parametric symmetries in exactly solvable real and pt symmetric complex potentials in this paper we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex pt symmetric potentials we focus our attention on the conventional potentials such as the generalized poschl teller gpt scarf i and pt symmetric scarf ii which are invariant under certain parametric transformations the resulting set of potentials is shown to yield a completely different behavior of the bound state solutions further the supersymmetric partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and pt symmetric complex potentials these potentials are also observed to be shape invariant si in nature we subsequently take up a study of the newly discovered rationally extended si potentials corresponding to the above mentioned conventional potentials whose bound state solutions are associated with the exceptional orthogonal polynomials eops we discuss the transformations of the corresponding

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

generalized quantum nonlinear oscillators exact solutions and rational extensions we construct exact solutions and rational extensions to quantum systems of generalized nonlinear oscillators our method is based on a connection between nonlinear oscillator systems and schrodinger models for certain hyperbolic potentials the rationally extended models admit discrete spectrums and corresponding closed form solutions are expressed through jacobi type xm exceptional orthogonal polynomials

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice we address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice as kay and moses showed in 1956 reflectionless potentials in one dimensional quantum mechanics are exactly solvable with an additional time dependence these potentials are identified as the soliton solutions of the korteweg de vries kdv hierarchy an n soliton potential has the time t and 2n positive parameters k1 kn and cj j 1 n corresponding to n discrete eigenvalues kj2 the eigenfunctions are elementary functions expressed by the ratio of determinants the darboux crum krein adler transformations or the abraham moses transformations based on eigenfunction deletions produce lower soliton number potentials with modified parameters cj we explore various identities satisfied by the eigenfunctions of the soliton potentials which reflect the uniqueness theorem of gel fand levitan marchenko equations for separable degenerate kernels

extending romanovski polynomials in quantum mechanics some extensions of the third class romanovski polynomials also called romanovski pseudo jacobi polynomials which appear in bound state wavefunctions of rationally extended scarf ii and rosen morse i potentials are considered for the former potentials the generalized polynomials satisfy a finite orthogonality relation while for the latter an infinite set of relations among polynomials with degree dependent parameters is obtained both types of relations are counterparts of those known for conventional polynomials in the absence of any direct information on the zeros of the romanovski polynomials present in denominators the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of second order differential equations of schrodinger type it is also shown that on going from scarf i to scarf ii or from rosen morse ii to rosen morse i potentials the variety of rational extensions is narrowed down from types i ii and iii to type iii only

the discrete memoryless multiple access channel with partially cooperating encoders corresp we introduce the communication situation in which the encoders of a multiple access channel are partially cooperating these encoders are connected by communication links with finite capacities which permit both encoders to communicate with each other first we give a general definition of such a communication process conference then by proving a converse and giving an achievability proof we establish the capacity region of the multiple access channel with partially cooperating encoders it turns out that the optimal conference is very simple

extensions of solvable potentials with finitely many discrete eigenstates we address the problem of rational extensions of six examples of shape invariant potentials having finitely many discrete eigenstates the overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions which are an essential ingredient for multi indexed and iso spectral extensions of these potentials they have exactly the same form as the eigenfunctions but their degrees are much higher than nmax so that their energies are lower than the groundstate energy

the sobolev orthogonality and spectral analysis of the laguerre polynomials l n k for positive integers k for k n we consider the analysis of the classical laguerre differential expression l k y x 1 x ke x x k 1e xy x rx ke xy x x 0 where r 0 is fixed in several nonisomorphic hilbert and hilbert sobolev spaces in one of these spaces specifically the hilbert space l2 0 x ke x it is well known that the glazman krein naimark theory produces a self adjoint operator a k generated by l k that is bounded below by rl where i is the identity operator on l2 0 x ke x consequently as a result of a general theory developed by littlejohn and wellman there is a continuum of left definite hilbert spaces h s k v s k s k s 0 and left definite self adjoint operators b s k s 0 associated with the pair l2 0 x ke x a k for a k and each of the operators b s k it is the case that the tail end sequence l n k n k of laguerre polynomials form a complete set of eigenfunctions in the corresponding hilbert spaces in 1995 kwon and littlejohn introduced a hilbert sobolev space w k 0 in which the entire sequence of laguerre polynomials is orthonormal in this paper we construct a self adjoint operator in this space generated by the second order laguerre differential expression l k having l n k n 0 as a complete set of eigenfunctions the key to this construction is in identifying a certain closed subspace of w k 0 with the kth left definite vector space v k k

