tridiagonal pairs and the askey wilson relations abstract the notion of a tridiagonal pair was introduced by ito tanabe and terwilliger let v denote a nonzero finite dimensional vector space over a field f a tridiagonal pair on v is a pair a a where a v v and a v v are linear transformations that satisfy some conditions assume a a is a tridiagonal pair on v recently terwilliger and vidunas showed that if a is multiplicity free on v then a a satisfy the following askey wilson relation for some scalars a 2 a aa a a a 2 aa a a a a 2 a i a 2 a a aa aa 2 a a aa a a 2 a i in the present paper we show that if a tridiagonal pair a a satisfy the askey wilson relations then the eigenspaces of a and the eigenspaces of a have one common dimension and moreover if f is algebraically closed then that common dimension is 1

on the shape of a tridiagonal pair abstract 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 the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d 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 v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 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 and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

the structure of some linear transformations abstract let f denote an algebraically closed field and let v denote a finite dimensional vector space over f recently ito and terwilliger considered a system of linear transformations a a a a on v which generalizes the notions of a tridiagonal pair and a q inverting pair in their paper they mentioned some open problems about this system in this paper we solve problem 1 2 with the following results let v i i 0 d denote the common eigenspaces of a a and let v i i 0 d denote the common eigenspaces of a a we show that each of a a a a is determined up to affine transformation by the sequences v i i 0 d v i i 0 d we also show that the following are equivalent i there exists a nonzero bilinear form on v such that a u v u a v and a u v u a v for all u v v ii there exist scalars in f with nonzero such that a a i and a a i and iii both a a and a a are tridiagonal pairs

evaluation modules for the q tetrahedron algebra let f denote an algebraically closed field and fix a nonzero q f that is not a root of unity we consider the q tetrahedron algebra q over f it is known that each finite dimensional irreducible q module of type 1 is a tensor product of evaluation modules this paper contains a comprehensive description of the evaluation modules for q this description includes the following topics given an evaluation module v for q we display 24 bases for v that we find attractive for each basis we give the matrices that represent the q generators we give the transition matrices between certain pairs of bases among the 24 it is known that the cyclic group z 4 acts on q as a group of automorphisms we describe what happens when v is twisted via an element of z 4 we discuss how evaluation modules for q are related to leonard pairs of q racah type

a family of tridiagonal pairs abstract let f denote a field and let v denote a vector space of finite positive dimension over f let a a denote a tridiagonal pair on v of diameter d 3 assume the eigenvalue and dual eigenvalue sequences of a a satisfy i q2i i q 2 d 2 i for some nonzero scalars q f where q is not a root of unity assume that not all eigenvalues of a and a have multiplicity one let m and m denote the subalgebras of end v generated by a and a respectively and assume that v mv m v for some eigenvectors v of a associated with 0 and v of a associated with d we find a nice basis for v and describe the action of a a on this basis in terms of six parameters

the equitable presentation for the quantum group uq g associated with a symmetrizable kac moody algebra g abstract we consider the quantum group u q g associated with a symmetrizable kac moody algebra g we display a presentation for u q g that we find attractive we call this the equitable presentation for g sl 2 the equitable presentation has generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1

two linear transformations each tridiagonal with respect to an eigenbasis of the other 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 satisfying both conditions below 1 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 2 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 we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

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 v0 v1 vd respectively v0 v1 vd denote a basis for v that satisfies i respectively ii for 0 i d let ai denote the coefficient of vi when we write avi as a linear combination of v0 v1 vd and let ai denote the coefficient of vi when we write a vi as a linear combination of v0 v1 vd in this paper we show a0 ad if and only if a0 ad moreover we show that for d 1 the following are equivalent i a0 ad and a1 ad 1 ii a0 ad and a1 ad 1 iii ai ad i and ai ad i for 0 i d these give a proof of a conjecture by the second author we say a a is balanced whenever ai ad i and ai ad i for 0 i d we say a a is essentially bipartite respectively essentially dual bipartite whenever ai respectively ai 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

redefining chance constrained programming in fuzzy environment this paper presents an alternative definition of probability of fuzzy event in analogy to probability of nonfuzzy event we view probability of fuzzy event in terms of cumulative distribution of a random variable a multi objective imprecise chance constrained programming problem has been defined here which is an extension of classical chance constrained programming problem in fuzzy environment an illustrative numerical example is given

