a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

orthogonality relations and generating functions for the generalized bessel polynomials abstract in this paper the author begins by presenting a systematic historical account of the orthogonality of the generalized bessel polynomials that arise naturally in a number of seemingly diverse contexts the related expansion problem and various recent results involving generating functions for these polynomials are discussed next some relevant generalizations of many of these generating functions are also considered

the extended heine stieltjes polynomials associated with a special lmg model the extended heine stieltjes polynomials associated with a special lipkin meshkov glick lmg model corresponding to the standard two site bose hubbard model are derived based on the stieltjes correspondence it is shown that there is a one to one correspondence between zeros of this new polynomial and solutions of the bethe ansatz equations for the lmg model a one dimensional classical electrostatic analog corresponding to the special lmg model is established according to stieltjes early work it shows that any possible configuration of equilibrium positions of the charges in the electrostatic problem corresponds uniquely to one set of roots of the bethe ansatz equations for the lmg model and the number of possible configurations of equilibrium positions of the charges equals exactly the number of energy levels in the lmg model some relations of sums of powers and inverse powers of zeros of the new polynomials related to the eigenenergies of the lmg model are derived

on sums of powers of zeros of polynomials abstract due to girard s sometimes called waring s formula the sum of the rth power of the zeros of every one variable polynomial of degree n pn x can be given explicitly in terms of the coefficients of the monic p n x polynomial this formula is closely related to a known n 1 variable generalisation of chebyshev s polynomials of the first kind tr n 1 the generating function of these power sums or moments is known to involve the logarithmic derivative of the considered polynomial this entails a simple formula for the stieltjes transform of the distribution of zeros perron stieltjes inversion can be used to find this distribution e g for n classical orthogonal polynomials are taken as examples the results for ordinary chebyshev tn x and un x polynomials are presented in detail this will correct a statement about power sums of zeros of chebyshev s t polynomials found in the literature for the various cases jacobi laguerre hermite these moment generating functions provide solutions to certain riccati equations

mixtures in nonstable levy processes we analyse the lprocesses produced by means of two interconnected classes of nonstable infinitely divisible distribution the variance gamma and the student laws while the variance gamma family is closed under convolution the student one is not this makes its time evolution more complicated we prove that at least for one particular type of student processes suggested by recent empirical results and for integral times the distribution of the process is a mixture of other types of student distributions randomized by means of a new probability distribution the mixture is such that along the time the asymptotic behaviour of the probability density functions always coincide with that of the generating student law we put forward the conjecture that this can be a general feature of the student processes we finally analyse the ornstein uhlenbeck process driven by our lnoises and show a few simulations of it

sums of kloosterman sums in arithmetic progressions and the error term in the dispersion method we prove a bound for quintilinear sums of kloosterman sums with congruence conditions on the smooth summation variables this generalizes classical work of deshouillers and iwaniec and is key to obtaining power saving error terms in applications notably the dispersion method as a consequence assuming the riemann hypothesis for dirichlet l functions we prove a power saving error term in the titchmarsh divisor problem of estimating sum p leq x tau p 1 unconditionally we isolate the possible contribution of siegel zeroes showing it is always negative extending work of fouvry and tenenbaum we obtain power saving in the asymptotic formula for sum n leq x tau k n tau n 1 reproving a result announced by bykovski u i and vinogradov by a different method the gain in the exponent is shown to be independent of k if a generalized lindel of hypothesis is assumed

distributions of zeros of discrete and continuous polynomials from their recurrence relation the hypergeometric polynomials in a continuous or a discrete variable whose canonical forms are the so called classical orthogonal polynomial systems are objects which naturally appear in a broad range of physical and mathematical fields from quantum mechanics the theory of vibrating strings and the theory of group representations to numerical analysis and the theory of sturm liouville differential and difference equations often they are encountered in the form of a three term recurrence relation ttrr which connects a polynomial of a given order with the polynomial of the contiguous orders this relation can be directly found in particular by use of lanczos type methods tight binding models or the application of the conventional discretization procedures to a given differential operator here the distribution of zeros and its asymptotic limit characterized by means of its moments around the origin is described for the continuous classical hermite laguerre jacobi bessel polynomials and for the discrete classical charlier meixner kravchuk hahn polynomials by means of a general procedure which i only requires the three term recurrence relation and ii avoids the often high brow subtleties of the potential theoretic considerations used in some recent approaches let us underline that the orthogonality condition is not required in our approach the moments are given in an explicit manner which at times allow us to recognize the analytical form of the corresponding distribution

