irreducible unitary representations of su 2 2 we give an explicit classification of the irreducible unitary representations of the simple lie group su 2 2

a basic analog of a theorem of polya we derive a q analog of a theorem of george polya 1918 0 q 1 concer ning the zeros of basic cosine and sine transforms the results are established without further restrictions on q

multiresolution wavelet analysis of integer scale bessel functions we identify multiresolution subspaces giving rise via hankel transforms to bessel functions they emerge as orthogonal systems derived from geometric hilbert space considerations the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of hilbert spaces we study the theory of representations of the c algebra o 1 arising from this multiresolution analysis a connection with markov chains and representations of o 1 is found projection valued measures arising from the multiresolution analysis give rise to a markov trace for the quantum groups soq

q deformed harmonic and clifford analysis and the q hermite and laguerre polynomials we define a q deformation of the dirac operator inspired by the one dimensional q derivative this implies a q deformation of the partial derivatives by taking the square of this dirac operator we find a q deformation of the laplace operator this allows us to construct q deformed schrodinger equations in higher dimensions the equivalence of these schrodinger equations with those defined on q euclidean space in quantum variables is shown we also define the m dimensional q clifford hermite polynomials and show their connection with the q laguerre polynomials these polynomials are orthogonal with respect to an m dimensional q integration which is related to integration on q euclidean space the q laguerre polynomials are the eigenvectors of an suq 1 1 representation

analysis and design of tuned turbo codes it has been widely observed that there exists a fundamental tradeoff between the minimum hamming distance properties and the iterative decoding convergence behavior of turbo like codes while capacity achieving code ensembles typically are asymptotically bad in the sense that their minimum distance does not grow linearly with block length and they therefore exhibit an error floor at moderate to high signal to noise ratios asymptotically good codes usually converge further away from channel capacity in this paper we introduce the concept of tuned turbo codes a family of asymptotically good hybrid concatenated code ensembles where asymptotic minimum distance growth rates convergence thresholds and code rates can be tradedoff using two tuning parameters and by decreasing the asymptotic minimum distance growth rate is reduced in exchange for improved iterative decoding convergence behavior while increasing raises the asymptotic minimum distance growth rate at the expense of worse convergence behavior and thus the code performance can be tuned to fit the desired application by decreasing a similar tuning behavior can be achieved for higher rate code ensembles

a q analogue of the weighted bergman space on the disk and associated second q bargmann transform abstract we provide a q analogue of the classical weighted bergman space on the complex unit disk and we give the explicit expression of the corresponding reproducing kernel function moreover we give a q analogue of the second bargmann integral transform introduced by v bargmann in 4 p 203 we show that it defines a unitary integral transform from the hilbert space on the nonnegative real half line spanned by the q laguerre polynomials with respect to the weight function x exp q 1 q x onto the considered weighted q bergman hilbert space

on interleaved differentially encoded convolutional codes we study a serially interleaved concatenated code construction where the outer code is a standard convolutional code and the inner code is a recursive convolutional code of rate 1 we focus on the ubiquitous inner differential encoder used in particular to resolve phase ambiguities double differential encoder used to resolve both phase and frequency ambiguities and another rate 1 recursive convolutional code of memory 2 we substantiate analytically the rather surprising result that the error probabilities corresponding to a maximum likelihood ml coherently detected antipodal modulation over the additive white gaussian noise awgn channel for this construction are advantageous as compared to the stand alone outer convolutional code this is in spite of the fact that the inner code is of rate 1 the analysis is based on the tangential sphere upper bound of an ml decoder incorporating the ensemble weight distribution wd of the concatenated code where the ensemble is generated by all random and uniform interleavers this surprising result is attributed to the wd thinning observed for the concatenated scheme which shapes the wd of the outer convolutional code to resemble more closely the binomial distribution typical of a fully random code of the same length and rate this gain is maintained regardless of a rather dramatic decrease as demonstrated here in the minimum distance of the concatenated scheme as compared to the minimum distance of the outer stand alone convolutional code the advantage of the examined serially interleaved concatenated code given in terms of bit and or block error probability which is decoded by a practical suboptimal decoder over the optimally decoded standard convolutional code is demonstrated by simulations and some insights into the performance of the iterative decoding algorithm are also discussed though we have investigated only specific constructions of constituent inner rate 1 and outer codes we trust hinging on the rational of the arguments here that these results extend to many other constituent convolutional outer codes and rate 1 inner recursive convolutional codes

orthogonal polynomials and some q beta integrals of ramanujan abstract two integrals of ramanujan are used to define a q analogue of the euler beta integral on the real line and of the cauchy beta integral on the complex unit circle such integrals are connected to orthogonal biorthogonal and laurent polynomials explicit examples of laurent orthogonal polynomials are given on the real line and on the circle

a note on p q oscillators and bibasic hypergeometric functions the authors examine the relation between the representation theory of a two parameter deformation of the oscillator algebra and certain bibasic laguerre functions and polynomials