tridiagonal pairs of height one let a a denote a tridiagonal pair on a vector space v over a field k let v0 vd denote a standard ordering of the eigenspaces of a on v and let 0 d denote the corresponding eigenvalues of a we assume d 3 let q denote a scalar taken from the algebraic closure of k such that q2 q 2 1 3 0 2 1 we assume q is not a root of unity let i denote the dimension of vi the sequence 0 1 d is called the shape of the tridiagonal pair it is known there exists a unique integer h 0 h d 2 such that i 1 i for d h i d the integer h is known as the height of the tridiagonal pair in this paper we show that the shape of a tridiagonal pair of height one with 0 1 is either 1 2 2 2 1 or 1 3 3 1 in each case we display a basis for v and give the action of a a on this basis

fuzzy stochastic linear programming survey and future research directions in many real life problems one has to base decision on information which is both fuzzily imprecise and probabilistically uncertain although consistency indexes providing a union nexus between possibilistic and probabilistic representation of uncertainty exist there are no reliable transformations between them this calls for new paradigms for incorporating the two kinds of uncertainty into mathematical models fuzzy stochastic linear programming is an attempt to fulfill this need it deals with modelling and problem solving issues related to situations where randomness and fuzziness co occur in a linear programming framework in this paper we provide a survey of the essential elements methods and algorithms for this class of linear programming problems along with promising research directions being a survey the paper includes many references to both give due credit to results in the field and to help readers obtain more detailed information on issues of interest

a refinement of the split decomposition of a tridiagonal pair let v denote a nonzero finite dimensional vector space over a field k and let a a denote a tridiagonal pair on v of diameter d let v u0 ud denote the split decomposition and let i denote the dimension of ui in this paper at first we show there exists a unique integer h 0 h d 2 such that i 1 i for d h i d we call h the height of the tridiagonal pair for 0 r h we define subspaces ui r r i d r by ui r ri r ur kerrd 2r 1 where r denotes the rasing map we show v is decomposed as a direct sum v r 0h i rd rui r this gives a refinement of the split decomposition define u r i rd rui r and observe v r 0hu r we show lu r u r 1 u r u r 1 for 0 r h where we set u 1 u h 1 0 let f r v u r denote the projection we show the lowering map l is decomposed as l l l 0 l where l r 1hf r 1 lf r l 0 r 0hf r lf r and l r 0h 1f r 1 lf r these maps satisfy l u r u r 1 l 0 u r u r and l u r u r 1 for 0 r h the main results of this paper are the following i for 0 r h 1 and r 2 i d r 1 rl l r holds on ui r for some scalar ii for 1 r h and r i d r 1 rl l r holds on ui r for some scalar iii for 0 r h and r 1 i d r 1 rl 0 l 0 r i holds on ui r for some scalars moreover we give explicit expressions of

towards a classification of the tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension let end v denote the k algebra consisting of all k linear transformations from v to v we consider a pair a a end v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 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 let e0 denote the element of end v such that e0 i v0 0 and e0 vi 0 for 1 i d let d resp d denote the k subalgebra of end v generated by a resp a in this paper we prove that the span of e0 dd de0 equals the span of e0 de0 de0 and that the elements of e0 de0 mutually commute we relate these results to some conjectures of tatsuro ito and the second author that are expected to play a role in the classification of tridiagonal pairs

normalized leonard pairs and askey wilson relations abstract 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 a 2 a aa a a a 2 aa a a a a 2 a i a 2 a a aa aa 2 a a aa a a 2 a i for some scalars the scalar sequence is unique if the dimension of v is at least 4 if c c t t are scalars and t t are not zero then ta c t a c is a leonard pair on v as well these affine transformations can be used to bring the leonard pair or its askey wilson relations into a convenient form this paper presents convenient normalizations of leonard pairs by the affine transformations and exhibits explicit askey wilson relations satisfied by them