new families of superintegrable systems from hermite and laguerre exceptional orthogonal polynomials in recent years many exceptional orthogonal polynomials eop were introduced and used to construct new families of 1d exactly solvable quantum potentials some of which are shape invariant in this paper we construct from hermite and laguerre eop and their related quantum systems new 2d superintegrable hamiltonians with higher order integrals of motion and the polynomial algebras generated by their integrals of motion we obtain the finite dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum we also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of eop

information entropy of conditionally exactly solvable potentials we evaluate shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials the bialynicki birula mycielski inequality has also been tested for a number of states

from krall discrete orthogonal polynomials to krall polynomials abstract we show how to get krall polynomials from krall discrete polynomials using a procedure of passing to the limit in some of the parameters of the family we also show that this procedure has to be different to the standard one used in the askey scheme to go from the classical discrete polynomials to the classical polynomials

real time coding with limited lookahead a real time coding system with lookahead consists of a memoryless source a memoryless channel an encoder which encodes the source symbols sequentially with knowledge of future source symbols up to a fixed finite lookahead d with or without feedback of the past channel output symbols and a decoder which sequentially constructs the source symbols using the channel output the objective is to minimize the expected per symbol distortion for a fixed finite lookahead d 1 we invoke the theory of controlled markov chains to obtain an average cost optimality equation acoe the solution of which denoted by d d is the minimum expected per symbol distortion with increasing d d d bridges the gap between causal encoding d 0 where symbol by symbol encoding decoding is optimal and the infinite lookahead case d where shannon theoretic arguments show that separation is optimal we extend the analysis to a system with finite state decoders with or without noise free feedback for a bernoulli source and binary symmetric channel under hamming loss we compute the optimal distortion for various source and channel parameters and thus obtain computable bounds on d d we also identify regions of source and channel parameters where symbol by symbol encoding decoding is suboptimal finally we demonstrate the wide applicability of our approach by applying it in additional coding scenarios such as the case where the sequential decoder can take cost constrained actions affecting the quality or availability of side information about the source

two step rational extensions of the harmonic oscillator exceptional orthogonal polynomials and ladder operators the type iii hermite xm exceptional orthogonal polynomial family is generalized to a double indexed one with m1 even and m2 odd such that m2 m1 and the corresponding rational extensions of the harmonic oscillator are constructed by using second order supersymmetric quantum mechanics the new polynomials are proved to be expressible in terms of mixed products of hermite and pseudo hermite ones while some of the associated potentials are linked with rational solutions of the painlev iv equation a novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial heisenberg algebra of order m2 m1 1 which may alternatively be interpreted in terms of a special type of m2 m1 2 th order shape invariance property

a spectral study of the second order exceptional x1 jacobi differential expression and a related non classical jacobi differential expression abstract the exceptional x 1 jacobi differential expression is a second order ordinary differential expression with rational coefficients it was discovered by gomez ullate kamran and milson in 2009 in their work they showed that there is a sequence of polynomial eigenfunctions p n n 1 called the exceptional x 1 jacobi polynomials there is no exceptional x 1 jacobi polynomial of degree zero these polynomials form a complete orthogonal set in the weighted hilbert space l 2 1 1 w where w is a positive rational weight function related to the classical jacobi weight among other conditions placed on the parameters and it is required that 0 in this paper we develop the spectral theory of this expression in l 2 1 1 w we also consider the spectral analysis of the extreme non exceptional case namely when 0 in this case the polynomial solutions are the non classical jacobi polynomials p n 2 n 2 we study the corresponding jacobi differential expression in several hilbert spaces including their natural l 2 setting and a certain sobolev space s where the full sequence p n 2 n 0 is studied and a careful spectral analysis of the jacobi expression is carried out