the unitary irreducible representations of overline rm so _ 0 4 2 an exhaustive classification of all irreducible harish chandra mathfrak s mathfrak o 4 2 modules integrable to unitarizable projective representations of the conformal group is established by infinitesimal methods the classification is based 1 on the reduction upon the maximal compact subalgebra associated with a lattice of points in 3 and 2 on a set of additional parameters upon which the eigenvalues of central elements of the enveloping algebra depend polynomially

the zeros of basic bessel functions the functions jv ax x and associated orthogonal polynomials abstract the modified lommel polynomials are generalized and their orthogonality relation is obtained as a byproduct we prove that the non zero roots of jv ax x are real simple and interlace with those of jv ax 1 x the q lommel polynomials are shown to play in the theory of basic bessel function the role played by the lommel polynomials in the bessel function theory the q lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support this is then used to prove that the prositive zeros of x j 2 v x q are real simple and interlace with the zeros of x 1j 2 v 1 x q when 1 we also establish the complete monotonicity of 1x 1 2 j v 1 2 i x q j v 2 i x q

structure of stieltjes classes of moment equivalent probability laws a stieltjes class is a one parameter family of moment equivalent distribution functions constructed by modulation of a given indeterminate distribution function f called the center of the class members of a stieltjes class are mutually absolutely continuous and conversely any pair of moment equivalent and mutually absolutely continuous distribution functions can be joined by a stieltjes class the center of a stieltjes class is an equally weighted mixture of its extreme members and this places restrictions on which distributions can belong to a stieltjes class with a given center the lognormal law provides interesting illustrations of the general ideas in particular it is possible for two moment equivalent infinitely divisible distributions to be joined by a stieltjes class and random scaling can be used to construct new stieltjes classes from a given stieltjes class

a note on tensor products of q algebra representations and orthogonal polynomials abstract we work out examples of tensor products of distinct generalized slq 2 algebras with a factor from the positive discrete series of representations of one algebra and a factor from the negative discrete series of the other we show that the equation for the common eigenfunctions of the casimir operator and the cartan subalgebra generator is just the three term recurrence relation corresponding to orthogonality for special cases of the askey wilson polynomials and this connection yields an almost immediate resolution of the tensor product representation into a direct integral of irreducible representation an identity for the matrix elements of the group representation operators with respect to the tensor product and the reduced bases follows easily cases where the measures for the orthogonal polynomials are not unique correspond to cases where the tensor products and their resolutions are also nonunique

on the asymptotic input output weight distributions and thresholds of convolutional and turbo like encoders we present a general method for computing the asymptotic input output weight distribution of convolutional encoders in some instances one can derive explicit analytic expressions in general though to determine the growth rate of the input output weight distribution for a particular normalized input weight spl kappa and output weight spl omega a system of polynomial equations has to be solved this method is then used to determine the asymptotic weight distribution of various concatenated code ensembles and to derive lower bounds on the thresholds of these ensembles under maximum likelihood ml decoding

on some indeterminate moment problems for measures on a geometric progression abstract we consider some indeterminate moment problems which all have a discrete solution concentrated on geometric progressions of the form cq k k z or cq k k z it turns out to be possible that such a moment problem has infinitely many solutions concentrated on the geometric progression

on p q μ ν ϕ1 ϕ2 generalized oscillator algebra and related bibasic hypergeometric functions this paper provides the generalization of the work by floreanini et al 1993 j phys a math gen 26 611 4 who generated bibasic hypergeometric functions from p q oscillators we consider a six parameter deformed oscillator algebra realized from the p q deformed boson oscillators we build the corresponding fock space representation in an infinite dimensional subspace of the hilbert space of a harmonic oscillator we also discuss the properties of a discrete spectrum of the hamiltonian of the deformed harmonic oscillator corresponding to this system we then define a realization of the deformed algebra in terms of a generalized derivative and investigate the relation between this representation and generalized bibasic laguerre functions and polynomials

permutations moments measures which combinatorial sequences correspond to moments of probability measures on the real line we present a generating function in the form of a continued fraction for a fourteenparameter family of such sequences and interpret these in terms of combinatorial statistics on the symmetric groups special cases include several classical and noncommutative probability laws along with a substantial subset of the orthogonalizing measures in the q askey scheme now given a new combinatorial interpretation in terms of elementary permutation statistics this framework further captures a variety of interesting combinatorial sequences including notably the moment sequences associated to distributions of the numbers of occurrences of classical and vincular permutation patterns of length three this connection between pattern avoidance and broader ideas in classical and noncommutative probability is among several intriguing new corollaries which generalize and unify results previously appearing in the literature while opening up new lines of inquiry the fourteen combinatorial statistics further generalize to signed and colored permutations and as an infinite family of statistics to the k arrangements permutations with k colored fixed points introduced here along with several related results and conjectures

