identities and exponential bounds for transfer matrices this paper is about analytic properties of single transfer matrices originating from general block tridiagonal or banded matrices such matrices occur in various applications in physics and numerical analysis the eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity if the block tridiagonal matrix is invertible it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain and the other half have an upper bound that is exponentially small this is a consequence of a theorem by demko moss and smith on the decay of matrix elements of the inverse of banded matrices this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

quantum 2 spheres and big q jacobi polynomials orthogonal bases for the algebras of functions of podles quantum 2 spheres are explicitly determined in terms of bigq jacobi polynomials this gives a group theoretic interpretation of the symmetric bigq jacobi polynomials and the symmetricq hahn polynomials

what s decidable about weighted automata abstract weighted automata map input words to values and have numerous applications in computer science a result by krob from the 90s implies that the universality problem is decidable for weighted automata over the tropical semiring with weights in n and is undecidable when the weights are in z we continue the study of the borders of decidability in weighted automata over the tropical semiring we give a complete picture of the decidability and complexity of various decision problems for them including non emptiness universality equality and containment for the undecidability results we provide direct proofs which stay in the terrain of state machines this enables us to tighten the results and apply them to a very simple class of automata in addition we provide a toolbox of algorithms and techniques for weighted automata on top of which we establish the complexity bounds

limitedness theorem on finite automata with distance functions an algebraic proof a distance function on a finite automaton m is defined by assigning to each transition a distance of nonnegative integer value m is said to be limited in distance if there is a nonnegative integer k such that for each accepted string w in the language of m there is an accepting path p for w on which the sum of distances is bounded by k the limitedness theorem 6 9 on finite automata with distance functions states that it is decidable if an arbitrary finite automaton with a distance function is limited in distance in this paper we give an algebraic proof of the theorem and derive from it an exponential time decision algorithm in addition we prove that the decision problem is pspace hard

determinants of block tridiagonal matrices an identity is proven that evaluates the determinant of a block tridiagonal matrix with or without corners as the determinant of the associated transfer matrix or a submatrix of it

infinite toeplitz and laurent matrices with localized impurities this paper is concerned with the change of the spectra of infinite toeplitz and laurent matrices under perturbations in a prescribed finite set of sites the main result says that the spectrum of a toeplitz matrix with a non constant rational symbol is not affected by small localized impurities while such impurities can nevertheless enlarge the spectrum of the corresponding laurent matrix we also study the spectra that may emerge when randomly perturbing toeplitz or laurent matrices in a randomly chosen single site

transfer matrices non hermitian hamiltonians and resolvents some spectral identities i consider the n step transfer matrix t for a general block hamiltonian with eigenvalue equation where and are matrices and provide its explicit representation in terms of blocks of the resolvent of the hamiltonian matrix for the system of length n with boundary conditions i then introduce the related hamiltonian for the case and and provide an exact relation between the trace of its resolvent and together with an identity of thouless type connecting tr with the hamiltonian eigenvalues for the results are then extended to by showing that it is itself a transfer matrix besides being of mathematical interest the identities should be useful for an analytical approach in the study of spectral properties of a physically relevant class of transfer matrices

transfer matrices and tridiagonal block hamiltonians with periodic and scattering boundary conditions general hamiltonian matrices with tridiagonal block structure and the associated transfer matrices are investigated in the cases of periodic and scattering boundary conditions they arise from tight binding models with finite range hopping in one or more dimensions of space in the presence of a aharonov bohm flux or in multichannel scattering an identity relating the characteristic equation of the periodic hamiltonian with that of the transfer matrix is found allowing a detailed analysis of the bands a velocity matrix is defined with properties relevant for the band structure or for the channel structure in the scattering problem

finite sections of random jacobi operators this article is about a problem in the numerical analysis of random operators we study a version of the finite section method for the approximate solution of equations ax b in infinitely many variables where a is a random jacobi i e tridiagonal operator in other words we approximately solve infinite second order difference equations with stochastic coefficients by reducing the infinite volume case to the large finite volume case via a particular truncation technique for most of the paper we consider non self adjoint operators a but we also comment on the self adjoint case when simplifications occur

