that D, D∗ is a totally bipartite Leonard pair on Kd+1 of q-Racah type in [5]. In the same paper, assume d ≥ 3, they determined all the matrices Dε such that D, D∗ , Dε form a Leonard triple on Kd+1 .

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From [17], there are only two families of totally almost bipartite Leonard pairs: Bannai/Ito and q-Racah types. Let ⎛

0 2d + 1 ⎜ 0 2d ⎜1 ⎜ ⎜ 2 0 ⎜ ⎜ 3 A = (−1)d ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0

0

2d − 1 . .

. . .

. . d−1

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ d+3 ⎟ ⎟ 0 d+2⎠ d d+1

(1)

and A∗ = diag((−1)d (2d + 1), . . . , −7, 5, −3, 1)

(2)

be two matrices in Matd+1 (K). Assume d ≥ 3, Brown showed that A, A∗ is a totally almost bipartite Leonard pair on Kd+1 of Bannai/Ito type in [2]. In this paper, we determine all the matrices Aε such that A, A∗ , Aε form a Leonard triple on Kd+1 . The study of Leonard triples extended from a given totally almost bipartite Leonard pair of q-Racah type will be given in another paper. The present paper is organized as follows. In Sections 2–4, we recall some background concerning Leonard pairs and Leonard systems. In Section 5, we recall the normalized Leonard pairs of Bannai/Ito type. In Section 6, we define a normalized Leonard pair B, B∗ from a given totally almost bipartite Leonard pair A, A∗ , and give both the parameter array and Askey–Wilson relations of B, B∗ . In Section 7, we recall Leonard triples and Leonard triple systems. In Section 8, we construct a normalized Leonard triple B, B∗ , Bε from B, B∗ . In Section 9, we determine all Aε ∈ Matd+1 (K) such that A, A∗ , Aε form a Leonard triple. 2. Leonard pairs and Leonard systems When working with a Leonard pair, it is often convenient to consider a closely related and somewhat more abstract object called a Leonard system. For the rest of the paper, let V denote a vector space over K with dimension d + 1. Let End(V ) denote the K-algebra consisting of all linear transformations from V to V . Let {vi }di=0 denote a basis for V . For X ∈ End(V ) and Y ∈ Matd+1 (K), we say Y represents d X with respect to {vi }di=0 whenever Xvj = i=0 Yij vi for 0 ≤ j ≤ d. For A ∈ End(V ), by an eigenvalue of A we mean a root of the characteristic polynomial of A. We say that A is multiplicity-free whenever it has d + 1 distinct eigenvalues. Assume A is multiplicity-free. Let {θi }di=0 denote an ordering of the eigenvalues of A. For 0 ≤ i ≤ d put

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Ei =

A − θj I , θ i − θj 0≤j≤d j=i

where I denotes the identity of End(V ). We refer to Ei as the primitive idempotent of A associated with θi . We now define a Leonard system. Definition 2.1. ([10, Definition 1.4]) By a Leonard system on V , we mean a sequence Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) that satisfies (i)–(v) below. (i) Each of A, A∗ is a multiplicity-free element in End(V ). (ii) {Ei }di=0 is an ordering of the primitive idempotents of A. (iii) {Ei∗ }di=0 is

an ordering of the primitive idempotents of A∗ . 0 if |i − j| > 1 (iv) Ei A∗ Ej = (0 ≤ i, j ≤ d). = 0 if |i − j| = 1

0 if |i − j| > 1 (v) Ei∗ AEj∗ = (0 ≤ i, j ≤ d). = 0 if |i − j| = 1 We refer to d as the diameter of Φ and say Φ is over K. Definition 2.2. ([10, Definition 1.8]) Let Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) denote a Leonard system on V . For 0 ≤ i ≤ d, let θi (resp. θi∗ ) denote the eigenvalue of A (resp. A∗ ) associated with Ei (resp. Ei∗ ). We refer to {θi }di=0 (resp. {θi∗ }di=0 ) as the eigenvalue sequence (resp. dual eigenvalue sequence) of Φ. Lemma 2.3. ([12, Lemma 1.2]) Let A, A∗ ∈ End(V ). Then A, A∗ is a Leonard pair if and only if the following (i), (ii) hold. (i) Each of A, A∗ is multiplicity-free. (ii) There exists an ordering E0 , E1 , · · · , Ed of the primitive idempotents of A and there exists an ordering E0∗ , E1∗ , · · · , Ed∗ of the primitive idempotents of A∗ such that the sequence Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) is a Leonard system. In this case we say that A, A∗ and the sequence Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) are associated. Suppose A, A∗ is a Leonard pair on V , and suppose Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) is an associated Leonard system. Then the only other Leonard systems associ∗ ated with A, A∗ are (A; {Ei }di=0 ; A∗ ; {Ed−i }di=0 ), (A; {Ed−i }di=0 ; A∗ ; {Ei∗ }di=0 ) and (A; ∗ {Ed−i }di=0 ; A∗ ; {Ed−i }di=0 ), denoted by Φ↓ , Φ⇓ and Φ↓⇓ , respectively. We refer to Φ↓ ⇓ (resp. Φ ) as the first inversion (resp. second inversion) of Φ.

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3. The parameter array of a Leonard system In this section we recall the notion of the parameter array of a Leonard system and some related results. Definition 3.1. ([12, Definition 5.4]) By a parameter array over K of diameter d we mean a sequence of scalars ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ) taken from K that satisfy the following conditions. (PA1) θi = θj , θi∗ = θj∗ if i = j, (0 ≤ i, j ≤ d).  0, φi = 0 (1 ≤ i ≤ d). (PA2) ϕi = i−1 θ −θ (PA3) ϕi = φ1 h=0 hθ0 −θd−h + (θi∗ − θ0∗ )(θi−1 − θd ) (1 ≤ i ≤ d). d i−1 θh −θd−h (PA4) φi = ϕ1 h=0 θ0 −θd + (θi∗ − θ0∗ )(θd−i+1 − θ0 ) (1 ≤ i ≤ d). (PA5) The expressions θi−2 − θi+1 , θi−1 − θi

∗ ∗ θi−2 − θi+1 ∗ θi−1 − θi∗

(3)

are equal and independent of i for 2 ≤ i ≤ d − 1. Let Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) denote a Leonard system on V . For 0 ≤ i ≤ d define Ui = (E0∗ V + E1∗ V + · · · + Ei∗ V ) ∩ (Ei V + Ei+1 V + · · · + Ed V ).