on weighted critical imbeddings of sobolev spaces our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of sobolev spaces we characterize weights which do not change an exponential space up to equivalence of norms specifically we first prove that l exp t alpha chi b l exp t alpha rho if and only if rho q in l q with some q 1 second we consider the sobolev space w 1 n varomega where is a bounded domain in mathbb r n with a sufficiently smooth boundary and its imbedding into a weighted exponential orlicz space l exp t p varomega rho where is a radial and non increasing weight function we show that there exists no effective weighted improvement of the standard target l exp t n varomega l exp t n varomega chi varomega in the sense that if w 1 n varomega is imbedded into l exp t p varomega rho then l exp t p varomega rho and l exp t n varomega coincide up to equivalence of the norms that is we show that there exists no effective improvement of the standard target space the same holds for critical cases of higher order sobolev spaces and even besov and lizorkin triebel spaces

computing the moments of the density of zeros for orthogonal polynomials abstract a representation formula by means of the generalized lucas polynomials of first kind for the moments of the density of zeros of orthogonal polynomial sets defined by a three term recurrence relation is used to derive information directly from the entries of the jacobi matrix numerical computations are explicitly developed in some particular case

current trends in asymptotics some problems and some solutions abstract i discuss five topics of current interest in asymptotic analysis the use of probabilistic methods to estimate the growth of combinatorial sequences asymptotic methods in the theory of random walks the asymptotic estimation of solutions of difference equations asymptotic expansions in generalized scales and the computation by asymptotic methods of distributions whose moments are known i give a number of examples and mention some outstanding unsolved problems

some results on the asymptotic distribution of the zeros of orthogonal polynomials abstract define a discrete measure that attributes masses of size 1 n at every zero of the polynomial of degree n orthogonal with respect to some positive measure on the real line the weak limit of this sequence of measures is studied by many authors in this paper we give some results on the rate of this asymptotic behaviour in particular the results show how the behaviour of the spectral measure with respect to which the polynomials are orthogonal affects this asymptotic zero behaviour

a note on certain generating functions for the generalized bessel polynomials abstract the main object of the present paper is to prove a general theorem on bilinear bilateral or mixed multilateral generating functions for the generalized bessel polynomials it is pointed out that this theorem and some other earlier results on the subject would provide unifications and generalizations of numerous generating functions which were obtained recently by using group theoretic techniques some interesting applications of these general results and their natural connections with the classical laguerre polynomials are also indicated

hankel transform of a series reversion of a certain rational function abstract starting from a sequence v n n n 0 defined by the series reversion of a certain rational function b x we use the method based on orthogonal polynomials to give closed form evaluation of the hankel transform of sequence v n n n 0 and corresponding shifted sequences that gives a proof the conjecture by barry 4 concerning the ratio equalities connecting those terms moreover in a special case we compute lu decomposition of the hankel matrix of a shifted sequence v n n n 0 which reproves the expression for the hankel transform of that sequence

melting and wetting transitions in the three state chiral clock model abstract the melting transition of the two dimensional three state asymmetric or chiral clock model is examined evidence from scaling arguments and analysis of perturbation series is presented indicating that the chiral symmetry breaking operator is relevant at the symmetric or pure potts critical point with a crossover exponent of o 0 2 the remainder of the commensurate disordered phase boundary therefore appears to be in a new universality class distinct from the pure three state potts transition an interfacial wetting transition that plays an important role in the crossover between the two types of critical behavior is discussed the location and exponents of this wetting transition are obtained both in a low temperature limit using generating function techniques and in a systematic low temperature expansion of the transfer matrix

