certain families of series associated with the hurwitz lerch zeta function abstract the history of problems of evaluation of series associated with the riemann zeta function can be traced back to christian goldbach 1690 1764 and leonhard euler 1707 1783 many different techniques to evaluate various series involving the zeta and related functions have since then been developed the authors show how elegantly certain families of series involving the hurwitz lerch zeta function can be evaluated by starting with a single known identity for the hurwitz lerch zeta function some of the special cases of these series identities are also shown to lead to certain known families of summation formulas involving the hurwitz zeta function

continuous right inverses for the asymptotic borel map in ultraholomorphic classes via a laplace type transform abstract a new construction of linear continuous right inverses for the asymptotic borel map is provided in the framework of general carleman ultraholomorphic classes in narrow sectors such operators were already obtained by v thilliez by means of whitney extension results for non quasianalytic ultradifferentiable classes due to j chaumat and a m chollet but our approach is completely different resting on the introduction of a suitable truncated laplace type transform this technique is better suited for a generalization of these results to the several variables setting moreover it closely resembles the classical procedure in the case of gevrey classes so indicating the way for the introduction of a concept of summability which generalizes k summability theory as developed by j p ramis

multisummability of formal solutions to the cauchy problem for some linear partial differential equations abstract the paper considers the cauchy problem for linear partial differential equations of non kowalevskian type in the complex domain it is shown that if the cauchy data are entire functions of a suitable order the problem has a formal solution which is multisummable the precise bound of the admissible order of entire functions is described in terms of the newton polygon of the equation

ramanujan formula for the generalized stirling approximation abstract the aim of this paper is to improve ramanujan s formula for approximation of the factorial function starting from burnside s formula in contradistinction with the classical formula that starts from stirling s formula

on hamiltonian perturbations of hyperbolic systems of conservation laws ii universality of critical behaviour hamiltonian perturbations of the simplest hyperbolic equation u t a u u x 0 are studied we argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution moreover this behaviour is described by a special solution to an integrable fourth order ode

asymptotics in a family of linked strip maps abstract we apply round off to planar rotations obtaining a one parameter family of invertible maps of a two dimensional lattice as the angle of rotation approaches 2 the fourth iterate of the map produces piecewise rectilinear motion which develops along the sides of convex polygons we characterise the dynamics which resembles outer billiards of polygons as the concatenation of so called strip maps each providing an elementary perturbation of an underlying integrable system significantly there are orbits which are subject to an arbitrarily large number of these perturbations during a single revolution resulting in the appearance of a novel discrete space version of near integrable hamiltonian dynamics we study the asymptotic regime of the limiting integrable system analytically and numerically some features of its very rich near integrable dynamics we unveil a dichotomy there is one regime in which the nonlinearity tends to zero and a second where it does not in the latter case numerical experiments suggest that the distribution of the periods of orbits is consistent with that of random dynamics in the former case the fluctuations result in an intricate structure of resonances

mathieu series and associated sums involving the zeta functions almost twelve decades ago mathieu investigated an interesting series s r in the study of elasticity of solid bodies since then many authors have studied various problems arising from the mathieu series s r in various diverse ways in this paper we present a relationship between the mathieu series s r and certain series involving the zeta functions by means of this relationship we then express the mathieu series s r in terms of the trigamma function j z or equivalently the hurwitz or generalized zeta function z s a accordingly various interesting properties of s r can be obtained from those of j z and z s a among other results certain integral representations of s r are deduced here by using the aforementioned relationships among s r j z and z s a

pade approximation and apostol bernoulli and apostol euler polynomials using the pade approximation of the exponential function we obtain recurrence relations between apostol bernoulli and between apostol euler polynomials as applications we derive some new lacunary recurrence relations for bernoulli and euler polynomials with gap of length 4 and lacunary relations for bernoulli and euler numbers with gap of length 6

sums associated with the zeta function abstract we evaluate the sums of certain classes of series involving the riemann zeta function by using the theory of the double gamma function which has recently been revived in the study of determinants of laplacians relevant connections with various known results are also pointed out