asymptotics of the largest zeros of some orthogonal polynomials we study the asymptotics of the largest zeros of the wilson hermite and q laguerre polynomials using two distinct techniques the first is based on the coulomb fluid technique developed in a previous paper where the primary input is the weight function while the second uses the method of chain sequences which supplies inequalities for the largest zeros using the recurrence coefficients we also investigate the asymptotics of the largest zeros of the polynomials orthogonal to the weight functions for c 0 and m a positive even integer

completeness special functions and uncertainty principles over q linear grids we derive completeness criteria for sequences of functions of the form f x n where n is the nth zero of a suitably chosen entire function using these criteria we construct complete nonorthogonal systems of fourier bessel functions and their q analogues as well as other complete sets of q special functions we discuss connections with uncertainty principles over q linear grids and the completeness of certain sets of q bessel functions is used to prove that if a function f and its q hankel transform both vanish at the points q n n 1 0 q 1 then f must vanish on the whole q linear grid qn n

inner products involving q differences the little q laguerre sobolev polynomials in this paper polynomials which are orthogonal with respect to the inner product p r s k 0 p qk r qk aq k aq q q q k k 0 dqp qk dqr qk aq k aq q q q k where dq is the q difference operator 0 0 q 1 and 0 aq 1 are studied for these polynomials algebraic properties and q difference equations are obtained as well as their relation with the monic little q laguerre polynomials some properties about the zeros of these polynomials are also deduced finally the relative asymptotics qn x pn x a q n on compact subsets of c 0 1 is given where qn x is the nth degree monic orthogonal polynomial with respect to the above inner product and pn x a q denotes the monic little q laguerre polynomial of degree n

on the variation with respect to a parameter of zeros of bessel and q bessel functions abstract we apply what is essentially the hellmann feynman theorem in a discrete setting to derive representations for the derivatives with respect to a parameter of the positive zeros of a family of entire functions these representations prove that the zeros are monotone functions of the parameters involved this family includes the bessel functions jv x and their basic analog jv 2 x q as a by product we show that j v 1 2 v 1 increases with v when v 1 jv 1 being the smallest positive zero of jv x we also use the lommel polynomials to derive improved lower bounds for the smallest positive zero of jv x

theq laguerre polynomials and related moment problems abstract we study two indeterminate hamburger moment problems associated with q laguerre polynomials the coefficients in their recurrence relations are of exponential growth this completes earlier work started by d moak some of the measures that the polynomials are orthogonal with respect to are found and lead to the evaluation of certain integrals a symmetric set of orthogonal polynomials is also considered

zeros of entire functions and a problem of ramanujan abstract we derive representations for certain entire q functions and apply our technique to the ramanujan entire function or q airy function and q bessel functions this is used to show that the asymptotic series of the large zeros of the ramanujan entire function and similar functions are also convergent series the idea is to show that the zeros of the functions under consideration satisfy a nonlinear integral equation

basic fourier series on a q linear grid convergence theorems abstract for 0 q 1 define the symmetric q linear operator acting on a suitable function f x by f x f q 1 2 x f q 1 2 x the q linear initial value problem f x x f x f 0 1 has two entire functions c q z and s q z as linearly independent solutions which are orthogonal on a discrete set sufficient conditions for pointwise convergence and for uniform convergence of the corresponding fourier expansion are given

addition formulas for q bessel functions the two dimensional euclidean algebra provides an algebraic setting for q bessel functions using a two variable realization of this algebra q analogs of the lommel and graf addition formulas for the q bessel functions are obtained

asymptotics of basic bessel functions and q laguerre polynomials abstract we establish a large n complete asymptotic expansion for q laguerre polynomials and a complete asymptotic expansion for a q bessel function of large argument these expansions are needed in our study of an exactly solvable random transfer matrix model for disordered electronic systems we also give a new derivation of an asymptotic formula due to littlewood 1907

contiguous relations basic hypergeometric functions and orthogonal polynomials i abstract the second order q difference equation of a 2 1 and the contiguous relations of 2 1 s are used to derive continued fraction expansions for quotients of 2 1 s we identify the cases when these continued fractions are positive definite j fractions i e the cases when the denominators of the approximants of the j fractions are orthogonal polynomials the measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail

on q analogues of the fourier and hankel transforms for h exton s q analogue of the bessel function going back to w hahn in a special case but different from f h jackson s q bessel functions we derive hansen lommel type orthogonality relations which by a symmetry turn out to be equivalent to orthogonality relations which are q analogues of the hankel integral transform pair these results are implicit in the context of quantum groups in a paper by vaksman and korogodskii as a specialization we get q cosines and q sines which admit q analogues of the fourier cosine and fourier sine transforms