(4)

By [10, Lemma 3.8] each of U0 , U1 , . . . , Ud has dimension one and V = U0 + U1 + · · · + Ud

(direct sum).

(5)

The elements A and A∗ act on {Ui }di=0 as follows. By [10, Lemma 3.9], both (A − θi I)Ui = Ui+1 ∗

(A −

θi∗ I)Ui

= Ui−1

(0 ≤ i ≤ d − 1), (1 ≤ i ≤ d),

(A − θd I)Ud = 0, ∗

(A −

θ0∗ I)U0

(6)

= 0.

Setting i = 0 in (4) we find U0 = E0∗ V . Combining this with (6) we find Ui = (A − θi−1 I) · · · (A − θ1 I)(A − θ0 I)E0∗ V

(0 ≤ i ≤ d).

(7)

Let v denote a nonzero vector in E0∗ V . By (7), for 0 ≤ i ≤ d the vector (A − θi−1 I) · · · (A − θ1 I)(A − θ0 I)v is a basis for Ui . By this and (5) the sequence vi = (A − θi−1 I) · · · (A − θ1 I)(A − θ0 I)v

(0 ≤ i ≤ d)

(8)

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is a basis for V . With respect to this basis the matrices representing A and A∗ are ⎛

θ0 ⎜ 1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0

0 θ1 1

θ2 .

. .

. 1 θd

⎞

⎛

⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

θ0∗

0

ϕ1 θ1∗

0

ϕ2 θ2∗

. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ . ⎟ . ϕd ⎠ θd∗

(9)

respectively, where ϕ1 , ϕ2 , . . . , ϕd are appropriate scalars in K. By a Φ-split basis for V we mean a sequence of the form (8). We call {ϕi }di=1 the first split sequence of Φ. We let {φi }di=1 denote the first split sequence of Φ⇓ and call this the second split sequence of Φ. For notational convenience define ϕ0 = 0, ϕd+1 = 0, φ0 = 0, φd+1 = 0. Definition 3.2. ([15, Definition 13.3]) Let Φ denote a Leonard system on V . Define a map  : End(V ) → Matd+1 (K) as follows. For all X ∈ End(V ) let X  denote the matrix in Matd+1 (K) which represents X with respect to a Φ-split basis for V . We observe  : End(V ) → Matd+1 (K) is a K-algebra isomorphism. We call  the natural map for Φ. Definition 3.3. ([12, Definition 5.5]) Let Φ denote a Leonard system on V . By the parameter array of Φ we mean the sequence ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ), where {θi }di=0 (resp. {θi∗ }di=0 ) is the eigenvalue sequence (resp. dual eigenvalue sequence) of Φ and {ϕi }di=1 (resp. {φi }di=1 ) is the first split sequence (resp. second split sequence) of Φ. Lemma 3.4. ([10, Theorem 1.9]) Let Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) denote a Leonard system on V . Then the parameter array ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ) of Φ is a parameter array over K of diameter d. Conversely, let ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ) denote a parameter array over K of diameter d. Then there exists a Leonard system Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) on V such that ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ) is the parameter array of Φ. In this case, we say Φ and ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ) correspond. Let Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) be a Leonard system on V and let u be a nonzero vector in E0 V . Then the sequence {Ei∗ u}di=0 is a basis of V . We call it a Φ-standard basis. Lemma 3.5. ([12, Lemma 3.4]) Let Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) denote a Leonard system and let {θi }di=0 (resp. {θi∗ }di=0 ) denote the eigenvalue sequence (resp. dual eigenvalue sequence) of Φ. Let v0 , v1 , . . . , vd denote a basis for V . Let B (resp. B ∗ ) denote the matrix that represents A (resp. A∗ ) with respect to this basis. Then v0 , v1 , . . . , vd is a Φ-standard basis for V if and only if both (i) and (ii) hold below.

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(i) B has constant row sum θ0 . (ii) B ∗ = diag(θ0∗ , θ1∗ , . . . , θd∗ ). Lemma 3.6. ([12, Theorem 5.6, Theorem 5.7]) Let Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) denote a Leonard system and let ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ) denote the parameter array of Φ. Then the matrices representing A and A∗ with respect to a Φ-standard basis are ⎛

a0 ⎜ ⎜ c1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0

b0 a1 c2

0

b1 a2 c3

b2 . .

. . .

. . cd−1

bd−2 ad−1 cd

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ bd−1 ⎠ ad

(10)

and diag(θ0∗ , θ1∗ , . . . , θd∗ ), respectively, where ϕi ϕi+1 + ∗ (0 ≤ i ≤ d), ∗ ∗ θi∗ − θi−1 θi − θi+1 i−1 ∗ (θ − θh∗ ) bi = ϕi+1 i h=0 i (0 ≤ i ≤ d − 1), ∗ ∗ h=0 (θi+1 − θh ) d−i−1 ∗ ∗ h=0 (θi − θd−h ) ci = φi d−i (1 ≤ i ≤ d). ∗ ∗ h=0 (θi−1 − θd−h )

ai = θ i +

(11)

For notational convenience define c0 = 0 and bd = 0. Moreover the matrix in (10) is irreducible tridiagonal and has constant row sum θ0 , i.e. ci + ai + bi = θ0 for 0 ≤ i ≤ d. Definition 3.7. With reference to Lemma 3.6, we call the matrix in (10) the intersection matrix of Φ. We call the scalars ai , bi , ci (0 ≤ i ≤ d) the intersection numbers of Φ. Let a∗i , b∗i , c∗i (0 ≤ i ≤ d) denote the intersection numbers of Φ∗ . We call a∗i , b∗i , c∗i (0 ≤ i ≤ d) the dual intersection numbers of Φ. 4. The Askey–Wilson relations for a Leonard pair In this section we recall the notion of the Askey–Wilson relations for a Leonard pair and some related results.