integrals involving a function associated with the euler maclaurin summation formula the authors first evaluate sums of certain series involving the riemann zeta function by using the theory of the double gamma function with or without the use of these results the authors then establish seberal integrals involving a certain periodic function which is associated for example with the euler maclaurin summation formula finally by introducing some new mathematical constants several families of definite integrals involving logarithms of gamma functions are considered relevant connections of the results presented here with those given in earlier works are also pointed out

a note on some alternating series involving zeta and multiple zeta values in this article we consider various generalizations of euler s famous relation gamma sum n geq 2 1 n frac zeta n n linking euler s constant gamma to special values of the riemann zeta function at positive integers among other things we highlight the existence of a very similar relation for the apostol vu harmonic zeta function which have never been noticed before

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

on parametric gevrey asymptotics for some nonlinear initial value cauchy problems abstract we study a nonlinear initial value cauchy problem depending upon a complex perturbation parameter with vanishing initial data at complex time t 0 and whose coefficients depend analytically on t near the origin in c 2 and are bounded holomorphic on some horizontal strip in c w r t the space variable this problem is assumed to be non kowalevskian in time t therefore analytic solutions at t 0 cannot be expected in general nevertheless we are able to construct a family of actual holomorphic solutions defined on a common bounded open sector with vertex at 0 in time and on the given strip above in space when the complex parameter belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood of 0 in c these solutions are achieved by means of laplace and fourier inverse transforms of some common depending function on c r analytic near the origin and with exponential growth on some unbounded sectors with appropriate bisecting directions in the first variable and exponential decay in the second when the perturbation parameter belongs to moreover these solutions satisfy the remarkable property that the difference between any two of them is exponentially flat for some integer order w r t with the help of the classical ramis sibuya theorem we obtain the existence of a formal series generally divergent in which is the common gevrey asymptotic expansion of the built up actual solutions considered above

distribution of periodic orbits for the casati prosen map on rational lattices abstract we study numerically the periodic orbits of the casati prosen map a two parameter reversible map of the torus with zero entropy for rational parameter values this map preserves rational lattices and each lattice decomposes into periodic orbits we consider the distribution function of the periods over prime lattices and its dependence on the parameters of the map based on extensive numerical evidence we conjecture that asymptotically almost all orbits are symmetric and that for a set of rational parameters having full density the distribution function approaches the gamma distribution r x 1 e x 1 x these properties which have been proved to hold for random reversible maps were previously thought to require a stronger form of deterministic randomness such as that displayed by rational automorphisms over finite fields furthermore we show that the gamma distribution is the limit of a sequence of singular distributions which are observed on certain lines in parameter space our experiments reveal that the convergence rate to r is highly non uniform in parameter space being slowest in sharply defined regions reminiscent of resonant zones in hamiltonian perturbation theory

existence and uniqueness for a class of nonlinear higher order partial differential equations in the complex plane we prove existence and uniqueness results for nonlinear third order partial differential equations of the form

certain classes of series involving the zeta function the authors apply the theory of the double gamma function which was recently revived in the study of determinants of laplacians to evaluate some families of series involving the riemann zeta function introducing a presumably new mathematical constant in the theory of the double gamma function they also systematically evaluate a definite integral of the double gamma function and various classes of series associated with zeta functions some of these definite integrals are expressed in terms of quotients of double gamma functions

a family of p adic twisted interpolation functions associated with the modified bernoulli numbers the main purpose of this paper is to construct a family of modified p adic twisted functions which interpolate the modified twisted q bernoulli polynomials and the generalized twisted q bernoulli numbers at negative integers we also give some applications and examples related to these functions and numbers

sums of certain series of the riemann zeta function abstract the object of the present paper is to investigate systematically several interesting families of summation formulas involving infinite series of the riemann zeta function many of the various results which are unified and generalized here in a remarkably simple manner have received considerable attention in recent years we also present a brief account of a number of analogous results associated with the hurwitz s generalized zeta function