pseudospectra of the convection diffusion operator the spectrum of the simplest 1d convection diffusion operator is a discrete subset of the negative real axis but the pseudospectra are regions in the complex plane that approximate parabolas put another way the norm of the resolvent is exponentially large as a function of the peclet number throughout a certain parabolic region these observations have a simple physical basis and suggest that conventional spectral analysis for convection diffusion operators may be of limited value in some applications

random fibonacci sequences solutions to the random fibonacci recurrence xn 1 xn xn 1 decrease increase exponentially xn exp n for sufficiently small large in the limits 0 and we expand the lyapunov exponent in powers of and 1 respectively for the classical case of 1 we obtain exact non perturbative results in particular an invariant measure associated with ricatti variable rn xn 1 xn is shown to exhibit plateaux around all rational r

computation of growth rates of random sequences with multi step memory we extend the generating function approach to the computation of growth rate of random fibonacci sequences with long memory functional iteration equations are obtained and its general form is conjectured and proved based on which an asymptotic representation of the growth rate is obtained the validity of both the derived and the conjectured formula are verified upon comparison with monte carlo simulation a numerical scheme of the functional iteration is designed and implemented successfully

on a generalization of the rogers generating function abstract we derive a generalization of the rogers generating function for the continuous q ultraspherical rogers polynomials whose coefficient is a 1 2 from that expansion we derive corresponding specialization and limit transition expansions for the continuous q hermite continuous q legendre laguerre and chebyshev polynomials of the first kind using a generalized expansion of the rogers generating function in terms of the askey wilson polynomials by ismail simeonov whose coefficient is a 7 8 we derive corresponding generalized expansions for the wilson continuous q jacobi and jacobi polynomials by comparing the coefficients of the askey wilson expansion to our continuous q ultraspherical rogers expansion we derive a new quadratic transformation for basic hypergeometric functions which relates an 7 8 to a 1 2 we also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality

spectral value sets a graphical tool for robustness analysis in this note we present a method to determine and visualize the set of all complex numbers to which at least one eigenvalue of a matrix a can be shifted by real or complex perturbations of the form a a a d e where d and e are fixed matrices and the unknown disturbance matrix satisfies 0 both real and complex perturbations are studied the graphical method may be considered as an extension of the classical root locus method to multivariable systems and multiparameter perturbations the results are illustrated by various examples

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

the spectra of large toeplitz band matrices with a randomly perturbed entry this paper is concerned with the union sp j k tn a of all possible spectra that may emerge when perturbing a large n n toeplitz band matrix tn a in the j k site by a number randomly chosen from some set the main results give descriptive bounds and in several interesting situations even provide complete identifications of the limit of sp j k tn a as n also discussed are the cases of small and large sets as well as the discontinuity of the infinite volume case which means that in general sp j k tn a does not converge to something close to sp j k tn a as n where t a is the corresponding infinite toeplitz matrix illustrations are provided for tridiagonal toeplitz matrices a notable special case

on large toeplitz band matrices with an uncertain block abstract this paper investigates the possible spectra of large finite dimensional toeplitz band matrices with perturbations impurities uncertainties in the upper left m m block the main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations provided they have sufficiently small norm this follows from analysis of structured pseudospectra structured spectral value sets in contrast for typical non hermitian toeplitz matrices there exist certain rank one perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large dimensional limit

on the remarkable spectrum of a non hermitian random matrix model a non hermitian random matrix model proposed a few years ago has a remarkably intricate spectrum various attempts have been made to understand the spectrum but even its dimension is not known using the dyson schmidt equation we show that the spectrum consists of a non denumerable set of lines in the complex plane each line is the support of the spectrum of a periodic hamiltonian obtained by the infinite repetition of any finite sequence of the disorder variables our approach is based on the theory of words we make a complete study of all four letter words the spectrum is complicated because our matrix contains everything that will ever be written in the history of the universe including this particular paper