orthogonal polynomials associated with coulomb wave functions abstract a new class of orthogonal polynomials associated with coulomb wave functions is introduced these polynomials play a role analogous to that the lommel polynomials have in the theory of bessel functions the orthogonality measure for this new class is described in detail in addition the orthogonality measure problem is discussed on a more general level apart from this various identities derived for the new orthogonal polynomials may be viewed as generalizations of certain formulas known from the theory of bessel functions a key role in these derivations is played by a jacobi tridiagonal matrix j l whose eigenvalues coincide with the reciprocal values of the zeros of the regular coulomb wave function f l the spectral zeta function corresponding to the regular coulomb wave function or more precisely to the respective tridiagonal matrix is studied as well

new connection formulae for the q orthogonal polynomials via a series expansion of the q exponential using a realization of the q exponential function as an infinite multiplicative series of the ordinary exponential functions we obtain new nonlinear connection formulae of the q orthogonal polynomials such as q hermite q laguerre and q gegenbauer polynomials in terms of their respective classical analogues

on the zeros of basic finite hankel transforms in this article we prove that the basic finite hankel transform whose kernel is the third type jackson q bessel function has only infinitely many real and simple zeros provided that q satisfies a condition additional to the standard one we also study the asymptotic behavior of the zeros the obtained results are applied to investigate the zeros of q bessel functions as well as the zeros of q trigonometric functions a basic analog of a theorem of g polya 1918 on the zeros of sine and cosine transformations is also given

some properties of q biorthogonal polynomials abstract almost four decades ago konhauser introduced and studied a pair of biorthogonal polynomials y n x k and z n x k 1 k n 1 2 3 which are suggested by the classical laguerre polynomials the so called konhauser biorthogonal polynomials z n x k of the second kind were indeed considered earlier by toscano without their biorthogonality property which was emphasized upon in konhauser s investigation many properties and results for each of these biorthogonal polynomials such as generating functions rodrigues formulas recurrence relations and so on have since been obtained in several works by others the main object of this paper is to present a systematic investigation of the general family of q biorthogonal polynomials several interesting properties and results for the q konhauser polynomials are also derived

diagonalizaton of certain integral operators ii abstract we establish an integral representation of a right inverse of the askey wilson finite difference operator in an l 2 space weighted by the weight function of the continuous q jacobi polynomials we characterize the eigenvalues of this integral operator and prove a q analog of the expansion of e ixy in jacobi polynomials of argument x we also outline a general procedure of finding integral representations for inverses of linear operators

the serial concatenation of rate 1 codes through uniform random interleavers until the analysis of repeat accumulate codes by divsalar et al 1998 few people would have guessed that simple rate 1 codes could play a crucial role in the construction of good binary codes we construct good binary linear block codes at any rate r 1 by serially concatenating an arbitrary outer code of rate r with a large number of rate 1 inner codes through uniform random interleavers we derive the average output weight enumerator we for this ensemble in the limit as the number of inner codes goes to infinity using a probabilistic upper bound on the minimum distance we prove that long codes from this ensemble will achieve the gilbert varshamov 1952 bound with high probability numerical evaluation of the minimum distance shows that the asymptotic bound can be achieved with a small number of inner codes in essence this construction produces codes with good distance properties which are also compatible with iterative turbo style decoding for selected codes we also present bounds on the probability of maximum likelihood decoding mld error and simulation results for the probability of iterative decoding error

new q laguerre polynomials having factorized permutation interpretations abstract in this paper we generalize the laguerre polynomials in terms of q analogue for riordan matrices to be more specific for n 0 we introduce new q laguerre polynomials l n x q by defining the eulerian generating function for l n x q as j 0 1 1 q j 1 z e q x z 1 z interestingly it turns out that l n x q have combinatorial descriptions in the aspect of the inversions of factorized permutations and q rook numbers we locate their zeros and develop their algebraic properties as well

the minimum distance of turbo like codes worst case upper bounds are derived on the minimum distance of parallel concatenated turbo codes serially concatenated convolutional codes repeat accumulate codes repeat convolute codes and generalizations of these codes obtained by allowing nonlinear and large memory constituent codes it is shown that parallel concatenated turbo codes and repeat convolute codes with sub linear memory are asymptotically bad it is also shown that depth two serially concatenated codes with constant memory outer codes and sublinear memory inner codes are asymptotically bad most of these upper bounds hold even when the convolutional encoders are replaced by general finite state automata encoders in contrast it is proven that depth three serially concatenated codes obtained by concatenating a repetition code with two accumulator codes through random permutations can be asymptotically good

an integral representation and some transformation properties of q bessel functions a poisson type integral representation for jackson s q bessel function is obtained by using askey and wilson s q beta integral and nassrallah and rahman s integral formula for an 8 7 series this representation along with some transformtion formulas for basic hypergeometric series help express the q bessel functions as a 3 2 series in base q a 2 2 series in base q2 and a 2 1 series in base qq12 where 0 q 1 as an application q analogues are found for gegenbauer s degenerate addition formulas for bessel functions

q laguerre and wall polynomials are related by the fourier gauss transform it is shown that the q laguerre and wall or little q laguerre polynomials are interrelated by the fourier gauss transform in the limit when the degree of these polynomials tends to infinity this integral transform provides the relation between jackson s second and third q bessel functions

