characterization of the dunkl classical symmetric orthogonal polynomials in this paper we introduce the notion of dunkl classical orthogonal polynomials then we show that generalized hermite and generalized gegenbauer polynomials are the only dunkl classical symmetric orthogonal polynomials by solving a suitable differential difference equation

decomposition formulas for some triple hypergeometric functions abstract with the help of some techniques based upon certain inverse pairs of symbolic operators the authors investigate several decomposition formulas associated with srivastava s hypergeometric functions h a h b and h c in three variables many operator identities involving these pairs of symbolic operators are first constructed for this purpose by means of these operator identities as many as 15 decomposition formulas are then found which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the gauss and appell hypergeometric functions other closely related results are also considered briefly

connection coefficients between boas buck polynomial sets abstract in this paper a general method to express explicitly connection coefficients between two boas buck polynomial sets is presented as application we consider some generalized hypergeometric polynomials from which we derive some well known results including duplication and inversion formulas

associated laguerre konhauser polynomials quasi monomiality and operational identities abstract in this paper we exploit the monomiality principle to discuss and introduce a new class of laguerre konhauser polynomials we study their properties differential equations generating functions recurrence relations expansions and so on and discuss the link with ordinary case

linearization and krein like functionals of hypergeometric orthogonal polynomials the krein like r functionals of the hypergeometric orthogonal polynomials pn x with the kernel of the form xs x pm1 x pmr x being x the weight function on the interval r are determined by means of the srivastava linearization method the particular 2 functionals which are particularly relevant in quantum physics are explicitly given in terms of the degrees and the characteristic parameters of the polynomials they include the well known power moments and the novel krein like moments moreover various related types of exponential and logarithmic functionals are also investigated

clebsch gordan type linearisation relations for the products of laguerre polynomials and hydrogen like functions two series of clebsch gordan type linearisation relations are derived for the most general product of the laguerre polynomials ln1alpha 1 u1x ln2alpha 2 u2x which differ in orders n weights alpha and scaling multipliers u the general form and particular cases of coefficients in the expansion of the polynomial xkln1alpha 1 u1x lnnalpha n unx in terms of the laguerre polynomials are established the applications to hydrogen like functions and morse oscillators are indicated connection with an earlier carlitz expansion the technical links with the hyperspherical harmonics formalism and different approaches to the important koornwinder s positivity theorems are discussed briefly

direct spreading measures of laguerre polynomials the direct spreading measures of the laguerre polynomials l n a x which quantify the distribution of its rakhmanov probability density r n a x 1d n 2x ae x l n a x 2 along the positive real line in various complementary and qualitatively different ways are investigated these measures include the familiar root mean square or standard deviation and the information theoretic lengths of fisher renyi and shannon types the fisher length is explicitly given the renyi length of order q such that email protected n is also found in terms of n a by means of two error free computing approaches one makes use of the lauricella function f a 2 q 1 1q 1q 1 which is based on the srivastava niukkanen linearization relation of laguerre polynomials and another one utilizes the multivariate bell polynomials of combinatorics the shannon length cannot be exactly calculated because of its logarithmic functional form but its asymptotics is provided and sharp bounds are obtained by the use of an information theoretic optimization procedure finally all these spreading measures are mutually compared and computationally analyzed in particular it is found that the apparent quasilinear relation between the shannon length and the standard deviation becomes rigorously linear only asymptotically i e for email protected 1

generalised operator reduction formula for multiple hypergeometric series nf x1 xn the reduction formula obtained in a previous paper niukkanen 1983 has been generalised as follows if nf and nf are generalised hypergeometric series of n variables ghs n then the operation on nf x1t1 xntn nf delta delta t1 delta delta tn gives at t1 t2 tn 0 a function nf x1 xn which is again a ghs n the differentiation procedure can be regarded as an algebraic omega multiplication which gives rise to a group theoretical interpretation of the method a concept of omega equivalent relations has been introduced which allows systematisation of numerous results obtained in special functions theory as the functions nf comprise a number of physically interesting series the operator factorisation method seems to be applicable to many physical problems providing a possibility of reducing any nf to simpler functions of the same class