conditionally exactly solvable potentials and exceptional orthogonal polynomials it is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials some properties of such polynomials e g recurrence relation ladder operators differential equations etc have been obtained

solvable rational extensions of the morse and kepler coulomb potentials we show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential this is made by using darboux backlund transformations based on unphysical regular riccati schrodinger functions which are obtained from specific symmetries associated with the considered family of potentials

equivalences of the multi indexed orthogonal polynomials multi indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion multi indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions for multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types two different index sets may give equivalent multi indexed orthogonal polynomials we clarify these equivalences multi indexed orthogonal polynomials with both type i and ii indices are proportional to those of type i indices only or type ii indices only with shifted parameters

a generalization of a theorem of bochner we describe all polynomial solutions to the general second order operator equation in the askey wilson operators thus extending a theorem of bochner from differential operators to askey wilson divided difference operators

extended calogero models a construction for exactly solvable kn body systems we propose a systematic procedure for the construction of exactly solvable kn body systems which are natural generalisations of calogero models as examples we present two new 3n body models and determine explicit expressions for their eigenvalues and eigenfunctions

tridiagonal pairs of q racah type abstract let f denote an algebraically closed 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 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 a v 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 a w w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d for 0 i d let i resp i denote the eigenvalue of a resp a associated with v i resp v i the pair a a is said to have q racah type whenever i a b q 2 i d c q d 2 i and i a b q 2 i d c q d 2 i for 0 i d where q a b c a b c are scalars in f with q b c b c nonzero and q 2 1 1 this type is the most general one we classify up to isomorphism the tridiagonal pairs over f that have q racah type our proof involves the representation theory of the quantum affine algebra u q sl 2

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

extending sdl and lmc complexity measures to quantum states an extension of sdl shiner davison landsberg and lmc lopez ruiz mancini calbet complexity measures is proposed for the quantum information context considering that von neumann entropy is a natural disorder quantifier for quantum states as a first example of application the simple qubit was studied presenting results similar to that obtained by applying sdl and lmc measures to a classical probability distribution then for the werner state a mixture of bell states sdl and lmc measures were calculated depending on the mixing factor providing some conjectures concerning quantum systems

successive refinement with decoder cooperation and its channel coding duals we study cooperation in multiterminal source coding models involving successive refinement specifically we study the case of a single encoder and two decoders where the encoder provides a common description to both the decoders and a private description to only one of the decoders the decoders cooperate via cribbing i e the decoder with access only to the common description is allowed to observe in addition a deterministic function of the reconstruction symbols produced by the other we characterize the fundamental performance limits in the respective settings of noncausal strictly causal and causal cribbing we use a coding scheme referred to as forward encoding and block markov decoding which builds on one recently used by cuff and zhao for coordination via implicit communication finally we use the insight gained to introduce and solve some dual channel coding scenarios involving multiple access channels with cribbing

deformed oscillator algebra approach of some quantum superintegrable lissajous systems on the sphere and of their rational extensions we extend the construction of 2d superintegrable hamiltonians with separation of variables in spherical coordinates using combinations of shift ladder and supercharge operators to models involving rational extensions of the two parameter lissajous systems on the sphere these new families of superintegrable systems with integrals of arbitrary order are connected with jacobi exceptional orthogonal polynomials of type i or ii and supersymmetric quantum mechanics moreover we present an algebraic derivation of the degenerate energy spectrum for the one and two parameter lissajous systems and the rationally extended models these results are based on finitely generated polynomial algebras casimir operators realizations as deformed oscillator algebras and finite dimensional unitary representations such results have only been established so far for 2d superintegrable systems separable in cartesian coordinates which are related to a class of polynomial algebras that display a simpler structure we also point out how the structure function of these deformed oscillator algebras is directly related with the generalized heisenberg algebras spanned by the nonpolynomial integrals

multi indexed q racah polynomials as the second stage of the project multi indexed orthogonal polynomials we present in the framework of discrete quantum mechanics with real shifts in one dimension the multi indexed q racah polynomials they are obtained from the q racah polynomials by the multiple application of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state vectors in a similar way to the multi indexed laguerre and jacobi polynomials reported earlier the virtual state vectors are the solutions of the matrix schrodinger equation with negative eigenvalues except for one of the two boundary points