coding for parallel channels gallager bounds and applications to turbo like codes the transmission of coded communication systems is widely modeled to take place over a set of parallel channels this model is used for transmission over block fading channels rate compatible puncturing of turbo like codes multicarrier signaling multilevel coding etc new upper bounds on the maximum likelihood ml decoding error probability are derived in the parallel channel setting we focus on the generalization of the gallager type bounds and discuss the connections between some versions of these bounds the tightness of these bounds for parallel channels is exemplified for structured ensembles of turbo codes repeat accumulate ra codes and some of their recent variations e g punctured accumulate repeat accumulate codes the bounds on the decoding error probability of an ml decoder are compared to computer simulations of iterative decoding the new bounds show a remarkable improvement over the union bound and some other previously reported bounds for independent parallel channels this improvement is exemplified for relatively short block lengths and it is pronounced when the block length is increased in the asymptotic case where we let the block length tend to infinity inner bounds on the attainable channel regions of modern coding techniques under ml decoding are obtained based solely on the asymptotic growth rates of the average distance spectra of these code ensembles

coding theorems for turbo code ensembles this paper is devoted to a shannon theoretic study of turbo codes we prove that ensembles of parallel and serial turbo codes are good in the following sense for a turbo code ensemble defined by a fixed set of component codes subject only to mild necessary restrictions there exists a positive number spl gamma sub 0 such that for any binary input memoryless channel whose bhattacharyya noise parameter is less than spl gamma sub 0 the average maximum likelihood ml decoder block error probability approaches zero at least as fast as n sup spl beta where spl beta is the interleaver gain exponent defined by benedetto et al in 1996

yet another basic analogue of graf s addition formula an identity involving basic bessel functions and al salam chihara polynomials is proved for which we recover graf s addition formula for the bessel function as the base q tends to 1 the corresponding product formula is derived some known identities for jackson s q bessel functions are obtained as limiting cases as special cases we prove identities for q charlier polynomials

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

tableaux combinatorics for the asymmetric exclusion process and askey wilson polynomials introduced in the late 1960 s 33 43 the asymmetric exclusion process asep is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one dimensional lattice of n sites it has been cited as a model for traffic flow and protein synthesis in the most general form of the asep with open boundaries particles may enter and exit at the left with probabilities and and they may exit and enter at the right with probabilities and in the bulk the probability of hopping left is q times the probability of hopping right the first main result of this paper is a combinatorial formula for the stationary distribution of the asep with all parameters general in terms of a new class of tableaux which we call staircase tableaux this generalizes our previous work 14 15 for the asep with parameters 0 using our first result and also results of uchiyama sasamoto wadati 48 we derive our second main result a combinatorial formula for the moments of askey wilson polynomials since the early 1980 s there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials e g hermite charlier laguerre meixner however this is the first such formula for the askey wilson polynomials which are at the top of the hierarchy of classical orthogonal polynomials

elementary representations and intertwining operators for su 2 2 i the structure of the group su 2 2 and of its lie algebra is studied in detail the results will be applied in subsequent parts devoted to the explicit construction of elementary representations of su 2 2 induced from different parabolic subgroups and of the intertwining operators between these representations a summary of some results of parts ii and iii is given

radii of starlikeness and convexity of some q bessel functions abstract geometric properties of the jackson and hahn exton q bessel functions are studied for each of them three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane for each of the six functions we determine the radii of starlikeness and convexity precisely by using their hadamard factorization these are q generalizations of some known results for bessel functions of the first kind the characterization of entire functions from the laguerre polya class via hyperbolic polynomials plays an important role in this paper moreover the interlacing property of the zeros of jackson and hahn exton q bessel functions and their derivatives is also useful in the proof of the main results we also deduce a necessary and sufficient condition for the close to convexity of a normalized jackson q bessel function and its derivatives some open problems are proposed at the end of the paper

restricted growth functions rank row matchings of partition lattices and q stirling numbers abstract in this paper we use restricted growth functions to give a direct algebraic description of the incidence relations of the lattice of partitions of an n set that is independent of the partitions themselves this description of the incidence relations not only gives new information about rank row matchings of partition lattices but is of independent interest since every finite lattice is a sublattice of some partition lattice as a further application of restricted growth functions we show how the polynomials in q known as q stirling numbers of the second kind may be viewed as generating functions

indeterminate moment problems and the theory of entire functions the set of solutions to an indeterminate hamburger moment problem is given by the nevanlinna parametrization involving four entire functions a b c d we review the properties of these functions and discuss recent progress in the still open problem of characterizing the class of entire functions which occur in this way they belong to the krein class and are of minimal exponential type we deduce the analogous parametrization of the set of solutions to an indeterminate stieltjes moment problem from the corresponding symmetric hamburger problem the survey ends with a discussion of three special cases where the functions a b c d are known explicitly

