supersymmetry and algebraic darboux transformations we describe a class of algebraically solvable susy models by considering the deformation of invariant polynomial flags by means of the darboux transformation the algebraic deformations corresponding to the addition of a bound state to a shape invariant potential are particularly interesting the polynomial flags in question are indexed by a deformation parameter m 1 2 and lead to new algebraically solvable models we illustrate these ideas by considering deformations of the hyperbolic p schl teller potential

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

orthogonal polynomials satisfying fourth order differential equations since they are rather important and quite accessible we repeat the general theoretical facts concerning weights moments and polynomials pertaining to fourth order differential equations we then briefly discuss the squares of the differential equations of the second order giving a number of easily derived examples of fourth order problems this is followed by three new orthogonal polynomial sets satisfying fourth order differential equations but which do not satisfy second order differential equations

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

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

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

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

quasi exact solvability and the direct approach to invariant subspaces we propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of this approach is used here to construct new exactly solvable and quasi exactly solvable quantum hamiltonians on the line which are not lie algebraic it is also applied to generate potentials with multiple algebraic sectors we discuss two illustrative examples of these two applications we show that the generalized lame potential possesses four algebraic sectors and describe a quasi exactly solvable deformation of the morse potential which is not lie algebraic

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

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

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

rational spectral transformations and orthogonal polynomials abstract we show that generic rational transformations of the stieltjes function with polynomial coefficients st can be presented as a finite superposition of four fundamental elementary transforms christoffel transform ct geronimus transform gt and forward and backward associated transformations a t a t it is shown that the laguerre hahn polynomials lhp on arbitrary nonuniform lattice are covariant with respect to st i e st of a lhp yields another lhp whereas the semi classical polynomials are covariant with respect to a subclass of linear st some applications of these results to the theory of the semi classical polynomials are considered

the octahedral algorithm a new simplicial fixed point algorithm a new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented it uses the restrart technique of merrill to improve the accuracy of the solution the algorithm is shown to converge quadratically under certain conditions the algorithm should be efficient and relatively easy to implement

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

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

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

on the zeros of a class of generalized hypergeometric polynomials we obtain the asymptotic behavior of the zeros of a class of generalized hypergeometric polynomials for this purpose we make use of a mehler heine type formula for these polynomials we illustrate these results with numerical experiments and some figures

darboux transformation and perturbation of linear functionals abstract let l be a quasi definite linear functional defined on the linear space of polynomials with real coefficients in the literature three canonical transformations of this functional are studied xl l c x and 1 x l c x where x denotes the linear functional x xk k 0 and k 0 is the kronecker symbol let us consider the sequence of monic polynomials orthogonal with respect to l this sequence satisfies a three term recurrence relation whose coefficients are the entries of the so called monic jacobi matrix in this paper we show how to find the monic jacobi matrix associated with the three canonical perturbations in terms of the monic jacobi matrix associated with l the main tools are darboux transformations in the case that the lu factorization of the monic jacobi matrix associated with xl does not exist and darboux transformation does not work we show how to obtain the monic jacobi matrix associated with x 2 l as a limit case we also study perturbations of the functional l that are obtained by combining the canonical cases finally we present explicit algebraic relations between the polynomials orthogonal with respect to l and orthogonal with respect to the perturbed functionals

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

alpha plane representation for type 2 fuzzy sets theory and applications this paper 1 reviews the alpha plane representation of a type 2 fuzzy set t2 fs which is a representation that is comparable to the alpha cut representation of a type 1 fs t1 fs and is useful for both theoretical and computational studies of and for t2 fss 2 proves that set theoretic operations for t2 fss can be computed using very simple alpha plane computations that are the set theoretic operations for interval t2 it2 fss 3 reviews how the centroid of a t2 fs can be computed using alpha plane computations that are also very simple because they can be performed using existing karnik mendel algorithms that are applied to each alpha plane 4 shows how many theoretically based geometrical properties can be obtained about the centroid even before the centroid is computed 5 provides examples that show that the mean value defuzzified value of the centroid can often be approximated by using the centroids of only 0 and 1 alpha planes of a t2 fs 6 examines a triangle quasi t2 fuzzy logic system q t2 fls whose secondary membership functions are triangles and for which all calculations use existing t1 or it2 fs mathematics and hence they may be a good next step in the hierarchy of flss from t1 to it2 to t2 and 7 compares t1 it2 and triangle q t2 flss to forecast noise corrupted measurements of a chaotic mackey glass time series

asymptotics for varying discrete sobolev orthogonal polynomials abstract we consider a varying discrete sobolev inner product such as f g s f x g x d m n f j c g j c where is a finite positive borel measure supported on an infinite subset of the real line c is adequately located on the real axis j 0 and mn n 0 is a sequence of nonnegative real numbers satisfying a very general condition our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this sobolev inner product in this way we focus our attention on mehler heine type formulae as they describe in detail the asymptotic behavior of these polynomials around c just the point where we have located the perturbation of the standard inner product moreover we pay attention to the asymptotic behavior of the scaled zeros of these varying sobolev polynomials and some numerical experiments are shown finally we provide other asymptotic results which strengthen the idea that mehler heine asymptotics describe in a precise way the differences between sobolev orthogonal polynomials and standard ones

the darboux transformation and algebraic deformations of shape invariant potentials we investigate the backward darboux transformations addition of the lowest bound state of shape invariant potentials on the line and classify the subclass of algebraic deformations those for which the potential and the bound states are simple elementary functions a countable family m 0 1 2 of deformations exists for each family of shape invariant potentials we prove that the m th deformation is exactly solvable by polynomials meaning that it leaves invariant an infinite flag of polynomial modules p m m p m 1 m where p n m is a codimension m subspace of in particular we prove that the first m 1 algebraic deformation of the shape invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules p n 1 by construction these algebraically deformed hamiltonians do not have an sl 2 hidden symmetry algebra structure

zeros of wronskians of hermite polynomials and young diagrams abstract for a certain class of partitions a simple qualitative relation is observed between the shape of the young diagram and the pattern of zeros of the wronskian of the corresponding hermite polynomials in the case of the two term wronskian w h n h n k we give an explicit formula for the asymptotic shape of the zero set as n some empirical asymptotic formulas are given for the zero sets of three term and four term wronskians

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

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

a better triangulation for wright s 2n ray algorithm abstract we give a triangulation for wright s 2n ray algorithm to compute solutions of nonlinear equations according to measures of efficiency of triangulations it is better than any other available triangulation for the 2n ray algorithm

moment representations of exceptional x1 orthogonal polynomials abstract we generalize the representations of x 1 exceptional orthogonal polynomials through determinants of matrices that have certain adjusted moments as entries we start out directly from the darboux transformation allowing for a universal perspective rather than one dependent upon the particular system jacobi or type of laguerre polynomials we include a recursion formula for the adjusted moments and provide the initial adjusted moments for each system throughout we relate to the various examples of x 1 exceptional orthogonal polynomials we especially focus on and provide complete proofs for the jacobi and the type iii laguerre case as they are less prevalent in literature lastly we include a preliminary discussion explaining that the higher codimension setting becomes more involved the number of possibilities and choices is exemplified but only starts with the lack of a canonical flag

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

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