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Lemma 4.1. ([14, Theorem 1.5]) Let A, A∗ denote a Leonard pair on V . Then there exists a sequence of scalars β, γ, γ ∗ , , ∗ , ω, η, η ∗ taken from K such that both A2 A∗ − βAA∗ A + A∗ A2 − γ(AA∗ + A∗ A) − A∗ = γ ∗ A2 + ωA + ηI, ∗2

∗

∗

A A − βA AA + AA

∗2

∗

∗

∗

∗

∗2

− γ (A A + AA ) −  A = γA

∗

∗

+ ωA + η I.

(12) (13)

The sequence is uniquely determined by the pair A, A∗ provided the dimension of V is at least 4. We refer to (12) and (13) as the Askey–Wilson relations. We call the sequence β, γ, γ ∗ , , ∗ , ω, η, η ∗ satisfying (12) and (13) an Askey–Wilson parameter sequence for A, A∗ . Lemma 4.2. ([14, Theorem 4.5; Theorem 5.3 ]) Let Φ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) denote a Leonard system over K with eigenvalue sequence {θi }di=0 and dual eigenvalue sequence {θi∗ }di=0 . Recall that A, A∗ is a Leonard pair. Let β, γ, γ ∗ , , ∗ , ω, η, η ∗ denote a sequence of scalars taken from K. Then this sequence is an Askey–Wilson parameter sequence for A, A∗ if and only if the following (i)–(viii) hold. (i) The expressions θi−2 − θi+1 , θi−1 − θi

∗ ∗ θi−2 − θi+1 ∗ θi−1 − θi∗

are both equal to β + 1 for 2 ≤ i ≤ d − 1. γ = θi−1 − βθi + θi+1 (1 ≤ i ≤ d − 1). ∗ ∗ − βθi∗ + θi+1 (1 ≤ i ≤ d − 1). γ ∗ = θi−1 2 2  = θi−1 − βθi−1 θi + θi − γ(θi−1 + θi ) (1 ≤ i ≤ d). ∗2 ∗ ∗ − βθi−1 θi∗ + θi∗2 − γ ∗ (θi−1 + θi∗ ) (1 ≤ i ≤ d). ∗ = θi−1 ∗ ∗ ∗ ∗ ω = ai (θi − θi+1 ) + ai−1 (θi−1 − θi−2 ) − γ ∗ (θi + θi−1 ) = ai (θi∗ − θi+1 ) + ai−1 (θi−1 − ∗ ∗ ∗ θi−2 ) − γ(θi + θi−1 ) (1 ≤ i ≤ d). (vii) η = a∗i (θi − θi−1 )(θi − θi+1 ) − γ ∗ θi2 − ωθi (0 ≤ i ≤ d). ∗ ∗ (viii) η ∗ = ai (θi∗ − θi−1 )(θi∗ − θi+1 ) − γθi∗2 − ωθi∗ (0 ≤ i ≤ d). (ii) (iii) (iv) (v) (vi)

∗ ∗ and θd+1 ) denote scalars in K In the above lines (vi)–(viii), θ−1 and θd+1 (resp. θ−1 that satisfy (ii) (resp. (iii)) for i = 0 and i = d. The scalars ai , a∗i (0 ≤ i ≤ d) are from Lemma 3.6 and Definition 3.7, respectively.

5. Normalized Leonard pairs of Bannai/Ito type In [16], Vid˜ unas introduced the definitions of normalized Leonard pairs and normalized Leonard systems. In this section, we only recall the definition of normalized Leonard pair of Bannai/Ito type and display the parameter array of normalized Leonard systems of Bannai/Ito type.

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From [13, Example 5.14], a Leonard system is said to be of Bannai/Ito type whenever the common value of (3) is equal to −1. A Leonard pair A, A∗ is said to be of Bannai/Ito type whenever each of Leonard systems associated with A, A∗ is of Bannai/Ito type. Definition 5.1. Let Φ denote a Leonard system of Bannai/Ito type, we say that Φ is of normalized whenever there exist u, u∗ ∈ K such that d θi = (−1)i (i + u − ) (0 ≤ i ≤ d), 2 d θi∗ = (−1)i (i + u∗ − ) (0 ≤ i ≤ d). 2

(14) (15)

Lemma 5.2. ([16, Lemma 7.1]) Let Φ be a normalized Leonard system of Bannai/Ito type. Let ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ) denote the parameter array of Φ. Then there exist scalars u, u∗ , v in K such that (14) and (15) hold, and for 1 ≤ i ≤ d the following hold. ⎧ d+1 ∗ ⎪ for i even, d even, ⎪ −i(i + u + u + v − 2 ), ⎪ ⎨ −(i − d − 1)(i + u + u∗ − v − d+1 ), for i odd, d even, 2 ϕi = ⎪ −i(i − d − 1), for i even, d odd, ⎪ ⎪ ⎩ v 2 − (i + u + u∗ − d+1 )2 , for i odd, d odd, 2 ⎧ ⎪ for i even, d even, i(i − u + u∗ − v − d+1 ⎪ 2 ), ⎪ ⎨ (i − d − 1)(i − u + u∗ + v − d+1 ), for i odd, d even, 2 φi = ⎪ −i(i − d − 1), for i even, d odd, ⎪ ⎪ 2 ⎩ d+1 2 ∗ for i odd, d odd. v − (i − u + u − 2 ) ,

(16)

(17)

Definition 5.3. A Leonard pair A, A∗ of Bannai/Ito type is said to be of normalized whenever there exists a Leonard system associated with A, A∗ is of normalized. To end this section, we recall the notion of an affine transformation of Leonard pairs. Definition 5.4. [8] Let A, A∗ denote a Leonard pair over K and let η, μ, η ∗ , μ∗ denote scalars in K with η, η ∗ nonzero. Then the pair ηA + μI,

η ∗ A∗ + μ∗ I

(18)

is a Leonard pair over K. We call it the affine transformation of A, A∗ associated with η, μ, η ∗ , μ∗ . Note that for any Leonard pair A, A∗ , there exist some scalars η, μ, η ∗ , μ∗ in K with η, η ∗ nonzero, such that the affine transformation of A, A∗ associated with above η, μ, η ∗ , μ∗ is of normalized.