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

the generalized quantum isotonic oscillator recently it has been proved that a nonlinear quantum oscillator generalization of the isotonic one is exactly solvable for certain values of its parameters here we show that the schrodinger equation for such an oscillator can be transformed into a confluent heun equation we give a very simple and efficient algorithm to solve it numerically no matter what the values of the parameters are algebraic quasi polynomial solutions for particular values of the parameters are found

the generalized confluent supersymmetry algorithm representations and integral formulas we extend the confluent version of the higher order supersymmetry susy formalism to general linear differential equations of second order closed form representations of transformation functions their wronskians and of the general solutions to susy transformed equations are derived we use these results to construct formulas for resolving multiple integrals of special functions in terms of wronskians

infinite families of non hermitian hamiltonians associated with exceptional xm jacobi polynomials using an appropriate change of variable the schr dinger equation is transformed into a second order differential equation satisfied by recently discovered jacobi type xm exceptional orthogonal polynomials this facilitates the derivation of infinite families of exactly solvable hermitian as well as non hermitian trigonometric scarf potentials and a finite number of hermitian and an infinite number of non hermitian symmetric hyperbolic scarf potentials the bound state solutions of all these potentials are associated with the aforesaid exceptional orthogonal polynomials these infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic scarf potentials by the addition of some rational terms characterized by the presence of classical jacobi polynomials all the members of a particular family of these rationally extended polynomial dependent potentials have the same energy spectrum and possess translational shape invariant symmetry the obtained non hermitian trigonometric scarf potentials are shown to be quasi hermitian in nature ensuring the reality of the associated energy spectra

probing capacity we consider the problem of optimal probing of states of a channel by transmitter and receiver for maximizing rate of reliable communication the channel is discrete memoryless dmc with i i d states the encoder takes probing actions dependent on the message it then uses the state information obtained from probing causally or noncausally to generate channel input symbols the decoder may also take channel probing actions as a function of the observed channel output and use the channel state information thus acquired along with the channel output to estimate the message we refer to the maximum achievable rate for reliable communication for such systems as the probing capacity we characterize this capacity when the encoder and decoder actions are cost constrained to motivate the problem we begin by characterizing the trade off between the capacity and fraction of channel states the encoder is allowed to observe while the decoder is aware of channel states in this setting of to observe or not to observe state at the encoder we compute certain numerical examples which exhibit a pleasing phenomenon where encoder can observe a relatively small fraction of states and yet communicate at maximum rate i e rate when observing states at encoder is not cost constrained

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

exceptional orthogonal polynomials and the darboux transformation we adapt the notion of the darboux transformation to the context of polynomial sturm liouville problems as an application we characterize the recently described xm laguerre polynomials in terms of an isospectral darboux transformation we also show that the shape invariance of these new polynomial families is a direct consequence of the permutability property of the darboux crum transformation

new ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two dimensional systems new ladder operators are constructed for a rational extension of the harmonic oscillator associated with type iii hermite exceptional orthogonal polynomials and characterized by an even integer m the eigenstates of the hamiltonian separate into m 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebra these ladder operators are used to construct a higher order integral of motion for two superintegrable two dimensional systems separable in cartesian coordinates the polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite dimensional unitary irreducible representations

orthogonality of the jacobi polynomials with negative integer parameters it is well known that the jacobi polynomials pn x are orthogonal with respect to a quasi definite linear functional whenever and 1 are not negative integer numbers recently sobolev orthogonality for these polynomials has been obtained for a negative integer and not a negative integer and also for the case negative integer numbers in this paper we give a sobolev orthogonality for the jacobi polynomials in the remainder cases

recurrence relations of the multi indexed orthogonal polynomials in paper ii we presented conjectures of the recurrence relations with constant coefficients for the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types in this paper we present a proof for the laguerre and jacobi cases their bispectral properties are also discussed which gives a method to obtain the coefficients of the recurrence relations explicitly this paper extends to the laguerre and jacobi cases the bispectral techniques recently introduced by gomez ullate et al j approx theory 204 1 2016 e print arxiv 1506 03651 math ca to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of hermite type

multi indexed jacobi polynomials and maya diagrams multi indexed jacobi polynomials are defined by the wronskian of four types of eigenfunctions of the poschl teller hamiltonian we give a correspondence between multi indexed jacobi polynomials and pairs of maya diagrams and we show that any multi indexed jacobi polynomial is essentially equal to some multi indexed jacobi polynomial of two types of eigenfunction as an application we show a wronskian type formula of some special eigenstates of the deformed poschl teller hamiltonian