spectra pseudospectra and localization for random bidiagonal matrices there has been much recent interest initiated by work of the physicists hatano and nelson in the eigenvalues of certain random non hermitian periodic tridiagonal matrices and their bidiagonal limits these eigenvalues cluster along a bubble with wings in the complex plane and the corresponding eigenvectors are localized in the wings delocalized in the bubble here in addition to eigenvalues pseudospectra are analyzed making it possible to treat the nonperiodic analogues of these random matrix problems inside the bubble the resolvent norm grows exponentially with the dimension outside it grows subexponentially in a bounded region that is the spectrum of the infinite dimensional operator localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase respectively with distance from the diagonal this article presents theorems that characterize the spectra pseudospectra and numerical range for the four cases of finite bidiagonal matrices infinite bidiagonal matrices stochastic toeplitz operators finite periodic matrices and doubly infinite bidiagonal matrices stochastic laurent operators c 2001 john wiley sons inc

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

regular spacings of complex eigenvalues in the one dimensional non hermitian anderson model we prove that in dimension one the non real eigenvalues of the non hermitian anderson nha model with a selfaveraging potential are regularly spaced the class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials with probability one as well as all quasi periodic potentials it should be emphasized that our approach here is much simpler than the one we used before it allows us a to investigate the above mentioned spacings b to establish certain properties of the integrated density of states of the hermitian anderson models with selfaveraging potentials and c to obtain as a by product much simpler proofs of our previous results concerned with non real eigenvalues of the nha model

quadratic transformation formulas for basic hypergeometric series starting with some of the known transformation formulas for well poised 2 1 and very well poised 8 7 basic hypergeometric series we obtain q analogues of 36 quadratic transformation formulas given in 2 11 of higher transcendental functions vol 1 edited by erdelyi et al we also derive some new quadratic transformation formulas that give rise to identities connecting very well poised but unbalanced 10 9 series in base q with very well poised and balanced 12 11 series in base q 2 a rogers ramanujan type identity is also found as a limiting case

automatic generation of hypergeometric identities by the beta integral method in this article hypergeometric identities or transformations for p 1fp series and for kampe de feriet series of unit arguments are derived systematically from known transformations of hypergeometric series and products of hypergeometric series respectively using the beta integral method in an automated manner based on the mathematica package hyp as a result we obtain some known and some identities which seem to not have been recorded before in literature

spectral properties of random non self adjoint matrices and operators i describe some numerical experiments which determine the degree of spectral instability of medium sized randomly generated matrices which are far from self adjoint the conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size i also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace but the eigenvalues do not determine the spectrum my results imply that the spectrum of the non self adjoint anderson model changes suddenly on passing to the infinite volume limit

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

non hermitian localization and delocalization we study localization and delocalization in a class of non hermitian hamiltonians inspired by the problem of vortex pinning in superconductors in various simplified models we are able to obtain analytic descriptions in particular of the nonperturbative emergence of a forked structure the appearance of wings in the density of states we calculate how the localization length diverges at the localization delocalization transition we map some versions of this problem onto a random walker problem in two dimensions for a certain model we find an intricate structure in its density of states

convergence analysis of krylov subspace iterations with methods from potential theory krylov subspace iterations are among the best known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices these methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix this leads to an extremal problem in polynomial approximation theory how small can a monic polynomial of a given degree be on the spectrum this survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices it is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure then depending on the number of iterations the lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region which is characterized by means of a constrained equilibrium problem from potential theory the same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

new upper bounds to the limitedness of distance automata abstract a distance automaton is a finite nondeterministic automaton with a distance function which assigns zero or one to each atomic transition and assigns a nonnegative integer to each accepted word by the plus min principle in this paper we prove that the distances of all accepted words of a distance automaton is bounded by some constant if and only if they are bounded by 2 4m 3 m log m 2 m where m is the number of states of the automaton