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

on ergodic and mixing properties of the triangle map abstract in this paper we study in detail both analytically and numerically the dynamical properties of the triangle map a piecewise parabolic automorphism of the two dimensional torus for different values of the two independent parameters defining the map the dynamics is studied numerically by means of two different symbolic encoding schemes both relying on the fact that it maps polygons to polygons in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons in the second we consider the standard binary partition of the torus induced by the discontinuity set these encoding schemes are studied in detail and shown to be compatible although not equivalent the ergodic properties of the triangle map are then investigated in terms of the markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the koopman operator in l 2 t 2 finally a stochastic version of the triangle map is introduced and studied a simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map

arithmetical properties of strongly chaotic motions abstract the orbits of the generalized arnold sinai cat maps or hyperbolic automorphisms of the two dimensional torus typify purely chaotic anosov motion we transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers classify all periodic orbits and show how to determine their periods and initial conditions the methods are based on ideal theory in quadratic fields which is reviewed it is shown that the structure of orbits rests upon some basic arithmetical notions such as unique factorization into prime ideals

some analytic continuations of the barnes zeta function in two and higher dimensions abstract formulas for the analytic continuation of the barnes zeta function and some affine extensions thereof in two and more dimensions are constructed the expressions are used to deal with determinants of multidimensional harmonic oscillators an example is therewith obtained of the multiplicative anomaly or defect associated with the most common definition due to ray and singer d b ray reidemeister torsion and the laplacian on lens spaces adv math 4 1970 109 126 d b ray i m singer r torsion and the laplacian on riemannian manifolds adv math 7 1971 145 201 d b ray i m singer analytic torsion for complex manifolds ann math 98 1973 154 177 of determinant of a pseudodifferential operator admitting a zeta function m kontsevich s vishik geometry of determinants of elliptic operators in functional analysis on the eve of the 21st century vol 1 1995 pp 173 197

two sided inequalities for the extended hurwitz lerch zeta function abstract recently srivastava et al 2011 2 unified and extended several interesting generalizations of the familiar hurwitz lerch zeta function z s a by introducing a fox wright type generalized hypergeometric function in the kernel for this newly introduced special function two integral representations of different kinds are investigated here by means of a known result involving a fox wright generalized hypergeometric function kernel which was given earlier by srivastava et al 2011 2 and by applying some mathieu a series techniques finally by appealing to each of these two integral representations two sets of two sided bounding inequalities are proved thereby extending and generalizing the recent work by jankov et al 2011 15

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

carlitz s q bernoulli and q euler numbers and polynomials and a class of generalized q hurwitz zeta functions in this paper we systematically recover the identities for the q eta numbers h k and the q eta polynomials h k x presented by carlitz l carlitz q bernoulli numbers and polynomials duke math j 15 1948 987 1000 which we define here via generating series rather than via the difference equations of carlitz following a method developed by kaneko et al m kaneko n kurokawa m wakayama a variation of euler s approach to the riemann zeta function kyushu j math 57 2003 175 192 for a canonical q extension of the riemann zeta function we investigate a similarly constructed q extension of the hurwitz zeta function the details of this investigation disclose some interesting connections among q eta polynomials carlitz s q bernoulli polynomials b k x e polynomials and the q bernoulli polynomials that emerge from the q extension of the hurwitz zeta function discussed here

some generalizations of the apostol bernoulli and apostol euler polynomials abstract the main object of this paper is to give analogous definitions of apostol type see t m apostol on the lerch zeta function pacific j math 1 1951 161 167 and h m srivastava some formulas for the bernoulli and euler polynomials at rational arguments math proc cambridge philos soc 129 2000 77 84 for the so called apostol bernoulli numbers and polynomials of higher order we establish their elementary properties derive several explicit representations for them in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical bernoulli numbers and polynomials of higher order

the borel summable solutions of heat operators on a real analytic manifold abstract we study heat type equations t u u where the operator is given by a sum of squares of commuting real analytic vector fields acting on a real analytic manifold we give necessary and sufficient conditions for convergence and borel summability of formal power series solutions in terms of generalized integral means of the initial data the results are also valid for affine perturbations of