expansion and asymptotic in terms of basic bessel functions abstract this work aims to study the expansion and asymptotic for solutions of q difference equations in terms of the basic bessel functions namely j 2 x q for this purpose we will show that the constructive method introduced by olver f w j olver asymptotics and special functions academic press inc 1974 and exploited early by fitouhi et al h chebli a fitouhi m m hamza expansion in series of bessel functions and transmutations for perturbed bessel operator j math anal appl 181 3 1994 a fitouhi m m hamza uniform expansion for eigenfunction of singular second order differential operator siam j math anal 21 6 1990 a fitouhi m m hamza expansion in series of laguerre functions for solution of perturbed laguerre equations contemp math 183 1995 a fitouhi j el kamel expansion in series of gegenbauer polynomials int trans sp funct 5 3 4 1997 213 226 can be extended in this context as application new expansions of some basic special functions are established in particular these given by ismail y chen m e h ismail k a muttalib asymptotics of basic bessel functions and q laguerre polynomilas j comput appl math 54 1994 263 272 m e h ismail classical and quantum orthogonal polynomials in one variable cambridge university press 2005

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

spectra and minimum distances of repeat multiple accumulate codes in this paper the ensembles of repeat multiple accumulate codes ram which are obtained by interconnecting a repeater with a cascade of m accumulate codes through uniform random interleavers are analyzed it is proved that the average spectral shapes of these code ensembles are equal to 0 below a threshold distance epsivm and moreover they form a nonincreasing sequence in m converging uniformly to the maximum between the average spectral shape of the linear random ensemble and 0 consequently the sequence epsivm converges to the gilbert varshamov gv distance a further analysis allows to conclude that if m ges 2 the ram are asymptotically good and that epsivm is the typical normalized minimum distance when the interleaver length goes to infinity combining the two results it is possible to conclude that the typical distance of the ensembles ram converges to the gilbert varshamov bound

length biasing and laws equivalent to the log normal abstract let x 0 denote a generic lifetime of a renewal process having unit mean lifetime let hat x denote the stationary total lifetime and let q 0 1 be a fixed constant we consider anew the scale invariance problem for which laws does q hat x have the same distribution as x our setting is more probabilistic than those presented hitherto and we explore connections with the log normal moment problem in particular it is shown that all explicitly known laws which have log normal moments solve our problem the notion of remaining lifetime is generalized and its scaling invariance is investigated using the notion of total lifetime two moment equivalent laws of askey are shown to have a simple representation in terms of laws equivalent to the log normal the representation involves a q gamma law which we explore in its own right an affine extension of our basic scale invariance relation arising in the theory of orthogonal polynomials is shown to be equivalent to the latter

generalized q hermite polynomials we consider two operators a and a in a hilbert space of functions on the exponential lattice where 0 q 1 the operators are formal adjoints of each other and depend on a real parameter we show how these operators lead to an essentially unique symmetric ground state 0 and that a and a are ladder operators for the sequence the sequence n 0 is shown to be a family of orthogonal polynomials which we identify as symmetrized q laguerre polynomials we obtain in this way a new proof of the orthogonality for these polynomials when 0 the polynomials are the discrete q hermite polynomials of type ii studied in several papers on q quantum mechanics

on a ramanujan type entire function and its zeros abstract in this paper we derive some properties of a ramanujan type entire function a mild generalization of the garret ismail stanton m version of the rogers ramanujan identities is obtained moreover we investigate the zeros of the ramanujan type entire function and our results generalize those for the zeros of the ramanujan function finally an integral equation related to the ramanujan type entire function is also derived

nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes abstract we consider two classes of birth and death processes which lead to indeterminate stieltjes moment problems and we calculate the entire functions c and d from the corresponding nevanlinna matrix this permits a direct computation of a nevanlinna extremal orthogonality measure for each of the processes

the impact of stieltjes work on continued fractions and orthogonal polynomials additional material abstract in the recent new edition of the collected works of t j stieltjes one of us gave an impression of the impact of stieltjes work a century after his death 43 in this paper we give an update and mention some observations which were missing from 43 and some results which appeared during the last two years and which are directly related to stieltjes work in particular there is a large section on special polynomials which were already considered by stieltjes but which were rediscovered later often without the knowledge of stieltjes work

minimum distance and convergence analysis of hamming accumulate accumulate codes in this letter we consider the ensemble of codes formed by the serial concatenation of a hamming code and two accumulate codes we show that this ensemble is asymptotically good in the sense that most codes in the ensemble have minimum distance growing linearly with the block length thus the resulting codes achieve high minimum distances with high probability about half or more of the minimum distance of a typical random linear code of the same rate and length in our examples the proposed codes also show reasonably good iterative convergence thresholds which makes them attractive for applications requiring high code rates and low error rates such as optical communications and magnetic recording