polynomial poisson algebras for classical superintegrable systems with a third order integral of motion we present polynomial poisson algebras for the eight classical potentials in two dimensional euclidian space that separate in cartesian coordinates and allow a third order integral of motion some of the classical superintegrable potentials do not coincide with quantum ones but are their singular limits we show that all bounded trajectories in these potentials are periodic

renyi entropies lq norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions the quantification of the spreading of the orthogonal polynomials p n x can be investigated by means of the renyi entropies r q r q being a positive integer number of the associated rakhmanov probability densities r x email protected x p n 2 x where w x is the corresponding weight function the renyi entropies are closely related to the l q norms of the polynomials in this manuscript the l q norms and the associated renyi entropies of the real hypergeometric orthogonal polynomials i e hermite laguerre and jacobi polynomials and the generalized hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of lauricella and srivastava daoust types which are evaluated at some specific values of 2q variables these functions depend on 4q 1 and 6q 2 parameters respectively which are determined by the order q the degree n of the polynomial and the parameters of the orthogonality weight function w x the key idea is based on some extended linearization formulas for these polynomials these results open the way to determine the renyi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials

operational rules and arbitrary order hermite generating functions operational identities often exploited in the theory of exponential operators are used to derive generating functions of hermite and kampe de feriet polynomials the method we propose allows us to obtain unilateral bilinear and bilateral generating functions by using the same procedure possible extensions of the technique are also discussed

orthogonality of some polynomial sets via quasi monomiality the lowering operator s and the raising operator t associated with a polynomial set p n n 0 are two operators which are independent of n and satisfy the relationships s p n x np n 1 x and t p n x p n 1 x n n 0 in this paper we use these operators to study the orthogonality of some polynomial sets more precisely we state sufficient conditions in terms of the s and t operators to ensure the orthogonality we also express explicitly by means of the s operator the linear functional for which the orthogonality holds true we obtain some well known results as particular cases of the results presented in this paper

a class of generalised multiple hypergeometric series arising in physical and quantum chemical applications the multivariable hypergeometric function nf x1 xn considered recently by niukkanen 1984 is a straightforward generalisation of certain well known hypergeometric functions of n variables indeed it provides a unification of the generalised hypergeometric function pfq of one variable appell and kampe de feriet functions of two variables and lauricella functions of n variables as well as of many other hypergeometric series which arise naturally in physical and quantum chemical applications the author derives several interesting properties of this multivariable hypergeometric function including for example many which were not given by niukkanen as useful consequences of substantially more general results available in the literature

transformation of certain generalized kampe de feriet functions new transformations of generalized kampe de feriet functions of several variables which occur in physical and quantum chemical contexts are given unlike many of the known results for these functions the variables are not fixed

connection problems via lowering operators the lowering operator associated with a polynomial set pn n 0 is an operator not depending on n and satisfying the relation pn npn 1 in this paper we express explicitly the connection coefficients between two polynomial sets using their corresponding lowering operators we obtain some well known results as particular cases including some duplication and addition formulas

on euler superintegrable systems we discuss the euler approach to the construction and investigation of the superintegrable systems with additional quadratic and cubic integrals of motions

unified framework for the effective rate analysis of wireless communication systems over miso fading channels this paper proposes a unified framework for the effective rate analysis over arbitrary correlated and not necessarily identical multiple input single output miso fading channels which uses the moment generating function mgf based approach and h transform representation the proposed framework has the potential to simplify the cumbersome analysis procedure compared with the probability density function based approach moreover the effective rates over two specific fading scenarios are investigated namely independent but not necessarily identical distributed i n i d miso hyper fox s h fading channels and arbitrary correlated generalized k fading channels the exact analytical representations for these two scenarios are also presented by substituting corresponding parameters the effective rates in various practical fading scenarios such as rayleigh nakagami m weibull gamma and generalized k fading channels are readily available in addition asymptotic approximations are provided for the proposed h transform and mgf based approach as well as for the effective rate over i n i d miso hyper fox s h fading channels simulations under various fading scenarios are also presented which support the validity of the proposed method

