Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A^∗ : V → V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned by

     

The switching element for a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let (resp. ) denote a basis for V referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation S:V→V that sends v₀ to a scalar multiple of v_d, fixes w₀, and sends w_i to a scalar multiple of w_i for 1?i?d. We call S the switching element. We describe S from many points of view.     

Some trace formulae involving the split sequences of a Leonard pair

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A^∗ : V → V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let diag(θ₀, θ₁, … , θ_d) denote the diagonal matrix referred to in (ii) above and let denote the diagonal matrix referred to in (i) above. It is known that there exists a basis u₀, u₁, … , u_d for V and there exist scalars ?₁, ?₂, … , ?_d in K such that Au_i = θ_iu_i + u_i+1 (0 ? i ? d − 1), Au_d = θ_du_d, , . The sequence ?₁, ?₂, … , ?_d is called the first split sequence of the Leonard pair. It is known that there exists a basis v₀, v₁, … , v_d for V and there exist scalars ?₁, ?₂, … , ?_d in K such that Av_i = θ_d−iv_i + v_i+1 (0 ? i ? d − 1),Av_d = θ₀v_d, , . The sequence ?₁, ?₂, … , ?_d is called the second split sequence of the Leonard pair. We display some attractive formulae for the first and second split sequence that involve the trace function.     

Balanced Leonard pairs

Let K denote a field, and 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 the following two conditions:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

Let (respectively v₀, v₁, … , v_d) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ? i ? d, let a_i denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of v_i, when we write A^∗v_i as a linear combination of v₀, v₁, … , v_d.In this paper we show a₀ = a_d if and only if . Moreover we show that for d ? 1 the following are equivalent; (i) a₀ = a_d and a₁ = a_d−1; (ii) and ; (iii) a_i = a_d−i and for 0 ? i ? d. These give a proof of a conjecture by the second author. We say A, A^∗ is balanced whenever a_i = a_d−i and for 0 ? i ? d. We say A,A^∗ is essentially bipartite (respectively essentially dual bipartite) whenever a_i (respectively ) is independent of i for 0 ? i ? d. Observe that if A, A^∗ is essentially bipartite or dual bipartite, then A, A^∗ is balanced. For d ≠ 2, we show that if A, A^∗ is balanced then A, A^∗ is essentially bipartite or dual bipartite.     

Askey-Wilson relations and Leonard pairs

It is known that if (A,A^*) is a Leonard pair, then the linear transformations A, A^* satisfy the Askey-Wilson relations

     

Normalized Leonard pairs and Askey-Wilson relations

Let V denote a vector space with finite positive dimension, and let (A, A^∗) denote a Leonard pair on V. As is known, the linear transformations A, A^∗ satisfy the Askey-Wilson relations

     

Zonal spherical functions on the complex reflection groups and (n+1,m+1)-hypergeometric functions

The pair of groups, complex reflection group G(r,1,n) and symmetric group S_n, is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called (n+1,m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the (n+1,m+1)-hypergeometric functions.     

On a necessary but not sufficient condition for a γ-generating pair to be a nehari pair

This paper gives a negative answer to a question due to V.M. Adamjan, D.Z. Arov and M.G. Krein, and (what is the same) gives a counterexample to D.Sarason's conjecture^* concerning exposed points inH ¹.     

Mock tridiagonal systems

We introduce the notion of a mock tridiagonal system. This is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain nonvanishing condition. We show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications. This paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism.     

Explicit inverse of a tridiagonal k−Toeplitz matrix

Summary We obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k–Toeplitz matrix A. The proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of Chebyshev polynomials of the second kind. We also compute the characteristic polynomial of A which enables us to state some conditions for the existence of A^–1. Our results also extend known results for the case when the residue mod k of the order of A is equal to 0 or k–1 (Numer. Math., 10 (1967), pp. 153–161.).The work was supported by CMUC (Centro de Matemática da Universidade de Coimbra) and by Acção Integrada Luso-Espanhola E-6/03     

A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A^∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

Spectrum of the Kerzman-Stein Operator for the Ellipse

The skew-hermitian part of the Cauchy operator, defined with respect to arclength measure on the boundary, is known as the Kerzman-Stein operator. For an ellipse, the eigenvalues of this operator are shown to have multiplicity two. For an ellipse with small eccentricity, we compute the leading coefficient in the asymptotic expansion of the eigenvalues.     

Zeros of entire functions and a problem of Ramanujan

We derive representations for certain entire q-functions and apply our technique to the Ramanujan entire function (or q-Airy function) and q-Bessel functions. This is used to show that the asymptotic series of the large zeros of the Ramanujan entire function and similar functions are also convergent series. The idea is to show that the zeros of the functions under consideration satisfy a nonlinear integral equation.     

Null space distributions—A new approach to finite convolution equations with a Hankel kernel

In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation.     

The asymptotics of the generalised Hermite-Bell polynomials

The Hermite-Bell polynomials are defined by for n=0,1,2,… and integer r≥2 and generalise the classical Hermite polynomials corresponding to r=2. We obtain an asymptotic expansion for as n→∞ using the method of steepest descents. For a certain value of x, two saddle points coalesce and a uniform approximation in terms of Airy functions is given to cover this situation. An asymptotic approximation for the largest positive zeros of is derived as n→∞. Numerical results are presented to illustrate the accuracy of the various expansions.     

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

We consider a bipartite distance-regular graph Γ with diameter D?4, valency k?3, intersection numbers b_i,c_i, distance matrices A_i, and eigenvalues θ₀>θ₁>?>θ_D. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of Mat_X(C) generated by , where A=A₁ and denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever for 0?i?D. By the endpoint of W we mean . Assume W is thin with endpoint 2. Observe is a one-dimensional eigenspace for ; let η denote the corresponding eigenvalue. It is known where , and d=⌊D/2⌋. To describe the structure of W we distinguish four cases: (i) ; (ii) D is odd and ; (iii) D is even and ; (iv) . We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694-1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is D-1-e where e=1 in case (iii) and e=0 in case (iv). Let v denote a nonzero vector in . We show W has a basis , where E_i denotes the primitive idempotent of A associated with θ_i and where the set S is {1,2,…,d-1}∪{d+1,d+2,…,D-1} in case (iii) and {1,2,…,D-1} in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W has a basis , and we find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square-norm of each basis vector. We find the transition matrix relating our two bases for W.     

Invertibility symbol for a Banach algebra generated by two idempotents and a shift

A Banach algebra generated by two idempotentsp, r, identitye and a shiftv which satisfy the conditionspv=vp andrv=v(e–r) is investigated. It is proved that all irreducible representations of algebraA are two- and four-dimensional. The explicit form of these representations is obtained. An Invertibility Symbol is constructed. Some examples are considered.