indeterminate symmetric moment problems let n x be a sequence of orthogonal polynomials with respect to a real distribution according to a classical theorem of reisz 191 x x is complete in l if and only if w is either the solution of a determined hamburger moment problem or an extremal solution of an indeterminate moment problem in the special case where the zeros of the orthogonal polynomials have a finite infimum a and the hamburger moment problem is indeterminate there is an extremal solution which is essentially the only solution whose spectrum is a subset of a co 4 this solution is in a sense the best solution from the viewpoint of orthogonal polynomials since its spectrum consists of the limit points of the zeros of the polynomials askey remarked to the author that in view of riesz theorem it is surprising that so little attention has been paid to the question of extremal solutions moak 8 has recently studied the extremal solutions associated with the q laguerre nee generalized stieltjes wigert polynomials askey suggested it would be desirable to investigate the existence of best extremal solutions in the general case or as a first step even in the symmetric case moments of odd order are 0 in this paper we will consider this special case and the related systems of orthogonal polynomials in particular we will show that an indeterminate symmetric moment problem has precisely two symmetric extremal solutions this result which we found surprizing at first appears after reflection quite natural and perhaps obvious since the two solutions are the symmetrizations of the best extremal solutions of the related moment problems on 0 co

performance analysis of 3 d turbo codes in this work we consider the minimum distance properties and convergence thresholds of 3 d turbo codes 3d tcs recently introduced by berrou here we consider binary 3d tcs while the original work of berrou considered double binary codes in the first part of the paper the minimum distance properties are analyzed from an ensemble perspective both in the finite length regime and in the asymptotic case of large block lengths in particular we analyze the asymptotic weight distribution of 3d tcs and show numerically that their typical minimum distance dmin may depending on the specific parameters asymptotically grow linearly with the block length i e the 3d tc ensemble is asymptotically good for some parameters in the second part of the paper we derive some useful upper bounds on the dmin when using quadratic permutation polynomial qpp interleavers with a quadratic inverse furthermore we give examples of interleaver lengths where an upper bound appears to be tight the best codes in terms of estimated dmin obtained by randomly searching for good pairs of qpps for use in the 3d tc are compared to a probabilistic lower bound on the dmin when selecting codes from the 3d tc ensemble uniformly at random this comparison shows that the use of designed qpp interleavers can improve the dmin significantly for instance we have found a 6144 2040 3d tc with an estimated dmin of 147 while the probabilistic lower bound is 69 higher rates are obtained by puncturing nonsystematic bits and optimized periodic puncturing patterns for rates 1 2 2 3 and 4 5 are found by computer search finally we give iterative decoding thresholds computed from an extrinsic information transfer chart analysis and present simulation results on the additive white gaussian noise channel to compare the error rate performance to that of conventional turbo codes

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

sampling theorems associated with biorthogonal q bessel functions this paper deals with the derivation of sampling theorems associated with q biorthogonal systems we derive interpolation expansions for qhankel transforms whose kernels are the second type q bessel functions j 2 z q 0 0 q 1 we investigate the eigenvalue problem whose solutions are the q bessel functions as well as its adjoint special cases and applications involving the associated q sine function are given the results are based on the conjecture that a family of q bessel functions of the second kind is a riesz basis clues are given to support our claim

attribute grammars are useful for combinatorics abstract the purpose of this paper is to show the use of attribute grammars in solving some combinatorics problems for example we give the generating function for the shape of skew ferrers diagrams according to the number of cells and the number of columns this result is new and proves that skew ferrer diagrams are related to new basic bessel functions

ladder operators for q orthogonal polynomials the q difference analog of the classical ladder operators is derived for those orthogonal polynomials arising from a class of indeterminate moments problem

transitions in spectral statistics we present long range statistical properties of a recently introduced unitary random matrix ensemble whose short range correlations were found to describe a transition from wigner to poisson type as a function of a single parameter we argue by evaluating the two level correlation function of a different solvable model that the transition is perhaps a quite general feature for a class of models our analytic results compare well with numerical studies on a variety of physical systems we discuss the possibility of observing experimentally the signature of such transitions

real paley wiener theorems for the koornwinder swarttouw q hankel transform abstract we derive two real paley wiener theorems in the setting of quantum calculus the first uses techniques due to tuan and zayed v k tuan a i zayed paley wiener type theorems for a class of integral transforms j math anal appl 266 1 2002 200 226 in order to describe the image of the space l q 2 0 r under koornwinder and swarttouw q hankel transform t h koornwinder r f swarttouw on q analogues of the fourier and hankel transforms trans amer math soc 333 1 1992 445 461 and contains as a special case a description of the domain of the q sampling theorem associated with the q hankel transform l d abreu a q sampling theorem related to the q hankel transform proc amer math soc 133 4 2005 1197 1203 the second characterizes the image of compactly supported q smooth functions under a rescaled version of the q hankel transform and is a q analogue of a recent result due to andersen n b andersen real paley wiener theorems for the hankel transform j fourier anal appl 12 1 2006 17 25