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

tridiagonal pairs and the q tetrahedron algebra abstract the q tetrahedron algebra q was recently introduced and has been studied in connection with tridiagonal pairs in this paper we further develop this connection let k denote an algebraically closed field and let q denote a nonzero scalar in k that is not a root of unity let v denote a vector space over k with finite positive dimension and let a a denote a tridiagonal pair on v let i i 0 d resp i i 0 d denote a standard ordering of the eigenvalues of a resp a ito and terwilliger have shown that when i q 2 i d and i q d 2 i 0 i d there exists an irreducible q module structure on v such that the q generators x 01 x 23 act as a a respectively in this paper we examine the case in which there exists a nonzero scalar c in k such that i q 2 i d and i q 2 i d cq d 2 i for 0 i d in this case we associate to a a a polynomial p in one variable and prove the following theorem as our main result theorem the following are equivalent 1 i there exists a q module structure on v such that x 01 acts as a and x 30 cx 23 acts as a where x 01 x 30 x 23 are standard generators for q 2 ii p q 2 d 2 q q 1 2 0 suppose i and ii hold then the q module structure on v is unique and irreducible

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

the structure of a tridiagonal pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k 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 vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 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 and for 0 i d the dimensions of vi vd i vi vd i coincide in this paper we show that the following i iv hold provided that k is algebraically closed i each of v0 v0 vd vd has dimension 1 ii there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v iii there exists a unique anti automorphism of end v that fixes each of a a iv the pair a a is determined up to isomorphism by the data i i 0d i i 0d i i 0d where i resp i is the eigenvalue of a resp a on vi resp vi and i i 0d is the split sequence of a a corresponding to i i 0d and i i 0d

dual polar graphs the quantum algebra uq sl2 and leonard systems of dual q krawtchouk type abstract in this paper we consider how the following three objects are related i the dual polar graphs ii the quantum algebra u q sl 2 iii the leonard systems of dual q krawtchouk type for convenience we first describe how ii and iii are related for a given leonard system of dual q krawtchouk type we obtain two u q sl 2 module structures on its underlying vector space we now describe how i and iii are related let denote a dual polar graph fix a vertex x of and let t t x denote the corresponding subconstituent algebra by definition t is generated by the adjacency matrix a of and a certain diagonal matrix a a x called the dual adjacency matrix that corresponds to x by construction the algebra t is semisimple we show that for each irreducible t module w the restrictions of a and a to w induce a leonard system of dual q krawtchouk type we now describe how i and ii are related we obtain two u q sl 2 module structures on the standard module of we describe how these two u q sl 2 module structures are related each of these u q sl 2 module structures induces a c algebra homomorphism u q sl 2 t we show that in each case t is generated by the image together with the center of t using the combinatorics of we obtain a generating set l f r k of t along with some attractive relations satisfied by these generators

the shape of some linear transformations abstract recently ito and terwilliger proposed a problem about some linear transformations a a a a on a finite dimensional vector space v over a field f which can be seen as a generalization of the notions of tridiagonal pair and q inverting pair see problem 1 2 in this paper we will introduce the notion of the shape of the linear transformations a a a a our main results are the properties of the shape i the shape is symmetric and unimodal ii let denote an anti isomorphism from end v to end v with v a vector space over f with the same dimension as v assume that the linear transformations a a a a on v satisfy all conditions of problem 1 2 then the linear transformations a a a a also satisfy all conditions of problem 1 2 moreover a a a a have the same shape as a a a a

the switching element for a leonard pair abstract 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 v i i 0 d resp w i i 0 d 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 v 0 to a scalar multiple of v d fixes w 0 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

structural and functional quantization of vagueness abstract this paper proposes two methods of quantizing vague categories in terms of quantities whose identities have been either widely recognized in the fuzzy set theory literature or established by bayesian decision theory the methods are called structural quantization and functional quantization respectively the idea behind structural quantization is that for a comparative vague category like good scores there exists a preference ordering among the possible scores and how good a particular score is depends on how large the portion of examinees whose scores are lower than that score is functional quantization adopts the conclusion of bayesian decision theory that a rational agent s belief can be represented by a probability to quantize the term good score the method first assesses the two probabilities that respectively represent a rational agent s beliefs about john s score before and after learning that john has got a good score it then derives a membership function by comparing those two probabilities the equivalence of the two quantization methods is hypothesized the logic connectives not and and or are discussed in the setting of structural quantization while aggregation of information is addressed in the setting of functional quantization