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

short sums of certain arithmetic functions uniformly for 1 6 q 6 y this result has turned out to be very useful in a wide range of applications a closer inspection of its proof reveals in the case q 1 that a g needs only be sub multiplicative i e g mn 6 g m g n for m n 1 with g 1 1 b the constant implicit in the sign depends only on a b and c given condition ii above need only hold for a particular shiu s result has been generalised by nair 5 to sub multiplicative functions of polynomial values in a short interval in this paper we weaken the property of sub multiplicativity significantly to appreciably widen the range of application of such a result consider for any fixed k 2 n the class mk a b of non negative arithmetic functions f n1 nk such that

scaling of levy student processes abstract student s t distributions are widely used in financial studies as heavy tailed alternatives to normal distributions as these distributions are not closed under convolution there exist no levy processes with student s t marginals at all points in time in this article we show that a student s t approximation of these marginals is still suitable while not exact using this approximation we are able to describe the scaling behavior of such levy student processes and the parameters of its marginal distributions by a simple analytical scaling law this scaling law drastically simplifies the use of levy student processes as a general diffusion process in various interdisciplinary applications we explicitly provide an application in the context of modelling high frequency price returns

characterization of generalized bessel polynomials in terms of polynomial inequalities abstract generalized bessel polynomials gbps are characterized as the extremal polynomials in certain inequalities in l 2 norm of markov type

rational jacobi matrices and certain quantum mechanical problems in several branches of physics the hamiltonian of a many body system can be reduced to a rational jacobi matrix i e a jacobi matrix whose elements are rational functions of the suffix by means of a method of the lanczos type here it is shown that one can calculate in a simple analytical way the asymptotic eigenvalue density of all these matrices by means of its moments without solving the corresponding eigenvalue problem the method is applied to a large class of quantum mechanical models of hamiltonians

uniform gateaux differentiability and weak uniform rotundity in musielak orlicz function spaces of bochner type equipped with the luxemburg norm abstract criteria for weak uniform rotundity and reflexivity of musielak orlicz function spaces of bochner type equipped with the luxemburg norm are given although criteria for uniform gateaux differentiability and weak uniform rotundity of musielak orlicz function spaces of real functions equipped with the luxemburg norm were known they can be also easily deduced from our main results

some families of generating functions for the multiple orthogonal polynomials associated with modified bessel k functions the main object of this paper is to derive several substantially more general families of bilinear bilateral and mixed multilateral finite series relationships and generating functions for the multiple orthogonal polynomials associated with the modified bessel k functions also known as macdonald functions some special cases of the above statements are also given

trilinear forms with kloosterman fractions abstract we give new bounds for a m n m n a e a m n where m n and a are arbitrary coefficients improving upon a result of duke friedlander and iwaniec 7 we also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations

le probleme de tchebychev pour le douzieme polynome cyclotomique let p n denote the largest prime factor of the integer n and 12 n n 4 n 2 1 we prove that for x large enough we have p x x 1 c with c 10 26531

on the zeros of eigenfunctions of polynomial differential operators it is shown that for polynomial eigenfunctions of an ordinary polynomial differential operator with coefficients depending only on the independent variable it is possible to determine the density of nodes around the mean without solving the corresponding eigenvalue problem this is done by means of the first few moments which can be directly expressed in terms of the above mentioned coefficients also very simple expressions for the asymptotic values i e when the degree of the polynomial becomes very large of these quantities are found for illustration these results are applied to various orthogonal polynomials which satisfy ordinary differential equations of second fourth and or sixth order

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

extremal polynomials on discrete sets we study asymptotics for orthogonal polynomials and other extremal polynomials on infinite discrete sets typical examples being the meixner polynomials and the charlier polynomials following ideas of rakhmanov dragnev and saff weshow that the asymptotic behaviour is governed by a constrained extremal energy problem for logarithmic potentials which can be solved explicitly we give formulas for the contracted zero distributions the n th root asymptotics and the asymptotics of the largest zeros

on two sets of orthogonal polynomial systems encountered in nonlinear physics two sets with an infinite number of new systems of orthogonal polynomials have recently been discovered by smith in connection with some non linear physical problems e g the dispersion of a buoyant contaminant in a fluid they appear as solutions of non linear differential equations let p x m k s and q x m k with n 0 k m k 2m 2m k denote a generic system of each set the positive integers k and m are restricted by k 1 k although the orthogonality interval of these polynomials is real their zeros are generally complex here the sum rules y x x r 1 2 for the zeros x i 1 2 n of the nth degree polynomials of these sets are studied it is found that all these quantities vanish except for r pm p being an arbitrary positive integer simple recurrent expressions for ypn are given