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6. Normalized Leonard pairs of Bannai/Ito type from A, A∗ In this section, we assume d ≥ 3 and let A, A∗ be as (1) and (2), respectively. By [2, Theorems 7.4 and 8.15], Brown showed that A, A∗ is a totally almost bipartite Leonard pair on Kd+1 of Bannai/Ito type. Lemma 6.1. The matrices A, A∗ satisfy the following relations. A2 A∗ + 2AA∗ A + A∗ A2 − 4A∗ = 0, A∗ 2 A + 2A∗ AA∗ + AA∗ 2 − 4A = 0. Proof. Since A, A∗ is a totally almost bipartite Leonard pair on Kd+1 of Bannai/Ito type, it follows from [2, Theorem 8.14] that A2 A∗ + 2AA∗ A + A∗ A2 − A∗ = 0,

(19)

A∗ 2 A + 2A∗ AA∗ + AA∗ 2 − ∗ A = 0.

(20)

Comparing the (0, 0)-entry of both sides of (19) and simplifying the result, we obtain  = 4. Similarly, comparing the (1, 0)-entry of both sides of (20) and simplifying the result, we obtain ∗ = 4. The results hold. 2 Definition 6.2. Let A, A∗ be as (1) and (2), respectively. If d is odd, we define B = 12 A and B∗ = 12 A∗ . If d is even, we define B = − 12 A and B∗ = − 12 A∗ . Lemma 6.3. With reference to Definition 6.2, the matrices B, B∗ satisfy the following relations. B2 B∗ + 2BB∗ B + B∗ B2 − B∗ = 0, ∗2

∗

∗

∗2

B B + 2B BB + BB

− B = 0.

Proof. Immediate from Lemma 6.1 and Definition 6.2.

(21) (22)

2

Theorem 6.4. With reference to Definition 6.2, there exists a unique Bε ∈ Matd+1 (K) such that B = B∗ Bε + Bε B∗ , ∗

ε

ε

(23)

B = B B + BB ,

(24)

Bε = BB∗ + B∗ B.

(25)

Proof. Define Bε such that (25) holds. Rewriting the relations (21), (22) using Bε , we get (23) and (24). It is clear from (25) that Bε is unique. 2 In the end of this section, we give the parameter array of B, B∗ for later use.

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Lemma 6.5. With reference to Definition 6.2, B, B∗ is a normalized Leonard pair of Bannai/Ito type. Moreover, there exists a normalized Leonard system associated with B, B∗ , denoted by Φ, whose parameter array is ({θi }di=0 , {θi∗ }di=0 , {ϕi }di=1 , {φi }di=1 ), where 1 θi = (−1)i (i − d − ), 2 1 θi∗ = (−1)i (i − d − ), 2

(26) (27)

for 0 ≤ i ≤ d, and

ϕi =

−i(i − d − 1), for i even, −(i − d − 1)(i − 2d − 2), for i odd,

i(i − d − 1), for d even, φi = −i(i − d − 1), for d odd,

(28)

(29)

for 1 ≤ i ≤ d. Proof. By [2, Proposition 5.2], the eigenvalues of B and B∗ are both {(−1)i (i−d− 12 )}di=0 . Obviously, B, B∗ is a normalized totally almost bipartite Leonard pair of Bannai/Ito type by the construction. Then there exists a normalized Leonard system Φ associated with B, B∗ , such that {θi = (−1)i (i − d − 12 )}di=0 (resp. {θi∗ = (−1)i (i − d − 12 )}di=0 ) is the eigenvalue sequence (resp. dual eigenvalue sequence) of Φ, respectively. So (26) and (27) hold. Since B has constant row sum θ0 and B∗ = diag(θ0∗ , θ1∗ , . . . , θd∗ ), the intersection number a0 is equal to B00 = 0 by Lemmas 3.5, 3.6. Setting i = 0 in (11), we obtain ϕ1 = −d(2d + 1). This gives (28) with i = 1. Evaluating the right-hand of (PA4) gives (29). Setting i = 1 in (29), we obtain that if d is odd, then φ1 = d; if d is even, then φ1 = −d. Evaluating the right-hand of (PA3) gives (28). Comparing (26)–(29) with (d+1)2 2 (14)–(17), we get u = u∗ = − d+1 if d is odd; and u = u∗ = − d+1 2 and v = 4 2 , d+1 v = 2 if d is even. 2 7. Leonard triples and Leonard triple systems In this section we recall the concept of Leonard triple systems and some basic facts. Definition 7.1. ([7, Definition 11.2]) By a Leonard triple system on V , we mean a sequence Ψ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ; Aε ; {Eiε }di=0 ) that satisfies (i)–(vii) below. (i) (ii) (iii) (iv)

Each of A, A∗ , Aε is a multiplicity-free element in End(V ). {Ei }di=0 is an ordering of the primitive idempotents of A. {Ei∗ }di=0 is an ordering of the primitive idempotents of A∗ . {Eiε }di=0 is an ordering of the primitive idempotents of Aε .

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(v) For B ∈ {A∗ , Aε },

Ei BEj =

0  0 =

if |i − j| > 1 if |i − j| = 1

(0 ≤ i, j ≤ d).

0 = 0

if |i − j| > 1 if |i − j| = 1

(0 ≤ i, j ≤ d).

0 = 0

if |i − j| > 1 if |i − j| = 1

(0 ≤ i, j ≤ d).