certain classes of series associated with the zeta function and multiple gamma functions abstract the authors apply the theory of multiple gamma functions which was recently revived in the study of the determinants of the laplacians in order to evaluate some families of series involving the riemann zeta function by introducing a certain mathematical constant they also systematically evaluate this constant and some definite integrals of the triple gamma function various classes of series associated with the zeta function are expressed in closed forms many of these results are also used here to compute the determinant of the laplacian on the four dimensional unit sphere s 4 explicitly

some generalizations of the apostol genocchi polynomials and the stirling numbers of the second kind abstract recently the authors introduced some generalizations of the apostol bernoulli polynomials and the apostol euler polynomials see q m luo h m srivastava j math anal appl 308 2005 290 302 and q m luo taiwanese j math 10 2006 917 925 the main object of this paper is to investigate an analogous generalization of the genocchi polynomials of higher order that is the so called apostol genocchi polynomials of higher order for these generalized apostol genocchi polynomials we establish several elementary properties provide some explicit relationships with the apostol bernoulli polynomials and the apostol euler polynomials and derive various explicit series representations in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function we also deduce their special cases and applications which are shown here to lead to the corresponding results for the genocchi and euler polynomials of higher order by introducing an analogue of the stirling numbers of the second kind that is the so called stirling numbers of the second kind we derive some basic properties and formulas and consider some interesting applications to the family of the apostol type polynomials furthermore we also correct an error in a previous paper q m luo h m srivastava comput math appl 51 2006 631 642 and pose two open problems on the subject of our investigation

universality of the break up profile for the kdv equation in the small dispersion limit using the riemann hilbert approach we obtain an asymptotic expansion for the solution of the cauchy problem for the korteweg de vries kdv equation u t 6uu x epsilon 2 u xxx 0 u x t 0 epsilon u 0 x for epsilon small near the point of gradient catastrophe x c t c for the solution of the dispersionless equation u t 6uu x 0 the sub leading term in this expansion is described by the smooth solution of a fourth order ode which is a higher order analogue to the painleve i equation this is in accordance with a conjecture of dubrovin suggesting that this is a universal phenomenon for any hamiltonian perturbation of a hyperbolic equation using the deift zhou steepest descent method applied on the riemann hilbert problem for the kdv equation we are able to prove the asymptotic expansion rigorously in a double scaling limit

a unified presentation of the generating functions of the generalized bernoulli euler and genocchi polynomials the goal of this paper is to unify and extend the generating functions of the generalized bernoulli polynomials the generalized euler polynomials and the generalized genocchi polynomials associated with the positive real parameters a and b and the complex parameter b by using this generating function we derive recurrence relations and other properties for these polynomials by applying the mellin transformation to the generating function of the unification of bernoulli euler and genocchi polynomials we construct a unification of the zeta functions furthermore we give many properties and applications involving the functions and polynomials investigated in this paper

weak mixing for interval exchange transformations and translation flows we prove that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation we also conclude that a typical translation flow on a typical translation surface of genus g 2 with prescribed singularity types is weakly mixing

on extended hurwitz lerch zeta function abstract this paper investigates an extended form of a beta function b p q x y we first study the convergence problem of the function b p q x y and consider the completely monotonic and log convex properties of this function as a result we obtain a pair of laguerre type inequalities next we provide a new double integral representation for the function b p q x y subsequently we consider the convergence problem of the extended hurwitz lerch zeta function z s a p q defined by its series representation upon using the series manipulation techniques we obtain two series identities we also find various integral representations for the function z s a p q lastly we apply fourier analysis to the function z a z s a p q and obtain a lindelof wirtinger type expansion some interesting and promising results are also illustrated

a unified presentation of certain meromorphic functions related to the families of the partial zeta type functions and the l functions abstract the aim of this paper is to construct a unified family of meromorphic functions which is related to many known functions such as a unified family of partial zeta type functions a unified family of l functions and so on we investigate and derive many properties of this family of meromorphic functions moreover we compute the residues of this family of meromorphic functions at their poles we also give some applications and remarks involving this family of meromorphic functions