nearest neighbour spacing distributions of the β hermite ensemble of random matrices the evolution with of the distributions of the spacing s between nearest neighbour levels of unfolded spectra of random matrices from the hermite ensemble he is investigated by monte carlo simulations the random matrices from the he are real symmetric and tridiagonal where which can take any positive value is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues the distribution of eigenvalues coincide with those of the three classical gaussian ensembles for 1 2 4 the use of the he ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices generalized gamma distributions are shown to be excellent approximations of the nearest neighbor spacing nns distributions for any while being still simple they account both for the level repulsion in s when s 0 and for the whole shape of the nns distributions in the range of s which is accessible to experiment or to most numerical simulations the exact nns distribution of the goe 1 is in particular significantly better described by a generalized gamma distribution than it is by the wigner surmise while the best generalized gamma approximation coincides essentially with the wigner surmise for 2 they describe too the evolution of the level repulsion between that of a poisson distribution and that of a goe distribution when increases from 0 to 1 the distribution of ln s related to the electrostatic interaction energy between neighbouring charges is accordingly well approximated by a generalized gumbel distribution for any 0 the distributions of the minimum nn spacing between eigenvalues of matrices from the he obtained both from as calculated eigenvalues and from unfolded eigenvalues are brody distributions which are classically used to characterize the spectral fluctuations of various physical systems

the δn statistic for the β hermite ensemble abstract the fluctuation n of the n th unfolded eigenvalue was recently characterized for the classical gaussian ensembles of n n random matrices goe gue gse it is investigated here for the hermite ensemble as a function of the reciprocal of the temperature by monte carlo simulations the ensemble averaged fluctuation n 2 and the autocorrelation function vary logarithmically with n for any 0 1 n n the simple logarithmic behavior of the higher order moments of n reported in the literature for the goe 1 and the gue 2 holds for any 0 and is accounted for by gaussian distributions whose variances depend linearly on ln n

bilinear forms with kloosterman sums and applications we prove nontrivial bounds for general bilinear forms in hyper kloosterman sums when the sizes of both variables may be below the range controlled by fourier analytic methods polya vinogradov range we then derive applications to the second moment of cusp forms twisted by characters modulo primes and to the distribution in arithmetic progressions to large moduli of certain eisenstein hecke coefficients on gl 3 our main tools are new bounds for certain complete sums in three variables over finite fields proved using methods from algebraic geometry especially l adic cohomology and the riemann hypothesis

the distribution of zeros of general q polynomials a general system of q orthogonal polynomials is defined by means of its three term recurrence relation this system encompasses many of the known families of q polynomials among them the q analogue of the classical orthogonal polynomials the asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation this result is applied to specific classes of polynomials known by the names q hahn q kravchuk q racah q askey and wilson al salam carlitz and the celebrated little and big q jacobi

trudinger moser type inequalities with logarithmic weights in dimension n abstract we consider borderline embeddings of trudinger moser type for weighted sobolev spaces in bounded domains in r n the embeddings go into orlicz spaces with exponential growth functions it turns out that the most interesting weights are powers of the logarithm for which an explicit dependence of the maximal growth functions can be established corresponding moser type results are also proved with explicit sharp exponents in the particular case of a logarithmic weight with the limiting exponent n 1 a maximal growth of double exponential type is obtained while for any larger exponent the embedding goes into l

on the kuznetsov trace formula for pgl2 c abstract in this note using a representation theoretic method of cogdell and piatetski shapiro we prove the kuznetsov trace formula for an arbitrary discrete group in pgl 2 c that is cofinite but not cocompact an essential ingredient is a kernel formula recently proved by the author on bessel functions for pgl 2 c this approach avoids the difficult analysis in the existing method due to bruggeman and motohashi

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

certain classes of finite series relationships and generating functions involving the generalized bessel polynomials recently by making use of the familiar group theoretic lie algebraic method a certain mixed trilateral finite series relationship was proven for the generalized bessel polynomials the main object of this paper is to derive several substantially more general families of bilinear bilateral and mixed multilateral finite series relationships and generating functions for these and other related classes of hypergeometric polynomials a duly corrected and modified analogue of the aforementioned trilateral finite series relationship is shown to follow by suitably specializing one of the general results presented here several closely related presumably new finite series relationships and generating functions some of which also involve for example the stirling numbers of the second kind are onsidered rather briefly