(vi) For B ∈ {A, Aε },

Ei∗ BEj∗

=

(vii) For B ∈ {A, A∗ },

Eiε BEjε =

We refer to d as the diameter of Ψ and say Ψ is over K. Lemma 7.2. ([7, Lemma 11.3]) Let Ψ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ; Aε ; {Eiε }di=0 ) denote a sequence of linear transformations in End(V ). Then Ψ is a Leonard triple system on V if and only if the following (i)–(iii) hold. (i) (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ) is a Leonard system on V . (ii) (A∗ ; {Ei∗ }di=0 ; Aε ; {Eiε }di=0 ) is a Leonard system on V . (iii) (Aε ; {Eiε }di=0 ; A; {Ei }di=0 ) is a Leonard system on V . Definition 7.3. ([7, Definition 11.4]) Let Ψ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ; Aε ; {Eiε }di=0 ) denote a Leonard triple system on V . For 0 ≤ i ≤ d, let θi , θi∗ , θiε denote the eigenvalues of A, A∗ , Aε associated with Ei , Ei∗ , Eiε , respectively. We call {θi }di=0 , {θi∗ }di=0 , {θiε }di=0 the first, second, third eigenvalue sequences of Ψ, respectively. Definition 7.4. ([7, Definition 11.5]) Let Ψ = (A; {Ei }di=0 ; A∗ ; {Ei∗ }di=0 ; Aε ; {Eiε }di=0 ) denote a Leonard triple system on V . Then the triple A, A∗ , Aε forms a Leonard triple on V . We say this triple is associated with Ψ. Observe that each Leonard triple system is associated with a unique Leonard triple. We remark that if A, A∗ , Aε is a Leonard triple on V , then the set of Leonard triple systems on V which are associated with A, A∗ , Aε has eight elements [7]. Definition 7.5. ([2, Definition 9.2]) A Leonard triple A, A∗ , Aε is said to be of Bannai/Ito type whenever any Leonard pair associated with A, A∗ , Aε is of Bannai/Ito type.

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Lemma 7.6. ([2, Lemma 9.3]) Let A, A∗ , Aε denote a Leonard triple. If any of the six Leonard pairs associated with A, A∗ , Aε is of Bannai/Ito type, then the Leonard triple A, A∗ , Aε is of Bannai/Ito type. Definition 7.7. A Leonard triple A, A∗ , Aε is said to be of normalized whenever any Leonard pair associated with A, A∗ , Aε is of normalized. To end this section, we recall the notion of an affine transformation of Leonard triples. Definition 7.8. Let A, A∗ , Aε denote a Leonard triple over K and let η, μ, η ∗ , μ∗ , η ε , με denote scalars in K with η, η ∗ , η ε nonzero. Then the triple ηA + μI,

η ∗ A∗ + μ∗ I,

η ε Aε + με I

(30)

is a Leonard triple over K. We call it the affine transformation of A, A∗ , Aε associated with η, μ, η ∗ , μ∗ , η ε , με . Note that for any Leonard triple A, A∗ , Aε , there exist some scalars η, μ, η ∗ , μ∗ , η ε , με in K with η, η ∗ , η ε nonzero, such that the affine transformation of A, A∗, Aε associated with above η, μ, η ∗ , μ∗ , η ε , με is of normalized. 8. A normalized Leonard triple B, B∗ , Bε Throughout this section, we assume d ≥ 3 and prove that B, B∗ , Bε is a normalized Leonard triple of Bannai/Ito type under the assumption of Theorem 6.4. We start by stating some linear algebraic results that will be useful in this section. Let V denote a finite-dimensional vector space over K. Let A denote a linear transformation on V . For scalar θ ∈ K we define VA (θ) = {v ∈ V |Av = θv}. Observe that θ is an eigenvalue of A if and only if VA (θ) = 0. In this case, VA (θ) is the corresponding eigenspace. We say that A is diagonalizable whenever V is spanned by the eigenspaces of A. Lemma 8.1. Let A and B denote linear transformations on V . Then for scalar θ ∈ K the following are equivalent: (i) The expression A3 B + A2 BA − ABA2 − BA3 + BA − AB vanishes on VA (θ). (ii) BVA (θ) ⊆ VA (−1 − θ) + VA (θ) + VA (1 − θ).

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Proof. For v ∈ VA (θ), we have (A3 B + A2 BA − ABA2 − BA3 + BA − AB)v = (A3 + θA2 − θ2 A − θ3 I + θI − A)Bv = (A3 + θA2 − (θ2 + 1)A − (θ3 − θ)I)Bv = (A − (−1 − θ)I)(A − θI)(A − (1 − θ)I)Bv, where I : V → V is the identity map. Then the results follow. 2 Definition 8.2. ([4, Definition 1.4]) Let A and B denote linear transformations on V . We say V is an irreducible A, B-module whenever there does not exist a subspace W ⊆ V such that W = 0, W = V , AW ⊆ W, BW ⊆ W . Lemma 8.3. With reference to Theorem 6.4, if d is odd, then the eigenvalues of Bε are 1 {(−1)i (i − d − )}di=0 ; 2 if d is even, then the eigenvalues of Bε are 1 {(−1)i (i + )}di=0 . 2 Moreover, Bε is multiplicity-free. Proof. Calculate the eigenvalues of Bε using (25). 2 Lemma 8.4. With reference to Theorem 6.4, the following (i)–(vi) hold. (i) (ii) (iii) (iv)

V is an irreducible B, B∗ -module. V is an irreducible B∗ , Bε -module. V is an irreducible Bε , B-module. B, B∗ satisfies the following relations. B3 B∗ + B2 B∗ B − BB∗ B2 − B∗ B3 + B∗ B − BB∗ = 0,

(31)

B∗3 B + B∗2 BB∗ − B∗ BB∗2 − BB∗3 + BB∗ − B∗ B = 0.

(32)

(v) B∗ , Bε satisfies the following relations. B∗3 Bε + B∗2 Bε B∗ − B∗ Bε B∗2 − Bε B∗3 + Bε B∗ − B∗ Bε = 0, ∗

∗

∗

B B +B B B −B B B ε3

ε2

ε

ε

ε2

∗

−B B

ε3

∗

∗

+ B B − B B = 0. ε

ε

(33) (34)

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(vi) Bε , B satisfies the following relations. Bε3 B + Bε2 BBε − Bε BBε2 − BBε3 + BBε − Bε B = 0,

(35)

B B + B B B − BB B − B B + B B − BB = 0.