further series representations for ζ 2n 1 for a natural number n we propose and develop yet another family of rapidly converging series representations for the riemann zeta function z 2n 1 these developments lead us naturally to the problem of computation of the derivatives z 1 2n and z n a of the riemann and hurwitz zeta functions respectively we also briefly indicate relevant connections of the results considered here with many other known series representations for z 2n 1

certain classes of series associated with the zeta and related functions a fascinatingly large number of seemingly independent solutions of the so called basler problem of evaluating the riemann zeta function z s when s 2 which was of vital importance to leonhard euler 1707 1783 and the bernoulli brothers jakob bernoulli 1654 1705 and johann bernoulli 1667 1748 have appeared in the mathematical literature ever since euler first solved this problem in the year 1736 the main object of this two part series of lectures is to present some recent developments on the evaluations and representations of z s when email protected email protected 1 n being the set of natural numbers we emphasize upon several interesting classes of rapidly convergent series representations for z 2n 1 email protected n which have been developed in recent years in two of many computationally useful special cases considered here it is observed that z 3 can be represented by means of series which converge much more rapidly than that in euler s celebrated formula as well as the series used recently by roger apery 1916 1994 in his proof of the irrationality of z 3 symbolic and numerical computations using mathematica version 4 0 for linux show among other things that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places

series involving the zeta function and multiple gamma functions the theory of multiple gamma functions which was recently revived in the study of the determinants of the laplacians was applied in several earlier works in order to evaluate some families of series involving the riemann zeta function as well as to compute the determinants of the laplacians here in the present paper the authors address the converse problem and apply various known or new formulas for series associated with the zeta and related functions with a view to developing the corresponding theory of multiple gamma functions and then using these series to compute the determinants of the laplacians on the n dimensional unit sphere s n n 5 6 7 explicitly

elliptic extensions of the apostol bernoulli and apostol euler polynomials in this paper we investigate the elliptic analogues of the apostol bernoulli and apostol euler polynomials and obtain the closed expressions of sums of products for these elliptic type polynomials some interesting special cases are also shown

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

some families of the hurwitz lerch zeta functions and associated fractional derivative and other integral representations the main object of this paper is to present in a unified manner a number of fractional derivative and other integral representations for several general families of the hurwitz lerch zeta functions relevant connections of the results presented here with those obtained in earlier works are also indicated precisely

multisummability of formal power series solutions of partial differential equations with constant coefficients we study gevrey properties and summability of power series in two variables that are formal solutions of a cauchy problem for general linear partial differential equations with constant coefficients in doing so we extend earlier results in two articles of balser and lutz miyake and schafke for the complex heat equation as well as in a paper of balser and miyake who have investigated the same questions for a certain class of linear pde with constant coefficients subject to some restrictive assumptions moreover we also present an example of a pde where the formal solution of the cauchy problem is not k summable for whatever value of k but instead is multisummable with two levels under corresponding conditions upon the cauchy data that this can occur has not been observed up to now

estimating the digamma and trigamma functions by completely monotonicity arguments abstract we establish new sharp bounds for the digamma and trigamma functions using the complete monotonicity of some class of functions

some two sided inequalities for multiple gamma functions and related results there is an abundant literature on inequalities for the euler s gamma function c and its various related functions yet only very recently several authors began to study inequalities for the barnes double gamma function c 2 here in this paper we aim at presenting several two sided inequalities for the multiple gamma functions c n n 2 3 4 5 in our investigation of these two sided inequalities for the multiple gamma functions c n n 2 3 4 5 we employ and extend a method based upon taylor s formula and express email protected n 1 x as series involving the zeta functions we also give a more convenient explicit form of the multiple gamma functions c n email protected n n being the set of positive integers the main two sided inequalities for the multiple gamma functions c n n 2 3 4 5 which we have presented in this paper are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple gamma functions c n when email protected 6

asymptotic expansions of the hurwitz lerch zeta function abstract the hurwitz lerch zeta function z s a is considered for large and small values of a c and for large values of z c with arg a z 1 and s c this function is originally defined as a power series in z convergent for z 1 s c and 1 a n an integral representation is obtained for z s a which define the analytical continuation of the hurwitz lerch zeta function to the cut complex z plane c 1 from this integral we derive three complete asymptotic expansions for either large or small a and large z these expansions are accompanied by error bounds at any order of the approximation numerical experiments show that these bounds are very accurate for real values of the asymptotic variables