state estimation for invariant systems on lie groups with delayed output measurements this paper proposes a state estimation methodology for invariant systems on lie groups where outputs of the system are measured with delay the proposed method is based on cascading an observer and a predictor the observer uses delayed measurements and provides estimates of delayed states the predictor uses those estimates together with the current inputs of the system to compensate for the delay and to provide a prediction of the current state of the system we consider three classes of left invariant right invariant and mixed invariant systems and propose predictors tailored to each class the key contribution of the paper is to exploit the underlying symmetries of systems to design novel predictors that are computationally simple and generic in the sense that they can be combined with any stable observer or filter we provide a rigorous stability analysis demonstrating that the prediction of the current state converges to the current system trajectory if the observer state converges to the delayed system trajectory the good performance of the proposed approach is demonstrated using a sophisticated software in the loop simulator indicating the robustness of the observer predictor methodology even when large measurement delays are present

modified clebsch gordan type expansions for products of discrete hypergeometric polynomials abstract starting from the second order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials p n we find the analytical expressions of the expansion coefficients of any polynomial rm x and of the product rm x qj x in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric polynomials here qj x denotes an arbitrary discrete hypergeometric polynomial of degree j the particular cases in which r m corresponds to the non orthogonal families x m the rising factorials or pochhammer polynomials x m and the falling factorial or stirling polynomials x m are considered in detail the connection problem between discrete hypergeometric polynomials which here corresponds to the product case with m 0 is also studied and its complete solution for all the classical discrete orthogonal hypergeometric cdoh polynomials is given also the inversion problems of cdoh polynomials associated to the three aforementioned nonorthogonal families are solved

duality between integrable stackel systems canonical transformations of the extended phase space are applied to the integrable stackel systems all these transformations may be associated with an ambiguity of the abel map on the corresponding hyperelliptic curve for some stackel systems with two degrees of freedom the 2 2 lax representations and the dynamical r matrix algebras are constructed

generalised hypergeometric series nf x1 xn arising in physical and quantum chemical applications functions nf x1 xn which are a straightforward generalisation of standard hypergeometric functions of n variables are introduced a convenient operator representation is established for these functions which permits one to consider in a uniform and simple way standard hypergeometric functions generalised hypergeometric series of one variable appell kampe de feriet and lauricella functions etc as well as many other nameless hypergeometric series arising in physical and quantum chemical applications generating functions generalisations of bailey 1935 and chaundy 1943 expansions recurrence relations reduction rules and some other topics are discussed

characterizations of generalized hermite and sieved ultraspherical polynomials a new characterization of the generalized hermite polynomials and of the orthogonal polynomials with respect to the measure ixy 1 x2 1 2dx is derived which is based on a reversing property of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of laguerre and jacobi polynomials a similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind these results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order

operational rules and a generalized hermite polynomials in this paper we use operational rules associated with three operators corresponding to a generalized hermite polynomials introduced by szego to derive as far as we know new proofs of some known properties as well as new expansions formulae related to these polynomials

extended information filter on matrix lie groups in this paper we propose a new state estimation algorithm called the extended information filter on lie groups the proposed filter is inspired by the extended kalman filter on lie groups and exhibits the advantages of the information filter with regard to multisensor update and decentralization while keeping the accuracy of stochastic inference on lie groups we present the theoretical development and demonstrate its performance on multisensor rigid body attitude tracking by forming the state space on the so 3 r3 group where the first and second component represent the orientation and angular rates respectively the performance of the filter is compared with respect to the accuracy of attitude tracking with parametrization based on euler angles and with respect to execution time of the extended kalman filter formulation on lie groups the results show that the filter achieves higher performance consistency and smaller error by tracking the state directly on the lie group and that it keeps smaller computational complexity of the information form with respect to high number of measurements

on the laplace transform of the g function it is pointed out that the main result of a recent letter on the laplace transform of meijer s g function is actually a special case of an already established transform

unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold in this paper we prove that the two dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the poisson algebra of the integrals of motion there are six general fundamental classes of superintegrable systems analytic formulas for the involved integrals are calculated in all the cases all the known superintegrable systems are classified as special cases of these six general classes

addition theorems and the drach superintegrable systems we propose a new construction of the polynomial integrals of motion related to the addition theorems as an example we reconstruct drach systems and get some new two dimensional superintegrable stackel systems with third fifth and seventh order integrals of motion