(36)

3

ε

2

ε

ε

2

ε

3

ε

ε

Proof. Since B, B∗ is a Leonard pair on V , V is an irreducible B, B∗ -module by Lemma 1.2 and Definition 8.2. Now we show V is an irreducible B∗ , Bε -module. Let W be a nonzero subspace of V such that B∗ W ⊆ W and Bε W ⊆ W . Using (23) we get BW ⊆ W . So W = V by (i), and hence (ii) holds. The proof of (iii) is similar to that of (ii). Eliminating B in (24) and (25) using (23), we obtain that the pair Bε , B∗ satisfies the Askey–Wilson relations Bε2 B∗ + 2Bε B∗ Bε + B∗ Bε2 − B∗ = 0, B∗2 Bε + 2B∗ Bε B∗ + Bε B∗2 − Bε = 0. It follows that [Bε , Bε2 B∗ + 2Bε B∗ Bε + B∗ Bε2 − B∗ ] = 0 and [B∗ , B∗2 Bε + 2B∗ Bε B∗ + Bε B∗2 − Bε ] = 0, where [a, b] = ab − ba. So we obtain (33) and (34), and hence (v) holds. The proofs of (iv) and (vi) are similar to that of (v). 2 Lemma 8.5. With reference to Theorem 6.4, for 0 ≤ i ≤ d, let Vi = VB ((−1)i (i − d − 12 )) and Vi∗ = VB∗ ((−1)i (i − d − 12 )). When d is odd, let Viε = VBε ((−1)i (i − d − 12 )). When d is even, let Viε = VBε ((−1)i (i + 12 )). Then the following (i)–(ix) hold. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

V = V0 + V1 + · · · + Vd is the decomposition of eigenspaces of B. V = V0∗ + V1∗ + · · · + Vd∗ is the decomposition of eigenspaces of B∗ . V = V0ε + V1ε + · · · + Vdε is the decomposition of eigenspaces of Bε . B∗ Vi ⊆ Vi−1 + Vi + Vi+1 for 0 ≤ i ≤ d. Bε Vi ⊆ Vi−1 + Vi + Vi+1 for 0 ≤ i ≤ d. ∗ ∗ + Vi∗ + Vi+1 for 0 ≤ i ≤ d. BVi∗ ⊆ Vi−1 ε ∗ ∗ ∗ ∗ B Vi ⊆ Vi−1 + Vi + Vi+1 for 0 ≤ i ≤ d. ε ε + Viε + Vi+1 for 0 ≤ i ≤ d. BViε ⊆ Vi−1 ∗ ε ε ε ε B Vi ⊆ Vi−1 + Vi + Vi+1 for 0 ≤ i ≤ d.

∗ ε For notational convenience, we set V0 = V0∗ = V0ε = Vd+1 = Vd+1 = Vd+1 = 0.

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Proof. By the construction, (i)–(iii) hold. We now prove (iv). Let {θi }di=0 be the eigenvalues of B. Note that θi = (−1)i (i − d − 12 ) for 0 ≤ i ≤ d. Then θi+1 = 1 − θi and θi−1 = −1 − θi when i is odd. And θi+1 = −1 − θi and θi−1 = 1 − θi when i is even. By these comments and by Lemma 8.1 and (31), we obtain B∗ VB (θi ) ⊆ VB (θi−1 ) + VB (θi ) + VB (θi+1 ) for 0 ≤ i ≤ d, and hence (iv) holds. The proofs of (v)–(ix) are similar to that of (iv). 2 Lemma 8.6. With reference to Theorem 6.4, the triple B, B∗ , Bε is a normalized Leonard triple of Bannai/Ito type. Proof. For 0 ≤ i ≤ d, let Vi , Vi∗ and Viε be as in Lemma 8.5. Let vi denote a nonzero vector in Vi for 0 ≤ i ≤ d. By Lemma 8.5(i) and since B is multiplicity-free, the sequence v0 , v1 , . . . , vd is a basis for V . With respect to the basis the matrix representing B is diagonal and the matrices representing B∗ and Bε are irreducible tridiagonal by Lemma 8.4(i), (iii) and Lemma 8.5(iv), (v). Let vi∗ denote a nonzero vector in Vi∗ for 0 ≤ i ≤ d. By Lemma 8.5(ii) and since B∗ is multiplicity-free, the sequence v0∗ , v1∗ , . . . , vd∗ is a basis for V . With respect to the basis the matrix representing B∗ is diagonal and the matrices representing B and Bε are irreducible tridiagonal by Lemma 8.4(i), (ii) and Lemma 8.5(vi), (vii). Let viε denote a nonzero vector in Viε for 0 ≤ i ≤ d. By Lemma 8.5(iii) and since ε B is multiplicity-free, the sequence v0ε , v1ε , . . . , vdε is a basis for V . With respect to the basis the matrix representing Bε is diagonal and the matrices representing B and B∗ are irreducible tridiagonal by Lemma 8.4(ii), (iii) and Lemma 8.5(viii), (ix). Thus the triple B, B∗ , Bε is a Leonard triple. Moreover, by Lemma 6.5, Definition 7.5 and Lemma 8.3, B, B∗ , Bε is a normalized Leonard triple of Bannai/Ito type. 2 Lemma 8.7. With reference to Theorem 6.4, suppose d is even, then B, B∗ , −Bε is a normalized Leonard triple of Bannai/Ito type. Proof. By Lemma 8.6, B, B∗ , Bε is a normalized Leonard triple of Bannai/Ito type. Note that the triple B, B∗ , −Bε is an affine transformation of B, B∗ , Bε . Then B, B∗ , −Bε is a Leonard triple of Bannai/Ito type by Definition 7.8 and Lemma 7.6. By Lemma 8.3 and since d is even, we obtain the eigenvalues of −Bε are {(−1)i (i − d − 12 )}di=0 . Thus B, B∗ , −Bε is a normalized Leonard triple of Bannai/Ito type. 2 9. Leonard triples extended from A, A∗ Throughout this section, we assume d ≥ 3. Let A, A∗ be as (1) and (2), respectively. In this section, we determine all Aε ∈ Matd+1 (K) such that A, A∗ , Aε form a Leonard triple. To do this, we first determine all Bε ∈ Matd+1 (K) such that B, B∗ , Bε form a normalized Leonard triple, where B, B∗ are from Definition 6.2.