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

admissibility condition for exceptional laguerre polynomials abstract we prove a necessary and sufficient condition for the integrability of the weight associated to the exceptional laguerre polynomials this condition is very much related to the fact that the associated second order differential operator has no singularities in 0

classical quantum correspondence for shape invariant systems a quantization procedure which has recently been introduced for the analysis of painleve equations is applied to a general time independent potential of a newton equation this analysis shows that the quantization procedure preserves the exact solvability property for the class of shape invariant potentials when a general potential is considered the quantization procedure involves the solution of a gambier xxvii transcendental equation explicit examples involving classical and exceptional orthogonal laguerre and jacobi polynomials are discussed

invariance properties of wronskian type determinants of classical and classical discrete orthogonal polynomials abstract given a finite set f f 1 f k of nonnegative integers written in increasing order of magnitude and a classical discrete family p n n of orthogonal polynomials charlier meixner krawtchouk or hahn we consider the casorati determinant det p f i x j 1 i j 1 k in this paper we prove an invariance property of this kind of casorati determinants when the set f is substituted by the set i f 0 1 2 max f max f f f f our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators krall discrete polynomials these polynomials are orthogonal with respect to certain christoffel transforms of the classical discrete measures by passing to the limit this invariance property is extended to wronskian type determinants whose entries are hermite laguerre and jacobi polynomials

heun s equation generalized hypergeometric function and exceptional jacobi polynomial we study heun s differential equation in the case that one of the singularities is apparent in particular we propose a conjecture that solutions of heun s equation in this case also satisfy a generalized hypergeometric equation which can be described in a more general form and establish it in some cases we demonstrate the application of our results to exceptional jacobi polynomials

infinitely many shape invariant potentials and cubic identities of the laguerre and jacobi polynomials we provide analytic proofs for the shape invariance of the recently discovered odake and sasaki phys lett b 679 414 2009 two families of infinitely many exactly solvable one dimensional quantum mechanical potentials these potentials are obtained by deforming the well known radial oscillator potential or the darboux poschl teller potential by a degree l l 1 2 eigenpolynomial the shape invariance conditions are attributed to new polynomial identities of degree 3l involving cubic products of the laguerre or jacobi polynomials these identities are proved elementarily by combining simple identities

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

combined state adding and state deleting approaches to type iii multi step rationally extended potentials applications to ladder operators and superintegrability type iii multi step rationally extended harmonic oscillator and radial harmonic oscillator potentials characterized by a set of k integers m1 m2 mk such that m1 m2 mk with mi even resp odd for i odd resp even are considered the state adding and state deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators the eigenstates of the hamiltonians are shown to separate into mk 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebras these ladder operators are then used to build a higher order integral of motion for seven new infinite families of superintegrable two dimensional systems separable in cartesian coordinates the finite dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one dimensional hamiltonian eigenstates and provide an algebr

recurrence relations of the multi indexed orthogonal polynomials iv closure relations and creation annihilation operators we consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types corresponding to the recurrence relations with constant coefficients for the m indexed orthogonal polynomials it is expected that the systems satisfy the generalized closure relations in fact we can verify this statement for small m examples the generalized closure relation gives the exact heisenberg operator solution of a certain operator from which the creation and annihilation operators of the system are obtained

a bochner type characterization theorem for exceptional orthogonal polynomials abstract it was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of darboux transformations in this paper we prove this conjecture which paves the road to a complete classification of all exceptional orthogonal polynomials in some sense this paper can be regarded as the extension of bochner s result for classical orthogonal polynomials to the exceptional class as a supplementary result we derive a canonical form for exceptional operators based on a bilinear formalism and prove that every exceptional operator has trivial monodromy at all primary poles