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

pseudozeros of polynomials and pseudospectra of companion matrices it is well known that the zeros of a polynomial p are equal to the eigenvalues of the associated companion matrix a in this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding the epsilon pseudozero set z epsilon p is the set of zeros of all polynomials hat p obtained by coefficientwise perturbations of p of size leq epsilon this is a subset of the complex plane considered earlier by mosier and is bounded by a certain generalized lemniscate the epsilon pseudospectrum lambda epsilon a is another subset of bbb c defined as the set of eigenvalues of matrices hat a a e with vert e vert leq epsilon it is bounded by a level curve of the resolvent of a we find that if a is first balanced in the usual eispack sense then z epsilon vert p vert p and lambda epsilon vert a vert a are usually quite close to one another it follows that the matlab roots algorithm of balancing the companion matrix then computing its eigenvalues is a stable algorithm for polynomial zerofinding experimental comparisons with the jenkins traub imsl and madsen reid harwell fortran codes confirm that these three algorithms have roughly similar stability properties

enumeration of simple random walks and tridiagonal matrices we present some old and new results in the enumeration of random walks in one dimension mostly developed in work on enumerative combinatorics the relation between the trace of the nth power of a tridiagonal matrix and the enumeration of weighted paths of n steps allows an easier combinatorial enumeration of paths it also seems promising for the theory of tridiagonal random matrices

the limitedness problem on distance automata hashiguchi s method revisited hashiguchi has studied the limitedness problem of distance automata da in a series of paper j comput system sci 24 1982 233 theoret comput sci 72 1990 27 theoret comput sci 233 2000 19 the distance of a da can be limited or unbounded given that the distance of a da is limited hashiguchi has proved in hashiguchi 2000 that the distance of the automaton is bounded by 24n3 n lg n 2 n where n is the number of states in this paper we study again hashiguchi s solution to the limitedness problem we have made a number of simplification and improvement on hashiguchi s method we are able to improve the upper bound to 23n3 n lg n n 1

eigenpairs of toeplitz and disordered toeplitz matrices with a fisher hartwig symbol toeplitz matrices have entries that are constant along diagonals they model directed transport are at the heart of correlation function calculations of the two dimensional ising model and have applications in quantum information science we derive their eigenvalues and eigenvectors when the symbol is singular fisher hartwig we then add diagonal disorder and study the resulting eigenpairs we find that there is a bulk behavior that is well captured by second order perturbation theory of non hermitian matrices the non perturbative behavior is classified into two classes runaways type i leave the complex valued spectrum and become completely real because of eigenvalue attraction runaways type ii leave the bulk and move very rapidly in response to perturbations these have high condition numbers and can be predicted localization of the eigenvectors are then quantified using entropies and inverse participation ratios eigenvectors corresponding to runaways type ii are most localized i e super exponential whereas runaways type i are less localized than the unperturbed counterparts and have most of their probability mass in the interior with algebraic decays the results are corroborated by applying free probability theory and various other supporting numerical studies

can spectral value sets of toeplitz band matrices jump abstract the spectral value set speb ca of a bounded linear operator a on l2 is the union of the spectra of all operators of the form a bkc where k

distribution of eigenvalues in non hermitian anderson models we develop a theory which describes the behavior of eigenvalues of a class of one dimensional random non hermitian operators introduced recently by hatano and nelson we prove that the eigenvalues are distributed along a curve in the complex plane an equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a reference hermitian disordered system the generic properties of the eigenvalue distribution are discussed

factorization forests of finite height abstract we introduce factorization forests it is shown that the vertex set of a dense factorization forest of finite height satisfies a bounded gap ramsey type property of words the main result is that every morphism from a free semigroup to a finite semigroup s admits a ramseyan factorization forest of height at most 9 s techniques for constructed factorization forests are developed

improved limitedness theorems on finite automata with distance functions abstract let a be a finite automaton with a distance function and id a be the set of distances associated with words accepted by a this paper presents an improved upper bound of id a when the upper limit of id a is finite it also presents one necessary and sufficient condition concerning word expressions for the upper limit of id a to be infinite