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

twisted p adic h q l functions by using the q volkenborn integral on z p in simsek 2006 33 and simsek 2007 34 generating functions for the h q bernoulli polynomials and numbers were defined by using these functions we define a new twisted h q partial zeta function which interpolates the twisted h q bernoulli polynomials and generalized twisted h q bernoulli numbers at negative integers we give a relation between twisted h q partial zeta functions and the twisted h q two variable l function we find the value of this function at s 0 we also find the residue of this function at s 1 we construct a p adic twisted h q l function which interpolates the twisted h q bernoulli polynomials l x p q h 1 n t g b n g n x h p t q email protected n p p n 1b n g n 1 h p 1p t q p n furthermore we construct an integral representation of the twisted h q two variable l function we give some applications related to the p adic twisted h q l function and the twisted h q bernoulli polynomials

some integral and asymptotic formulas associated with the hurwitz zeta function the main object of this paper is to present an easily accessible integral representation of the partial sum l u x a 0 n x n a u of the hurwitz zeta function z u a which entails a number of important implications for l u x a and z u a and for their derivatives furthermore a new proof is given for a fundamental summation formula contained in theorem 3 below which is shown to possess numerous further consequences and applications relevant connections of the results obtained here with those considered in many other earlier and recent works are also discussed

fourier expansions and integral representations for the apostol bernoulli and apostol euler polynomials we investigate fourier expansions for the apostol bernoulli and apostol euler polynomials using the lipschitz summation formula and obtain their integral representations we give some explicit formulas at rational arguments for these polynomials in terms of the hurwitz zeta function we also derive the integral representations for the classical bernoulli and euler polynomials and related known results

borel summability of the heat equation with variable coefficients abstract we prove borel summability in nonsingular directions of solutions of partial differential equations of the form u t a z u z z where a z is a quartic polynomial and the initial condition is analytic in the special case a z z we obtain the detailed resurgent structure even in such a simple case the structure of the singular manifolds is quite intricate

on gevrey solutions of threefold singular nonlinear partial differential equations abstract we study gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain we deal with both fuchsian and irregular singularities and allow the presence of a singular perturbation parameter by means of the borel laplace summation method we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their gevrey asymptotic expansion in the perturbation parameter this result rests on the malgrange sibuya theorem and it requires to prove that the difference between two neighboring solutions is exponentially small what in this case involves an asymptotic estimate for a particular dirichlet like series

the determinant of one dimensional polyharmonic operators of arbitrary order abstract we obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator p n 1 n x 2 n on 0 t with dirichlet boundary conditions and n a positive integer and show that it satisfies the asymptotics log det p n n 2 log n 7 3 2 2 3 2 log t 4 n 2 o n for large n this is a consequence of sharp upper and lower bounds for log det p n valid for all n and which coincide in the terms up to order n these results form the basis to analyse more general operators with nonconstant coefficients and show that the corresponding determinants have a similar asymptotic behaviour

some q extensions of the apostol bernoulli and the apostol euler polynomials of order n and the multiple hurwitz zeta function abstract in this paper we first investigate several further interesting properties of the multiple hurwitz lerch zeta function n z s a which was introduced recently by choi et al j choi d s jang h m srivastava a generalization of the hurwitz lerch zeta function integral transform spec funct 19 2008 we then introduce and investigate some q extensions of the multiple hurwitz lerch zeta function n z s a the apostol bernoulli polynomials b k n x of order n and the apostol euler polynomials e k n x of order n relevant connections of the results presented here with those obtained in earlier works are also indicated precisely

evaluation of series with hurwitz and lerch zeta function coefficients by using hankel contour integrals we introduce a new technique for evaluation of series with zeta coefficients and also for evaluation of certain integrals involving the loggamma function this technique is based on hankel integral representations of the hurwitz zeta the lerch transcendent the digamma and loggamma functions