time dependent potentials associated with exceptional orthogonal polynomials we introduce a method for generating rational extensions of time dependent potentials such that the associated schrodinger equation admits solutions in terms of exceptional orthogonal polynomials our method is applicable to position dependent schrodinger equations as well as to their conventional counterparts for constant mass

exceptional orthogonal polynomials and generalized schur polynomials we show that the exceptional orthogonal polynomials can be viewed as confluent limits of the generalized schur polynomials introduced by sergeev and veselov

a class of exactly solvable rationally extended non central potentials in two and three dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions

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

multi indexed meixner and little q jacobi laguerre polynomials as the fourth stage of the project multi indexed orthogonal polynomials we present the multi indexed meixner and little q jacobi laguerre polynomials in the framework of discrete quantum mechanics with real shifts defined on the semi infinite lattice in one dimension they are obtained in a similar way to the multi indexed laguerre and jacobi polynomials reported earlier from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state vectors the virtual state vectors are the solutions of the matrix schrodinger equation on all the lattice points having negative energies and infinite norm this is in good contrast to the q racah systems defined on a finite lattice in which the virtual state vectors satisfy the matrix schrodinger equation except for one of the two boundary points

perturbations around the zeros of classical orthogonal polynomials starting from degree n solutions of a time dependent schrodinger like equation for classical orthogonal polynomials a linear matrix equation describing perturbations around the n zeros of the polynomial is derived the matrix has remarkable diophantine properties its eigenvalues are independent of the zeros the corresponding eigenvectors provide the representations of the lower degree 0 1 n 1 polynomials in terms of the zeros of the degree n polynomial the results are valid universally for all the classical orthogonal polynomials including the askey scheme of hypergeometric orthogonal polynomials and its q analogues

whittaker hill equation and semifinite gap schrodinger operators a periodic one dimensional schrodinger operator is called semifinite gap if every second gap in its spectrum is eventually closed we construct explicit examples of semifinite gap schrodinger operators in trigonometric functions by applying darboux transformations to the whittaker hill equation we give a criterion for the regularity of the corresponding potentials and investigate the spectral properties of the new operators

rational extensions of the trigonometric darboux poschl teller potential based on para jacobi polynomials the possibility for the jacobi equation to admit in some cases general solutions that are polynomials has been recently highlighted by calogero and yi who termed them para jacobi polynomials such polynomials are used here to build seed functions of a darboux backlund transformation for the trigonometric darboux poschl teller potential as a result one step regular rational extensions of the latter depending both on an integer index n and on a continuously varying parameter are constructed for each n value the eigenstates of these extended potentials are associated with a novel family of dependent polynomials which are orthogonal on 1 1

the q version of a theorem of bochner askey and wilson 1985 found a family of orthogonal polynomials in the variable s k 1 2 k 1 k that satisfy a q difference equation of the form a k p n s qk p n s k b k p n s k q p n s k theta n p n s k n 0 1 we show here that this property characterizes the askey wilson polynomials the proof is based on an operator identity of independent interest this identity can be adapted to prove other characterization results indeed it was used in grunbaum and haine 1996 to give a new derivation of the result of bochner alluded to in the title of this paper we give the appropriate identity for the case of difference equations leading to the wilson polynomials but pursue the consequences only in the case of q difference equations leading to the askey wilson and big q jacobi polynomials this approach also works in the discrete case and should yield the results in leonard 1982

the capacity of the semideterministic relay channel corresp the capacity of the class of relay channels with sender x 1 a relay sender x 2 a relay receiver y 1 f x 1 x 2 and ultimate receiver y is proved to be c max min p x 1 x 2 i x 1 x 2 y h y 1 x 2 i x 1 y x 2 y 1

simple measure for complexity a measure of complexity is proposed based on appropriately defined notions of order and disorder which has a considerable degree of flexibility in its dependence on these concepts the possible functional dependencies which result encompass those of many earlier definitions of complexity the proposed measure is in principle easy to calculate and has the property of an intensive thermodynamic quantity with appropriate choices of parameters it behaves similarly to effective measure complexity for the logistic map it is also a generalization of the normalized complexity of lopez ruiz et al but does not suffer from over universality