transition maps between the 24 bases for a leonard pair abstract 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 1 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 2 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 in an earlier paper we described 24 special bases for v one feature of these bases is that with respect to each of them the matrices that represent a and a are i diagonal and irreducible tridiagonal or ii irreducible tridiagonal and diagonal or iii lower bidiagonal and upper bidiagonal or iv upper bidiagonal and lower bidiagonal for each ordered pair of bases among the 24 there exists a unique linear transformation from v to v that sends the first basis to the second basis we call this the transition map in this paper we find each transition map explicitly as a polynomial in a a

two linear transformations each tridiagonal with respect to an eigenbasis of the other the td d canonical form and the lb ub canonical form abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider an ordered pair of linear transformations a v v and b v v which satisfy both i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we call such a pair a leonard pair on v we introduce two canonical forms for leonard pairs we call these the td d canonical form and the lb ub canonical form in the td d canonical form the leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix subject to a certain normalization in the lb ub canonical form the leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix subject to a certain normalization we describe the two canonical forms in detail as an application we obtain the following results given square matrices a b over k with a tridiagonal and b diagonal we display a necessary and sufficient condition for a b to represent a leonard pair given square matrices a b over k with a lower bidiagonal and b upper bidiagonal we display a necessary and sufficient condition for a b to represent a leonard pair we briefly discuss how leonard pairs correspond to the q racah polynomials and some related polynomials in the askey scheme we present some open problems concerning leonard pairs

data based fuzzy rule test for fuzzy modelling in the field of fuzzy modelling the exclusive consideration of the modelling error leads to problems concerning the handling of high dimensional applications and the interpretability of the resulting rule base to solve those problems a statistically motivated fuzzy rule test is proposed it decides if a fuzzy if then statement is a relevant rule or not in this way the problem of finding a good rule base can be reduced to the problem of finding good relevant rules

characterization of continuous t norms compatible with zadeh s probability of fuzzy events in this work we study the concept of p z compatibility which holds for those t norms for which zadeh s probability of fuzzy events fulfills kolmogorov s axioms whether a probability satisfies kolmogorov s axioms or not depends almost exclusively on the t norm and dual t conorm used to define intersection and union respectively of fuzzy sets in order to better understand fuzzy probability theory we provide a complete characterization of the p z compatible continuous t norms

medical diagnosis fuzzy sets and degrees of belief abstract a model simulating the medical diagnostic process is presented diagnostic groups are considered as fuzzy sets partial knowledge is quantified by belief functions the process is considered as the evaluation by the clinician of his degree of belief concerning the belonging of his patient to a fuzzy set given fuzzy and partial informations

paper evidence measures based on fuzzy information an organized body of results pertaining to the analysis of evidence when the available knowledge is pervaded with imprecision is provided the evidence theory of dempster and shafer is extended to the case of fuzzy observations and fuzzy events upper and lower possibilities of such events are derived by iterating the generation process of upper and lower probabilities as done by dempster both probabilistic and possibilistic models are developed in parallel these evidence measures are used for decision evaluation when the available knowledge is poor the classical model of decision making under uncertainty is thus extended to the case when the consequences of a decision are only roughly described and their probabilities of occurrence modeled by intervals or fuzzy numbers

probability of a fuzzy event an axiomatic approach abstract axioms are proposed that could justify the natural definition of the probability of a fuzzy event initially given by zadeh they are based 1 on the postulate that the sum of the conditional probability of a fuzzy event and of its complement given any fuzzy event adds to one or 2 on soft independence for orthogonal sets with independent constitutive elements a general postulate is also required concerning the complement of a fuzzy set the classical definition of the operator representing the complement can also be deduced

finite dimensional irreducible uq sl2 modules from the equitable point of view abstract we consider the quantum algebra u q sl 2 with q not a root of unity we describe the finite dimensional irreducible u q sl 2 modules from the point of view of the equitable presentation

a new approach to the 2 variable subnormal completion problem we study the subnormal completion problem scp for 2 variable weighted shifts we use tools and techniques from the theory of truncated moment problems to give a general strategy to solve scp we then show that when all quadratic moments are known equivalently when the initial segment of weights consists of five independent data points the natural necessary conditions for the existence of a subnormal completion are also sufficient to calculate explicitly the associated berger measure we compute the algebraic variety of the associated truncated moment problem it turns out that this algebraic variety is precisely the support of the berger measure of the subnormal completion