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Lemma 9.1. With reference to Definition 6.2, assume d is odd. Let Bε ∈ Matd+1 (K) such that B, B∗ , Bε is a normalized Leonard triple of Bannai/Ito type. Then Bε = BB∗ + B∗ B. Proof. Let Ψ = (B; {Ei }di=0 ; B∗ ; {Ei∗ }di=0 ; Bε ; {Eiε }di=0 ) denote a Leonard triple system associated with B, B∗ , Bε , for 0 ≤ i ≤ d, whose first, second and third eigenvalue sequences are 1 θi = (−1)i (i − d − ), 2

1 θi∗ = (−1)i (i − d − ), 2

d θiε = (−1)i (i + x − ), 2

respectively, where x is a scalar in K. Note that Φ = (Bε ; {Eiε }di=0 ; B; {Ei }di=0 ) is a normalized Leonard system by Lemma 7.2. For the normalized Leonard pair Bε , B, replacing u by x and u∗ by − d+1 2 in (14)–(17), we get the parameter array of Φ . Then by Lemma 4.2, the Askey–Wilson relations of Bε , B are Bε 2 B + 2Bε BBε + BBε 2 − B = ωBε + η ε I,

(37)

B2 Bε + 2BBε B + Bε B2 − Bε = ωB + ηI,

(38)

where ω = −2x2 + 2v 2 ,

η ε = −x2 − v 2 +

(d + 1)2 , 2

η = x2 − v 2 .

(39)

By [9, Theorem 3.2] and the definition of Leonard triples, there exists a unique sequence e, f , f ∗ , g, g ∗ ∈ K such that Bε = eI + f B + f ∗ B∗ + gBB∗ + g ∗ B∗ B.

(40)

In order to find the values of e, f , f ∗ , g, g ∗ , we consider the relations under the natural map  for Φ. Recall that B (resp. B∗ ) is the matrix on the left (resp. right) in (9). Using this and (40) we find the matrix Bε is tridiagonal with entries ∗ (Bε  )i,i−1 = f + gθi−1 + g ∗ θi∗ ,

(Bε  )i−1,i = ϕi (f ∗ + gθi−1 + g ∗ θi ) for 1 ≤ i ≤ d and (Bε  )i,i = e + f θi + f ∗ θi∗ + g(θi θi∗ + ϕi ) + g ∗ (θi θi∗ + ϕi+1 ) for 0 ≤ i ≤ d, where {ϕi }di=1 is the first split sequence of Φ in (28).

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Applying  to (37) and (38), respectively, we obtain 2

2

2

2

Bε  B + 2Bε  B Bε  + B Bε  − B = ωBε  + η ε I, B Bε  + 2B Bε  B + Bε  B

− Bε  = ωB + ηI.

(41) (42)

Comparing the (2, 0)-entry of both sides of (42) and simplifying the result, we obtain 4e + 2f − 4df = 0.

(43)

Similarly, comparing (3, 1)-entry of both sides of (42) and simplifying the result, we obtain 4e − 6f + 4df = 0.

(44)

Subtracting (43) from (44) we find 8(d − 1)f = 0. Note that d − 1 = 0 by the fact d ≥ 3. Therefore f = 0. So (43) becomes e = 0. Comparing the (3, 0)-entry of both sides of (41) and simplifying the result, we obtain −2(g − g ∗ )2 = 0. Thus g = g ∗ . Comparing the (3, 1)-entry of both sides of (41) and simplifying the result, we obtain −2gf ∗ = 0, from which g = 0 or f ∗ = 0. Assume g = 0, for contradiction. Then by (40), we obtain Bε = f ∗ B∗ , which is a contradiction with the fact that B∗ , Bε is a Leonard pair. Therefore f ∗ = 0. Comparing the (0, 1)-entry of both sides of (41) and simplifying the result, we obtain 0 = ω(Bε )01 . Since (Bε )01 = −ϕ1 g = 0, we get ω = 0.

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Comparing the (1, 0)-entry of both sides of (41) and simplifying the result, we obtain (g − 1)(g + 1) = 0. Thus g=1

or g = −1.

Assume that g = −1 for contradiction. Then by (40), we obtain Bε = −(BB∗ + B∗ B).

(45)

Calculating the eigenvalues of Bε using (45), we get 1 θiε = (−1)i+1 (i + ) 2

1 or θiε = (−1)i+1 (i − d − ) 2

for 0 ≤ i ≤ d, which is a contradiction with the assumption that θiε = (−1)i (i + x − d2 ). Thus the result holds. 2 Lemma 9.2. With reference to Definition 6.2, assume d is even. Let Bε ∈ Matd+1 (K) such that B, B∗ , Bε is a normalized Leonard triple of Bannai/Ito type. Then Bε = BB∗ + B∗ B

or

Bε = −(BB∗ + B∗ B).

Proof. Let B, B∗ , Bε denote a normalized Leonard triple of Bannai/Ito type. Let {θi }di=0 , {θi∗ }di=0 , {θiε }di=0 be the eigenvalues of B, B∗ , Bε , respectively. By Lemma 6.5, for 0 ≤ i ≤ d, we have 1 θi = (−1)i (i − d − ), 2

1 θi∗ = (−1)i (i − d − ), 2

d θiε = (−1)i (i + x − ), 2

respectively, where x is a scalar in K. For the normalized Leonard pair Bε , B, there exists a normalized Leonard system whose parameter array is the form of (14)–(17), where u = x, u∗ = − d+1 2 . Then by ε Lemma 4.2, the Askey–Wilson relations of B , B are as follows. Bε 2 B + 2Bε BBε + BBε 2 − B = ωBε + η ε I,

(46)

B2 Bε + 2BBε B + Bε B2 − Bε = ωB + ηI,

(47)

where ω = −2(d + 1)(x + v),

η ε = 2xv +

(d + 1)2 , 2

η = −(d + 1)(x + v).

(48)

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By [9, Theorem 3.2] and the definition of Leonard triples, there exists a unique sequence e, f , f ∗ , g, g ∗ ∈ K such that Bε = eI + f B + f ∗ B∗ + gBB∗ + g ∗ B∗ B.