recurrence relations of the multi indexed orthogonal polynomials v racah and q racah types in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson and askey wilson types in this paper we explore those of the racah and q racah types for the m indexed q racah polynomials we derive 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi indexed q racah polynomials are obtained in appendix b and appendix c we present a proof and some data of the recurrence relations with constant coefficients for the multi indexed wilson and askey wilson polynomials in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson and askey wilson types in this paper we explore those of the racah and q racah types for the m indexed q racah polynomials we derive 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi indexed q racah polynomials are obtained in appendix b and appendix c we present a proof and some data of the recurrence relations with constant coefficients for the multi indexed wilson and askey wilson polynomials

n fold supersymmetry and quasi solvability associated with x2 laguerre polynomials we construct a new family of quasi solvable and n fold supersymmetric quantum systems where each hamiltonian preserves an exceptional polynomial subspace of codimension 2 we show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the x2 laguerre polynomials of the second kind in addition we find that the two kinds of the x2 laguerre polynomials are ingeniously connected with each other by the n fold supercharge

multiple access channel with partial and controlled cribbing encoders in this paper we consider a multiple access channel mac with partial cribbing encoders this means that each of the two encoders obtains a deterministic function of the output of the other encoder with or without delay the partial cribbing scheme is especially motivated by the additive noise gaussian mac where perfect cribbing results in the degenerated case of full cooperation between the encoders and requires an infinite entropy link we derive a single letter characterization of the capacity of the mac with partial cribbing for the cases of causal and strictly causal cribbing several numerical examples such as those of quantized cribbing are presented we further consider and derive the capacity region where the cribbing depends on actions that are functions of the previous cribbed observations in particular we consider a scenario where the action is taken to decide to crib or not to crib and show that a naive time sharing strategy is not optimal

complexity measures from interaction structures we evaluate information theoretic quantities that quantify complexity in terms of k text th order statistical dependences that cannot be reduced to interactions among k ensuremath 1 random variables using symbolic dynamics of coupled maps and cellular automata as model systems we demonstrate that these measures are able to identify complex dynamical regimes

quantization of the lienard ii equation and jacobi s last multiplier in this paper the role of jacobi s last multiplier in mechanical systems with a position dependent mass is unveiled in particular we map the lienard ii equation to a position dependent mass system the quantization of the lienard ii equation is then carried out using the point canonical transformation method together with the von roos ordering technique finally we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated laguerre and exceptional laguerre functions

exceptional laguerre and jacobi polynomials and the corresponding potentials through darboux crum transformations a simple derivation is presented of the four families of infinitely many shapeinvariant hamiltonians corresponding to the exceptional laguerre and jacobi polynomials the darboux crum transformations are applied to connect the well known shape invariant hamiltonians of the radial oscillator and the darboux p oschl teller potential to the shape invariant potentials of odake sasaki dutta and roy derived the two lowest members of the exceptional laguerre polynomials by this method the method is expanded to its full generality and many other ramifications including the aspects of the generalized bochner problem and the bispectral property of the exceptional orthogonal polynomials are discussed

disconjugacy regularity of multi indexed rationally extended potentials and laguerre exceptional polynomials the power of the disconjugacy properties of second order differential equations of schrodinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re examining the extensions of the isotonic oscillator or radial oscillator potential derived in kth order supersymmetric quantum mechanics or multistep darboux backlund transformation method the function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term whose value is calculated by induction over k the sign of this term being the same as that of the already known highest degree term the potential denominator has the same sign at both extremities of the definition interval a property that is shared by the seed eigenfunction used in the potential construction by virtue of disconjugacy such a property implies the nodeless character of both the eigenfunction and the resulting potential

multi indexed wilson and askey wilson polynomials as the third stage of the project multi indexed orthogonal polynomials we present in the framework of discrete quantum mechanics with pure imaginary shifts in one dimension the multi indexed wilson and askey wilson polynomials they are obtained from the original wilson and askey wilson polynomials by multiple applications of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state solutions of types i and ii in a similar way to the multi indexed laguerre jacobi and q racah polynomials reported earlier