square root sigma point information filtering the sigma point information filters employ a number of deterministic sigma points to calculate the mean and covariance of a random variable which undergoes a nonlinear transformation these sigma points can be generated by the unscented transform or stirling s interpolation which corresponds to the unscented information filter uif and the central difference information filter cdif respectively in this technical note we develop the square root extensions of uif and cdif which have better numerical properties than the original versions e g improved numerical accuracy double order precision and preservation of symmetry we also show that the square root unscented information filter sruif might lose the positive definiteness due to the negative cholesky update whereas the square root central difference information filter srcdif has only positive cholesky update therefore the srcdif is preferable to the sruif concerning the numerical stability

generating differential operators for the basis functions in the variational lcao type methods the analytical expressions for operators which allow us to generate the basis functions from the simplest ground states by means of differentiating by parameters are obtained some important properties of these operators are indicated this approach allows us to simplify the structure of molecular matrix elements whose calculation presents considerable difficulties in the variational lcao type methods

on the connection and linearization problem for discrete hypergeometric q polynomials abstract in the present paper starting from the second order difference hypergeometric equation on the non uniform lattice x s satisfied by the set of discrete hypergeometric orthogonal q polynomials p n we find analytical expressions of the expansion coefficients of any q polynomial r m x s on x s and of the product r m x s q j x s in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric q polynomials

some integrals involving three modified bessel functions ii the integrals 0z at k bt ik ct dt where z i k are calculated with the help of the factorization properties of the function f4 results are given for real parameters a b c both when they are and are not in a triangle configuration some generalizations using derivation with respect to the parameters are considered

on the superintegrable richelot systems we introduce the richelot class of superintegrable systems in n dimensions whose n n equations of motion coincide with the abel equations on the n 1 genus hyperelliptic curve corresponding additional integrals of motion are the second order polynomials of momenta and multiseparability of the richelot superintegrable systems is related to the classical theory of covers of the hyperelliptic curves

some clebsch gordan type linearisation relations and other polynomial expansions associated with a class of generalised multiple hypergeometric series arising in physical and quantum chemical applications in this sequel to srivastava s work 1987 a number of new expansions in series of various classes of hypergeometric polynomials are derived for the general multivariable hypergeometric function which provides an interesting and useful unification of numerous families of hypergeometric functions of one two or more variables encountered naturally in a wide variety of physical and quantum chemical applications by suitably specialising some of these polynomial expansions clebsch gordan type linearisation relations are deduced for the products of several jacobi or laguerre polynomials it is also shown how one of the linearisation relations involving laguerre polynomials would yield the corrected and modified versions of a couple of results given by niukkanen 1985

new properties of hypergeometric series derivable from feynman integrals i transformation and reduction formulae a study in the statistical mechanics of phase transitions has involved certain seemingly simple feynman integrals which have yielded surprisingly significant mathematical results the integrals enable the derivation of new transformation summation and reduction formulae for some single and multiple variable hypergeometric series the main result reported in this paper is a transformation formula a special case of which is a summation formula for a symmetric combination of two non terminating clausen s hypergeometric series of unit argument

quantum interference effects on the superposition of n displaced number states we describe properties of a quantum system prepared in a superposition of n displaced number states with emphasis on the interference effects among the state components explicit expressions for wavefunctions in the coordinate and fock state representations photon number distribution variances and quantum correlation coefficient of the quadrature operators and quasiprobability distributions are calculated in detail nonclassical effects due to such interference and the analogy with diffraction patterns arising in an n slit young type experiment are the main results in the present work in particular we show that interference and correlation effects in phase space are connected with the nondiagonal term of the quasiprobability distributions providing further insight into the phenomenon

a class of stable square root nonlinear information filters information filters can process nonlinear systems with uncertain prior knowledge and the particular square root form of adaptive filters can improve numerical stability based on a square root decomposition of information matrix and an extra positive definite matrix the unscented transform and the cubature rule are applied to the information filtering architecture for nonlinear estimation a class of stable square root nonlinear information filters is then proposed in this technical note in addition the boundedness of their estimation errors is also proven results from simulations of filtering a chaotic map demonstrate that the proposed square root nonlinear filters can improve numerical stability and has better filtering performance than other information filters

some reducible generalized kampe de feriet functions on montre qu une fonction de kampe de feriet generalisee f q 1 1 p 2 2 dont n variables sont toutes egales et dont des parametres satisfont certaines conditions est exprimable en fonction de la fonction hypergeometrique unique p 1fq

some results on quasi monomiality a polynomial set p n n 0 is called quasi monomial if and only if it is possible to define two operators p and m independent of n such thatp p n x np n 1 x andm p n x p n 1 x in this paper we show that every polynomial set is quasi monomial and we present some useful tools to explicitly express the p and m operators for some polynomial families given by their generating functions the obtained results are applied to boas buck polynomial sets