spectral zeta functions for q bessel equations explicit formulae for the zeta functions of the zeros of hahn exton and jackson s q bessel functions are derived they can be regarded as spectral sum rules for some discrete quantum billiards

quasi orthogonality and zeros of some 2ϕ2 and 3ϕ2 polynomials abstract we state and prove the q extension of a result due to johnston and jordaan cf johnston and jordaan 2015 and make use of this result the orthogonality of q laguerre little q jacobi q meixner and al salam carlitz i polynomials as well as contiguous relations satisfied by the polynomials to establish the quasi orthogonality of certain 2 2 and 3 2 polynomials the location and interlacing properties of the real zeros of these quasi orthogonal polynomials are studied interlacing properties of the zeros of q laguerre quasi orthogonal polynomials l n z q when 2 1 with those of l n 1 1 z q and l n 1 z q are also considered

summation formulae for the product of the q kummer functions from e q 2 using the representation of eq 2 on the non commutative space zz qz z q 0 summation formulae for the product of two three and four q kummer functions are derived

models of q algebra representations tensor products of special unitary and oscillator algebras this paper begins a study of one and two variable function space models of irreducible representations of q analogs of lie enveloping algebras motivated by recurrence relations satisfied by q hypergeometric functions the algebras considered are the quantum algebra uq su2 and a q analog of the oscillator algebra not a quantum algebra in each case a simple one variable model of the positive discrete series of finite and infinite dimensional irreducible representations is used to compute the clebsch gordan coefficients it is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and the corresponding matrix elements of the group operators on these representation spaces are computed it is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations it is also demonstrated that general q hypergeometric functions can occur as

the ratio of q like orthogonal polynomials abstract it is shown that the ratio of orthogonal polynomials with exponentially increasing recurrence coefficients is closely related to some new orthogonal q polynomials of which the recurrence coefficients converge exponentially fast to zero these q polynomials are investigated in detail

quantum algebra from generalized q hermite polynomials abstract in this paper we discuss new results related to the generalized discrete q hermite ii polynomials h n x q introduced by mezlini et al in 2014 our aim is to give a continuous orthogonality relation a q integral representation and an evaluation at unity of the poisson kernel for these polynomials furthermore we introduce q schrodinger operators and we give the raising and lowering operator algebra corresponding to these polynomials our results generate a new explicit realization of the quantum algebra su q 1 1 using the generators associated with a q deformed generalized para bose oscillator

combinatorics of the two species asep and koornwinder moments abstract in previous work 12 13 14 the first and third authors introduced staircase tableaux which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process asep and for the moments of the askey wilson weight function the fact that the asep and askey wilson moments are related at all is unexpected and is due to 45 the asep is a model of particles hopping on a one dimensional lattice of n sites with open boundaries particles can enter and exit at both left and right borders it was introduced around 1970 34 43 and is cited as a model for both traffic flow and translation in protein synthesis meanwhile the askey wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials so from this previous work we have the relationship asep staircase tableaux askey wilson moments the askey wilson polynomials can be viewed as the one variable case of the multivariate koornwinder polynomials which are also known as the macdonald polynomials attached to the non reduced affine root system c n c n it is natural then to ask whether one can generalize the relationships among the asep askey wilson moments and staircase tableaux in such a way that koornwinder moments replace askey wilson moments in 15 we made a precise link between koornwinder moments and the two species asep a generalization of the asep which has two species of particles with different weights in this article we introduce rhombic staircase tableaux and show that we have the relationship 2 species asep rhombic staircase tableaux koornwinder moments in particular we give formulas for the stationary distribution of the two species asep and for koornwinder moments in terms of rhombic staircase tableaux download download high res image 75kb download download full size image

octobasic laguerre polynomials and permutation statistics abstract a set of orthogonal polynomials with eight independent q s is defined which generalizes the laguerre polynomials the moments of the measure for these polynomials are the generating functions for permutations according to eight different statistics specializing these statistics gives many other well known sets of combinatorial objects and relevant statistics the specializations are studied with applications to classical orthogonal polynomials and equidistribution theorems for statistics

unveiling turbo codes some results on parallel concatenated coding schemes a parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver the input bits to the first encoder are scrambled by the interleaver before entering the second encoder the codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders this construction can be generalized to any number of constituent codes parallel concatenated schemes employing two convolutional codes as constituent codes in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes have been previously shown to yield remarkable coding gains close to theoretical limits they have been named and are known as turbo codes we propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length the analytical bounding technique is then used to shed some light on some crucial questions which have been floating around in the communications community since the proposal of turbo codes

quantum symmetries of q difference equations a general method is presented to determine the symmetry operators of linear q difference equations it is applied to q difference analogs of the helmoltz heat and wave equations in diverse dimensions the symmetries are found to generate q deformations of classical lie algebras the method of separation of variables is used to obtain solutions which are seen to involve many basic special functions this allows the derivation of various identities and formulas for these q functions