how likely is polya s drunkard to stay in x y z in his celebrated paper polya has considered the random walk in the three dimensional cubic lattice and showed that the probability of return to the origin is less than 1 subsequent authors have shown that the probability is 34 053 here we consider the same random walk with the restriction that the drunkard is only allowed to stay inx y z it is shown that his probability of returning to the originand staying in the allowed region is 6 4844

some general families of generating functions for the laguerre polynomials abstract the object of the present paper is to develop rather systematically some general families of bilinear bilateral or mixed multilateral generating functions for the classical laguerre polynomials numerous straightforward consequences of some of the results considered here frequently appear in the literature especially from the viewpoint of lie groups and lie algebras it is also pointed out how the main generating functions can be suitably applied to derive numerous further results involving laguerre polynomials and various other related polynomials

on the zeros of meixner polynomials we investigate the zeros of a family of hypergeometric polynomials m n x beta c beta n 2f 1 n x beta 1 frac 1 c n in mathbb n known as meixner polynomials that are orthogonal on 0 infty with respect to a discrete measure for beta 0 and 0 1 as well as the polynomials k n x p n with non hermitian orthogonality for 0 p 1 and n n 1 n 2 ldots we also show that the polynomials m n x beta c beta in mathbb r are real rooted when c rightarrow 0

on the representation of primes by cubic polynomials in two variables in an earlier paper see proc london math soc 3 84 2002 257 288 we showed that an irreducible integral binary cubic form f x y attains infinitely many prime values providing that it has no fixed prime divisor we now extend this result by showing that f m n still attains infinitely many prime values if m and n are restricted by arbitrary congruence conditions providing that there is still no fixed prime divisor two immediate consequences for the solvability of diagonal cubic diophantine equations are given

exponential generating functions for the associated bessel functions similar to the associated legendre functions the differential equation for the associated bessel functions bl m x is introduced so that its form remains invariant under the transformation l l 1 a rodrigues formula for the associated bessel functions as squared integrable solutions in both regions l 0 and l 0 is presented the functions with the same m but with different positive and negative values of l are not independent of each other while the functions with the same l m l m but with different values of l and m are independent of each other so all the functions bl m x may be taken into account as the union of the increasing decreasing infinite sequences with respect to l it is shown that two new different types of exponential generating functions are attributed to the associated bessel functions corresponding to these rearranged sequences

high order approximation of pearson diffusion processes this paper focuses on pearson diffusions and the spectral high order approximation of their related fokker planck equations the pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient they are widely used in the physical and chemical sciences engineering rheology environmental sciences and financial mathematics in recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations analytical solutions have been derived for some of the pearson diffusions including the ornstein uhlenbeck cox ingersoll ross and jacobi processes however analytical investigations and computations for diffusions with so called heavy tailed ergodic distributions are more difficult to perform the novelty of this research is the development of an accurate and efficient numerical method to solve the fokker planck equations associated with pearson diffusions with different boundary conditions comparisons between the numerical predictions and available time dependent and equilibrium analytical solutions are made the solution of the fokker planck equation is approximated using a reduced basis spectral method the advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving fokker planck type equations

sur une question d erdos et schinzel ii let f n denote a polynomial with integer coefficients and define h f x y z to be the number of integers not exceeding x for which f n has at least one divisor d such that y

some open problems of generalised bessel polynomials the solution of several open problems connected with the generalised bessel polynomials which appear in solving the wave equation in spherical coordinates and in network synthesis and design is shown in particular explicit and simple recursion formulae for sums of powers and product sums of the zeros of these polynomials are found also three different sets of sums of generalised bessel polynomials are analytically evaluated in a compact way

brownian motion quantum corrections and a generalization of the hermite polynomials the nonequilibrium evolution of a brownian particle in the presence of a heat bath at thermal equilibrium without imposing any friction mechanism from the outset is considered using a suitable family of orthogonal polynomials moments of the nonequilibrium probability distribution for the brownian particle are introduced which fulfill a recurrence relation we review the case of classical brownian motion in which the orthogonal polynomials are the hermite ones and the recurrence relation is a three term one after having performed a long time approximation in the recurrence relation the approximate nonequilibrium theory yields irreversible evolution of the brownian particle towards thermal equilibrium with the heat bath for quantum brownian motion which is the main subject of the present work we restrict ourselves to include the first quantum correction this leads us to introduce a new family of orthogonal polynomials which generalize the hermite ones some general properties of the new family are established the recurrence relation for the new moments of the nonequilibrium distribution including the first quantum correction turns out to be also a three term one which justifies the new family of polynomials a long time approximation on the new three term recurrence relation describes irreversible evolution towards equilibrium for the new moment of lowest order the standard smoluchowski equations for the lowest order moments are recovered consistently both classically and quantum mechanically