noncommutative bispectral darboux transformations we prove a general theorem establishing the bispectrality of noncommutative darboux transformations it has a wide range of applications that establish bispectrality of such transformations for differential difference and q difference operators with values in all noncommutative algebras all known bispectral darboux transformations are special cases of the theorem using the methods of quasideterminants and the spectral theory of matrix polynomials we explicitly classify the set of bispectral darboux transformations from rank one differential operators and airy operators with values in matrix algebras these sets generalize the classical calogero moser spaces and wilson s adelic grassmannian

extended krein adler theorem for the translationally shape invariant potentials considering successive extensions of primary translationally shape invariant potentials we enlarge the krein adler theorem to mixed chains of state adding and state deleting darboux backlund transformations it allows us to establish novel bi linear wronskian and determinantal identities for classical orthogonal polynomials

some functions that generalize the askey wilson polynomials we determine all biinfinite tridiagonal matrices for which some family of eigenfunctions are also eigenfunctions of a second order q difference operator the solution is described in terms of an arbitrary solution of a q analogue of gauss hypergeometric equation depending on five free parameters and extends the four dimensional family of solutions given by the askey wilson polynomials there is some evidence that this bispectral problem for an arbitrary order q difference operator is intimately related with some q deformation of the toda lattice hierarchy and its virasoro symmetries when tridiagonal matrices are replaced by the schroedinger operator and q 1 this statement holds with toda replaced by kdv in this context this paper determines the analogs of the bessel and airy potentials

exceptional askey wilson type polynomials through darboux crum transformations an alternative derivation is presented of the infinitely many exceptional wilson and askey wilson polynomials which were introduced by the present authors in 2009 darboux crum transformations intertwining the discrete quantum mechanical systems of the original and the exceptional polynomials play an important role infinitely many continuous hahn polynomials are derived in the same manner the present method provides a simple proof of the shape invariance of these systems as in the corresponding cases of the exceptional laguerre and jacobi polynomials

a new family of shape invariantly deformed darboux poschl teller potentials with continuous l we present a new family of shape invariant potentials which could be called a continuous version of the potentials corresponding to the exceptional x j1 jacobi polynomials constructed recently by the present authors in a certain limit it reduces to a continuous family of shape invariant potentials related to the exceptional x l1 laguerre polynomials the latter was known as one example of the conditionally exactly solvable potentials on a half line

families of superintegrable hamiltonians constructed from exceptional polynomials we introduce a family of exactly solvable two dimensional hamiltonians whose wave functions are given in terms of laguerre and exceptional jacobi polynomials the hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems additional higher order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable

recurrence relations of the multi indexed orthogonal polynomials vi meixner pollaczek and continuous hahn types in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson askey wilson racah and q racah types in this paper we explore those of the meixner pollaczek and continuous hahn types for the m indexed meixner pollaczek and continuous hahn polynomials we present 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation annihilation operators of the quantum mechanical systems described by the multi indexed meixner pollaczek and continuous hahn polynomials are obtained

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

non polynomial extensions of solvable potentials a la abraham moses abraham moses transformations besides darboux transformations are well known procedures to generate extensions of solvable potentials in one dimensional quantum mechanics here we present the explicit forms of infinitely many seed solutions for adding eigenstates at arbitrary real energy through the abraham moses transformations for typical solvable potentials e g the radial oscillator the darboux poschl teller and some others these seed solutions are simple generalisations of the virtual state wavefunctions which are obtained from the eigenfunctions by discrete symmetries of the potentials the virtual state wavefunctions have been an essential ingredient for constructing multi indexed laguerre and jacobi polynomials through multiple darboux crum transformations in contrast to the darboux transformations the virtual state wavefunctions generate non polynomial extensions of solvable potentials through the abraham moses transformations

rational extension and jacobi type xm solutions of a quantum nonlinear oscillator we construct a rational extension of a recently studied nonlinear quantum oscillator model our extended model is shown to retain exact solvability admitting a discrete spectrum and corresponding closed form solutions that are expressed through jacobi type xm exceptional orthogonal polynomials

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them