linear programming under randomness and fuzziness this paper is concerned with the use of fuzzy sets theory well suited for nuanced human behavior to extend the applicability of linear programming in a way to let both randomness and fuzziness uncertainties be taken into account a fuzzy stochastic linear programming problem is considered and ways to handle it suggested the basic idea behind our approaches is to translate the original problem in deterministic terms via tools provided by fuzzy sets theory to get a classical one some conditions for the resulting problems to be solved without extra computational effort are stressed and illustrative examples given

q inverting pairs of linear transformations and the q tetrahedron algebra abstract as part of our study of the q tetrahedron algebra q we introduce the notion of a q inverting pair roughly speaking this is a pair of invertible semisimple linear transformations on a finite dimensional vector space each of which acts on the eigenspaces of the other according to a certain rule our main result is a bijection between the following two sets i the isomorphism classes of finite dimensional irreducible q modules of type 1 ii the isomorphism classes of q inverting pairs

a mass assignment theory of the probability of fuzzy events abstract mass assignment theory techniques for processing uncertainty in fril are reviewed the notion of the probability of a fuzzy event is introduced together with the t norm definition of conditional probabilities the latter is then shown to be probability possibility inconsistent an alternative theory of conditional probabilities based on mass assignments is presented together with a number of results illustrating some intuitive properties in particular the mass assignment theory of conditional probabilities is shown to be probability possibility consistent

the quantum algebra uq sl2 and its equitable presentation abstract we show that the quantum algebra u q sl 2 has a presentation with generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1 we call this the equitable presentation we show that y respectively z is not invertible in u q sl 2 by displaying an infinite dimensional u q sl 2 module that contains a nonzero null vector for y respectively z we consider finite dimensional u q sl 2 modules under the assumption that q is not a root of 1 and char k 2 where k is the underlying field we show that y and z are invertible on each finite dimensional u q sl 2 module we display a linear operator that acts on finite dimensional u q sl 2 modules and satisfies 1 x y 1 y z 1 z x on these modules we define using the q exponential function

leonard pairs and the q racah polynomials abstract 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 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 diagonal and the matrix representing a is irreducible tridiagonal wecall such a pair a leonard pair on v in the appendix to linear algebra appl 330 2001 p 149 we outlined a correspondence between leonard pairs and a class of orthogonal polynomials consisting of the q racah polynomials and some related polynomials of the askey scheme we also outlined how for the polynomials in this class the 3 term recurrence difference equation askey wilson duality and orthogonality can be obtained in a uniform manner from the corresponding leonard pair the purpose of this paper is to provide proofs for the assertions which we made in that appendix

introduction to leonard pairs in this survey paper we give an elementary introduction to the theory of leonard pairs a leonard pair is defined as follows 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 b v v that satisfy conditions i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we give several examples of leonard pairs using these we illustrate how leonard pairs arise in representation theory combinatorics and the theory of orthogonal polynomials

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 0 1 d denote the diagonal matrix referred to in ii above and let diag 0 1 d denote the diagonal matrix referred to in i above it is known that there exists a basis u0 u1 ud for v and there exist scalars 1 2 d in k such that aui iui ui 1 0 i d 1 aud dud a ui iui 1 i ui 1 i d a u0 0 u0 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 vd for v and there exist scalars 1 2 d in k such that avi d ivi vi 1 0 i d 1 avd 0vd a vi ivi 1 i vi 1 i d a v0 0 v0 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

three mutually adjacent leonard pairs abstract let k denote a field of characteristic 0 and let v denote a vector space over k with positive finite dimension consider an ordered pair of linear transformations a v v and a v v that satisfies both conditions 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 we call such a pair a leonard pair on v let a a denote a leonard pair on v a basis for v is said to be standard for a a whenever it satisfies i or ii above a basis for v is said to be split for a a whenever with respect to this basis the matrix representing one of a a is lower bidiagonal and the matrix representing the other is upper bidiagonal let a a and b b denote leonard pairs on v we say these pairs are adjacent whenever each basis for v which is standard for a a resp b b is split for b b resp a a our main results are as follows theorem 1 there exist at most 3 mutually adjacent leonard pairs on v provided the dimension of v is at least 2 theorem 2 let a a b b and c c denote three mutually adjacent leonard pairs on v then for each of these pairs the eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression theorem 3 let a a denote a leonard pair on v whose eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression then there exist leonard pairs b b and c c on v such that a a b b and c c are mutually adjacent