on some results of cufaro petroni about student t processes this paper deals with student t processes as studied in cufaro petroni 2007 j phys a math theor 40 2227 50 we prove and extend some conjectures expressed by cufaro petroni about the asymptotical behavior of a student t process and the expansion of its density first the explicit asymptotic behavior of any real positive convolution power of a student t density with any real positive degrees of freedom is given in the multivariate case then the integer convolution power of a student t distribution with odd degrees of freedom is shown to be a convex combination of student t densities with odd degrees of freedom at last we show that this result does not extend to the case of non integer convolution powers

the largest prime factor of x3 2 the largest prime factor of x 3 2 was investigated in 1978 by hooley who gave a conditional proof that it is infinitely often at least as large as x 1 delta with a certain positive constant delta it is trivial to obtain such a result with delta 0 one may think of hooley s result as an approximation to the conjecture that x 3 2 is infinitely often prime the condition required by hooley his r conjecture gives a non trivial bound for short ramanujan kloosterman sums the present paper gives an unconditional proof that the largest prime factor of x 3 2 is infinitely often at least as large as x 1 delta though with a much smaller constant than that obtained by hooley in order to do this we prove a non trivial bound for short ramanujan kloosterman sums with smooth modulus it is also necessary to modify the chebychev method as used by hooley so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus 2000 mathematics subject classification 11n32

primes represented by binary cubic forms let f x y be a binary cubic form with integral rational coefficients and suppose that the polynomial f x y is irreducible in mathbb q x y and no prime divides all the coefficients of f we prove that the set f mathbb z 2 contains infinitely many primes unless f a b is even for each a b in mathbb z 2 in which case the set frac 1 2 f mathbb z 2 contains infinitely many primes 2000 mathematical subject classification primary 11n32 secondary 11n36 11r44

on the polynomial solutions of ordinary differential equations of the fourth order the density of zeros of polynomial solutions of ordinary differential equations of the fourth order with coefficients depending only on the independent variable is analyzed the first four moments of such a density are given directly in terms of the coefficients which characterize the differential operator application to the nonclassical orthogonal polynomials corresponding to the names krall legendre krall laguerre and krall jacobi is done global asymptotic properties of the zeros of these polynomials are also obtained

on trudinger moser type inequalities with logarithmic weights abstract trudinger moser type inequalities for radial sobolev spaces with logarithmic weights are considered the precise trudinger moser growths in dependence on the logarithmic terms and the corresponding sharp moser type exponents are determined in a particular case a critical trudinger moser growth of double exponential type is found

real zeros and size of rankin selberg l functions in the level aspect in this paper some asymptotic formulas are proved for the harmonic mollified second moment of a family of rankin selberg l functions one of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average a first consequence is a new subconvexity bound for rankin selberg l functions in the level aspect moreover infinitely many rankin selberg l functions having at most eight non trivial real zeros are produced and some new non trivial estimates for the analytic rank of the family studied are obtained

some families of generating functions for the bessel polynomials abstract the main object of this paper is to show how readily some general results on bilinear bilateral or mixed multilateral generating functions for the bessel polynomials would provide unifications and generalizations of numerous generating functions which were proven recently by using group theoretic techniques

density results for automorphic forms on hilbert modular groups ii we obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region for sl2 over a totally real number field f with a discrete subgroup of hecke type 0 i for a non zero ideal i in the ring of integers of f the weights are products of fourier coefficients this implies in particular the existence of infinitely many cuspidal automorphic representations with multi eigenvalues in various regions growing to infinity for instance in the quadratic case the regions include floating boxes floating balls sectors slanted strips see 1 2 4 1 2 13 and products of prescribed small intervals for all but one of the infinite places of f the main tool in the derivation is a sum formula of kuznetsov type sum formula for sl2 over a totally real number field theorem 2 1