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

on the linearization problem involving pochhammer symbols and their q analogues abstract in this paper we present a simple recurrent algorithm for solving the linearization problem involving some families of q polynomials in the exponential lattice x s c1qs c3 some simple examples are worked out in detail

on algebraic construction of certain integrable and super integrable systems abstract we propose a new construction of two dimensional natural bi hamiltonian systems associated with a very simple lie algebra the presented construction allows us to distinguish three families of super integrable monomial potentials for which one additional first integral is quadratic and the second one can be of arbitrarily high degree with respect to the momenta many integrable systems with additional integrals of degree greater than two in momenta are given moreover an example of a super integrable system with first integrals of degree two four and six in the momenta is found

linearization and connection coefficients for hypergeometric type polynomials we consider the problem of finding closed analytical formulas for both the linearization and connection coefficients for hypergeometric type polynomials directly in terms of the corresponding differential equations we illustrate the method by producing explicit formulas for hermite polynomials

differential equation of appell polynomials via the factorization method let pn x n 0 be a sequence of polynomials of degree n we define two sequences of differential operators n and n satisfying the following properties n pn x pn 1 x n pn x pn 1 x by constructing these two operators for appell polynomials we determine their differential equations via the factorization method introduced by infeld and hull rev mod phys 23 1951 21 the differential equations for both bernoulli and euler polynomials are given as special cases of the appell polynomials

general linearization formulae for products of continuous hypergeometric type polynomials the linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials hps of a continuous variable which consists of the expansion of the product of two arbitrary hps in series of an orthogonal hp set here this problem is algebraically solved directly in terms of the coefficients of the second order differential equations satisfied by the involved polynomials general expressions for the expansion coefficients are given in integral form and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis hermite laguerre and jacobi as well as several generalized linearization formulae involving these families the connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions which often reduce to a single function of the same type when possible these functions are evaluated in closed form in some cases sign properties of the coefficients such as positivity or non negativity conditions are derived as a by product from their resulting explicit representations

transformations of multipleq series with quasi arbitrary terms abstract the main object of this paper is to present several presumably new transformations of series whose terms involve basic or q pochhammer symbols together with an essentially arbitrary factor each of these general q series transformations is shown to be related rather closely to a certain basic hypergeometric summation or transformation formula relevant connections of many of the multiple q series transformations with various known results are also considered briefly

generalized linearization problems we prove that if pi x and pj x are two families of semi classical orthogonal polynomials all the linearization coefficients li j k occurring in the product of these two families satisfy a linear recurrence relation involving only the k index this property also extends to the linearization coefficients arising from an arbitrary number of products of semi classical orthogonal polynomials

transformation of the stackel matrices preserving superintegrability if we take a superintegrable stackel system and make variables faster or slower that is equivalent to a trivial transformation of the stackel matrix and potentials then we obtain an infinite family of superintegrable systems with explicitly defined additional integrals of motion we present some examples of such transformations associated with angle variables expressed via logarithmic functions if we take a superintegrable stackel system and make variables faster or slower that is equivalent to a trivial transformation of the stackel matrix and potentials then we obtain an infinite family of superintegrable systems with explicitly defined additional integrals of motion we present some examples of such transformations associated with angle variables expressed via logarithmic functions

superintegrable systems and riemann roch theorem in algebraic geometry there is a reduction algorithm that transforms the unreduced divisor into a unique reduced divisor which existence is guaranteed by the riemann roch theorem we discuss application of this algorithm to construction of finite dimensional superintegrable systems with n degrees of freedom identifying coordinates of the reduced divisor with integrals of motion