algorithms for determining relative star height and star height eggan l963 introduced the notion of the star height to each regular expression which is a nonnegative integer denoting the nestedness of star operators in this expression the star height of a regular language is the minimum of the star height of regular expressions denoting this language we remark here that to each regular language there exist generally infinitely many regular expressions denoting this language eggan l963 showed that for each nonnegative integer k there exists a regular language of star height k and posed the problem of determining the star height of any regular language dejean and schutzenberger 1966 showed that for each nonnegative integer k there exists a regular language of star height k over the two letter alphabet mcnaughton 1967 presented an algorithm for determining the loop complexity i e the star height of any regular language whose syntactic monoid is a group cohen 1970 1971 and cohen and brzozowski 1970 investigated many properties of star height some of which provide algorithms for determining the star height of any regular language of certain reset free type hashiguchi and honda 1979 presented an algorithm for determining the star height of any reset free language and any reset language hashiguchi 1982b presented an algorithm for deciding whether or not an arbitrary language is of star height one to obtain this result it uses the limitedness theorem on finite automata with distance functions in short d automata in hashiguci 1982a 1983

non hermitian tridiagonal random matrices and returns to the origin of a random walk we study a class of tridiagonal matrix models the q roots of unity models which includes the sign q 2 and the clock q models by feinberg and zee we find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides in the complex plane furthermore the averaged traces of mk are integers that count closed random walks on the line such that each site is visited a number of times multiple of q we obtain an explicit evaluation for them

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

coburn s lemma and the finite section method for random jacobi operators abstract we study the spectra and pseudospectra of semi infinite and bi infinite tridiagonal random matrices and their finite principal submatrices in the case where each of the three diagonals varies over a separate compact set say u v w c such matrices are sometimes termed stochastic toeplitz matrices a in the semi infinite case and stochastic laurent matrices a in the bi infinite case their spectra spec a and spec a are independent of a and a as long as a and a are pseudoergodic in the sense of davies 2001 20 which holds almost surely in the random case this was shown in davies 2001 20 for a that the same holds for a is one main result of this paper although the computation of and in terms of u v and w is intrinsically difficult we give upper and lower spectral bounds and we explicitly compute a set g that fills the gap between and in the sense that g we also show that the invertibility of one and hence all operators a implies the invertibility and uniform boundedness of the inverses of all finite tridiagonal square matrices with diagonals varying over u v and w this implies that the so called finite section method for the approximate solution of a system a x b is applicable as soon as a is invertible and that the finite section method for estimating the spectrum of a does not suffer from spectral pollution both results illustrate that tridiagonal stochastic toeplitz operators share important properties of classical toeplitz operators indeed one of our main tools is a new stochastic version of the coburn lemma for classical toeplitz operators saying that a stochastic tridiagonal toeplitz operator if fredholm is always injective or surjective in the final part of the paper we bound and compare the norms and the norms of inverses of bi infinite semi infinite and finite tridiagonal matrices over u v and w this in particular allows the study of the resolvent norms and hence the pseudospectra of these operators and matrices

the diffusion reaction d r hamiltonian and the solutions of certain types of linear and nonlinear d r equations in one dimension with a view to having further insight into the mathematical content of the non hermitian hamiltonian associated with the diffusion reaction d r equation in one dimension we investigate a the solitary wave solutions of certain types of its nonlinear versions and b the problem of real eigenvalue spectrum associated with its linear version or with this class of non hermitian hamiltonians for case a we use the standard techniques to handle the quadratic and cubic nonlinearities in the d r equation whereas for case b a newly proposed method based on an extended complex phase space is employed for a particular class of solutions an ermakov system of equations is also found for the linear case further corresponding to the classical version of the above one dimensional complex hamiltonian an equivalent integrable system of two two dimensional real hamiltonians is suggested

on the q analogues of some transformations of nearly poised hypergeometric series a number of transformation formulas for very well poised basic hypergeometric series have been obtained which in the limit q 1 approach the known transformation formulas for nearly poised ordinary hypergeometric series