(49)

In order to find the values of e, f , f ∗ , g, g ∗ , we consider the relations under the natural map  for Φ. Recall that B (resp. B∗ ) is the matrix on the left (resp. right) in (9). Using this and (49) we find the matrix Bε is tridiagonal with entries ∗ (Bε  )i,i−1 = f + gθi−1 + g ∗ θi∗ ,

(Bε  )i−1,i = ϕi (f ∗ + gθi−1 + g ∗ θi ) for 1 ≤ i ≤ d and (Bε  )i,i = e + f θi + f ∗ θi∗ + g(θi θi∗ + ϕi ) + g ∗ (θi θi∗ + ϕi+1 ) for 0 ≤ i ≤ d, where {ϕi }di=1 is the first split sequence of Φ in (28). Applying  to (46) and (47), respectively, we obtain 2

2

2

2

Bε  B + 2Bε  B Bε  + B Bε  − B = ωBε  + η ε I, B Bε  + 2B Bε  B + Bε  B

− Bε  = ωB + ηI.

(50) (51)

Comparing the (2, 0)-entry of both sides of (51) and simplifying the result, we obtain 4e + 2f − 4df = 0.

(52)

Similarly, comparing (3, 1)-entry of both sides of (51) and simplifying the result, we obtain 4e − 6f + 4df = 0.

(53)

Subtracting (52) from (53) we find 8(d − 1)f = 0. Note that d − 1 = 0 by the fact d ≥ 3. Therefore f = 0. So (52) becomes e = 0.

(54)

Comparing the (3, 0)-entry of both sides of (50) and simplifying the result, we obtain −2(g − g ∗ )2 = 0. Thus g = g ∗ .

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Comparing the (3, 1)-entry of both sides of (50) and simplifying the result, we obtain −2gf ∗ = 0, from which f ∗ = 0 or g = 0. Assume g = 0 for contradiction. Then by (49), we obtain Bε = f ∗ B∗ , which is a contradiction with the fact that B∗ , Bε is a Leonard pair. Therefore f ∗ = 0. Comparing the (0, 1)-entry of both sides of (50) and simplifying the result, we obtain 0 = ω(Bε )01 . Since (Bε )01 = −ϕ1 g = 0, we get ω = 0. Comparing the (1, 0)-entry of both sides of (50) and simplifying the result, we obtain (g − 1)(g + 1) = 0. Thus g=1

or g = −1.

By (49), we obtain Bε = BB∗ + B∗ B

or Bε = −(BB∗ + B∗ B).

The result holds. 2 Theorem 9.3. With reference to Definition 6.2, for Bε ∈ Matd+1 (K), if d is odd, then B, B∗ , Bε is a normalized Leonard triple of Bannai/Ito type if and only if Bε = BB∗ + B∗ B; if d is even, then B, B∗ , Bε is a normalized Leonard triple of Bannai/Ito type if and only if Bε = BB∗ + B∗ B

or

Bε = −(BB∗ + B∗ B).

Proof. Immediate from Lemmas 8.6, 8.7, 9.1 and 9.2. 2

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Theorem 9.4. Let A, A∗ be as (1) and (2), respectively, and Aε ∈ Matd+1 (K). Then A, A∗ , Aε is a Leonard triple if and only if Aε = α(AA∗ + A∗ A) + βI, where α, β ∈ K with α nonzero. Proof. We only give the proof of the case that d is even. Let A, A∗ , Aε be a Leonard triple. By the comments below Definition 7.8, there exists an affine transformation of A, A∗ , Aε associated with − 12 , 0, − 12 , 0, α , β  with α = 0, which is normalized. Namely, B, B∗ , α Aε + β  I is a normalized Leonard triple. By Lemma 7.6 and since B, B∗ is a Leonard pair of Bannai/Ito type, the normalized Leonard triple B, B∗ , α Aε + β  I is of Bannai/Ito type. Then by Theorem 9.3, we have α Aε + β  I = BB∗ + B∗ B or α Aε + β  I = −(BB∗ + B∗ B). If α Aε + β  I = BB∗ + B∗ B, then we have Aε =

1 β 1 β ∗ ∗ ∗ ∗ (BB + B B) − I = (AA + A A) − I. α α 4α α 

β 1  ε  ∗ ∗ Setting α = 4α  and β = − α , the result holds. Similarly, if α A +β I = −(BB +B B),  β 1 setting α = − 4α  and β = − α , the result holds. ε Conversely, let A = α(AA∗ +A∗ A) +βI with α = 0. By Theorem 9.3, B, B∗ , BB∗ + B∗ B is a normalized Leonard triple of Bannai/Ito type. Then the result follows from the fact that A, A∗ , Aε is the affine transformation of B, B∗ , BB∗ + B∗ B associated with −2, 0, −2, 0, 4α, β. 2

Acknowledgements The authors would like to thank the referee for giving this paper a careful reading and many valuable comments and useful suggestions. The authors are also grateful to Professor P. Terwilliger and Professor T. Ito for the advice they offered during their study of the q-tetrahedron algebra. This work was supported by the NSFC (No. 11271257 and No. 11471097), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20121303110005) and the Innovative Fund Project of Hebei Province (sj. 2015002). References [1] J.M.P. Balmaceda, J.P. Maralit, Leonard triples from Leonard pairs constructed from the standard basis of the Lie algebra sl2 , Linear Algebra Appl. 437 (2012) 1961–1977. [2] G.M.F. Brown, Totally Bipartite/ABipartite Leonard pairs and Leonard triples of Bannai/Ito type, Electron. J. Linear Algebra 26 (2013) 258–299. [3] B. Curtin, Modular Leonard triples, Linear Algebra Appl. 424 (2007) 510–539. [4] A. Godjali, Hessenberg pairs of linear transformations, Linear Algebra Appl. 431 (2009) 1579–1586. [5] B. Hou, J. Wang, S.G. Gao, Totally bipartite Leonard pairs and totally bipartite Leonard triples of q-Racah type, Linear Algebra Appl. 448 (2014) 168–204.

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