neumann expansions for a certain class of generalised multiple hypergeometric series arising in physical and quantum chemical applications a multivariable hypergeometric function which was studied recently by niukkanen 1984 and srivastava 1985 provides an interesting and useful unification of the generalised hypergeometric pfq function of one variable with p numerator and q denominator parameters appell and kampe de feriet s hypergeometric functions of two variables and lauricella s hypergeometric functions of n variables and also of many other classes of hypergeometric series which arise naturally in various physical and quantum chemical applications indeed as already observed by srivastava this multivariable hypergeometric function is an obvious special case of the generalised lauricella hypergeometric function of n variables which was first introduced and studied systematically by srivastava and daoust 1969 by employing such useful connections of this function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely srivastava presented several interesting and useful properties of this multivariable hypergeometric function most of which did not appear in the work of niukkanen the object of this sequel to srivastava s work is to derive a number of new neumann expansions in series of bessel functions for the multivariable hypergeometric function from substantially more general expansions involving for example multiple series with essentially arbitrary terms some interesting special cases of the neumann expansions presented here are also indicated

construction of new integrable hamiltonians in two degrees of freedom a new procedure for deriving integrable hamiltonians and their constants of the motion is introduced we term this procedure the truncation program integrable hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter e the relationship between this program and the whittaker program in two degrees of freedom is discussed integrable hamiltonians occurring in the whittaker program a generalization of whittaker s work possess constants of the motion which are polynomials in the momentum coordinates many previously known integrable hamiltonians are derived a new family of integrable double resonance hamiltonians and a new family of integrable hamiltonians of the form p21 p22 2 v q1 q2 are derived

superintegrable n 2 systems quadratic constants of motion and potentials of drach the properties of superintegrable systems in two degrees of freedom possessing three independent globally defined constants of motion are studied using as an approach the existence of hidden symmetries and the generalized noether s theorem the potentials are obtained as solution of a system of two partial differential equations first the case of standard lagrangians is studied and then the method is applied to the case of lagrangians with a pseudo euclidean kinetic term finally the results are related with other approaches and with a family of potentials admitting a second integral of motion cubic in the velocities obtained by drach

some multiple hypergeometric transformations and associated reduction formulas the main object of the present paper is to derive various classes of double series identities and to show how these general results would apply to yield some known or new reduction formulas for the appell kampe de feriet and lauricella hypergeometric functions of several variables a number of closely related linear generating functions for the classical jacobi polynomials are also investigated

reduction and summation formulae for certain classes of generalised multiple hypergeometric series arising in physical and quantum chemical applications a multivariable hypergeometric function considered recently by niukkanen and srivastava 1984 1985 provides an interesting unification of the generalised hypergeometric function pfq of one variable appell and kampe de feriet functions of two variables and lauricella functions of n variables and also of many other hypergeometric series which arise naturally in various physical and quantum chemical applications as pointed out by srivastava the multivariable hypergeometric function is an obvious special case of the generalised lauricella function of n variables which was first introduced and studied by srivastava and daoust 1969 by employing such connections of this multivariable hypergeometric function with the more general multiple hypergeometric functions several interesting and useful properties of this function have been studied many of which have not been given by niukkanen the author derives a number of new reduction formulae for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms some interesting summation formulae for the multivariable hypergeometric function with x1 xn 1 and x1 xn 1 are also presented

applications of the operator h α β to the humbert double hypergeometric functions by making use of some techniques based upon certain new inverse pairs of symbolic operators the authors investigate several decomposition formulas associated with humbert hypergeometric functions f 1 f 2 f 3 j 1 j 2 x 1 and x 2 these operational representations are constructed and applied in order to derive the corresponding decomposition formulas with the help of these inverse pairs of symbolic operators as many as 34 decomposition formulas are found euler type integrals connected with humbert functions are also presented

the stackel systems and algebraic curves we show how the abel jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves we prove that derivative of the abel jacobi map is just the stackel matrix which determines n orthogonal curvilinear coordinate systems in a flat space the lax pairs r matrix algebras and explicit form of the flat coordinates are constructed an application of the weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space

some discrete d orthogonal polynomial sets in this paper we characterize all polynomial sets which are at the same time d orthogonal and appell the resulting polynomials reduce to charlier polynomials for d 1 1 various properties of the obtained polynomials are singled out generating function recurrence relation of order d 1 and a difference equation of order d 1 we also explicitly express the d dimensional functional for which the d orthogonality holds