unitary representations and theta correspondence for type i classical groups in this paper we discuss the positivity of the hermitian form p introduced by li in invent math 27 1989 237 255 let g1 g2 be a type i dual pair with g1 the smaller group let p be an irreducible unitary representation in the semistable range of y mg1 mg2 see communications in contemporary mathematics vol 2 2000 pp 255 283 we prove that the invariant hermitian form p is positive semidefinite under certain restrictions on the size of g2 and a mild growth condition on the matrix coefficients of p therefore if p does not vanish y mg1 mg2 p is unitary theta correspondence over r was established by howe in j amer math soc 2 1989 535 552 li showed that theta correspondence preserves unitarity for dual pairs in stable range our results generalize the results of li for type i classical groups invent math 27 1989 237 the main result in this paper can be used to construct irreducible unitary

bessel models for gsp 4 methods of theta correspondence are used to analyze local and global bessel models for gsp 4 proving a conjecture of gross and prasad which describes these models in terms of local epsilon factors in the local case and the non vanishing of central critical l value in the global case

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

one dimensional representations of u p q and the howe correspondence abstract we explicitly determine the theta lifts of all one dimensional representations of u p q in terms of langlands parameters and determine exactly which lifts are unitary moreover we show that such a lift is unitary if and only if it is a weakly fair derived functor module of the form a q finally we show that the correspondence of unitary representations behaves well with respect to associated cycles

the np completeness of some edge partition problems we show that for each fixed n geqq 3 it is np complete to determine whether an arbitrary graph can be edge partitioned into subgraphs isomorphic to the complete graph k n the np completeness of a number of other edge partition problems follows immediately

the saito kurokawa lifting and functoriality certain nontempered liftings from pgl 2 pgl 2 to pgsp 4 are constructed using the theory of local and global theta lifts the resulting representations on pgsp 4 are the saito kurokawa representations the lifting is shown to be functorial under certain reasonable assumptions on the local langlands correspondence for pgsp 4

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

the asymptotic behavior of densities related to the supremum of a stable process p s1 x v a 1 x as x 1 and p s1 x v b 1 1 x as x 0 here p x1 0 and a and b are known constants it is also known that s1 has a continuous density m say the main point of this note is to show that m x v ax 1 as x 1 and m x v bx 1 as x 0 similar results are obtained for related densities

theta lifting of unitary lowest weight modules and their associated cycles we consider a reductive dual pair g g in the stable range with g the smaller member and of hermitian symmetric type we study theta lifting of holomorphic nilpotent k c orbits in relation to theta lifting of unitary lowest weight modules of g we determine the associated cycles of all such representations in particular we prove that the multiplicity in the associated cycle is preserved under theta lifting

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

the law of the supremum of a stable levy process with no negative jumps let x xt t 0 be a stable levy process of index fi 2 1 2 with no negative jumps and let st sup0 st xs denote its running supremum for t 0 we show that the density function ft of st can be characterized as the unique solution to a weakly singular volterra integral equation of the flrst kind or equivalently as the unique solution to a flrst order riemann liouville fractional difierential equation satisfying a boundary condition at zero this yields an explicit series representation for ft recalling the familiar relation between st and the flrst entry time x of x into x 1 this further translates into an explicit series representation for the density function of x

on howe s duality theorem abstract we adopt the langlands classification to the context of real reductive dual pairs and prove that howe s duality correspondence maps hermitian representations to hermitian representations

hardness and approximation of traffic grooming traffic grooming is a central problem in optical networks it refers to packing low rate signals into higher speed streams in order to improve bandwidth utilization and reduce network cost in wdm networks the most accepted criterion is to minimize the number of electronic terminations namely the number of sonet add drop multiplexers adms in this article we focus on ring and path topologies on the one hand we provide an inapproximability result for traffic grooming for fixed values of the grooming factor g answering affirmatively the conjecture of chow and lin t chow p lin the ring grooming problem networks 44 2004 194 202 more precisely we prove that ring traffic grooming for fixed g 1 and path traffic grooming for fixed g 2 are apx complete that is they do not accept a ptas unless p np both results rely on the fact that finding the maximum number of edge disjoint triangles in a tripartite graph and more generally cycles of length 2g 1 in a 2g 1 partite graph of girth 2g 1 is apx complete on the other hand we provide a polynomial time approximation algorithm for ring and path traffic grooming based on a greedy cover algorithm with an approximation ratio independent of g namely the approximation guarantee is o n 1 3log 2n for any g 1 n being the size of the network this is useful in practical applications since in backbone networks the grooming factor is usually greater than the network size finally we improve this approximation ratio under some extra assumptions about the request graph

singular unitary representations of classical groups on presente une description de la partie relativement singuliere du dual unitaire d un groupe classique de type i sur un corps local f de caracteristique autre que 2

archimedean zeta integrals on u 2 1 it is known that for a dual pair of unitary groups with equal size zeta integrals arising from rallis inner product formula give the central values of certain au tomorphic l functions which have applications to arithmetic in this paper we explicitly calculate archimedean zeta integrals of this type for u 2 1 in particular we compute the matrix coefficients of weil representations using joint harmonics in the fock model and those of discrete series using schmid operators in the theory of automorphic forms to obtain deep arithmetic applications it is often necessary to have effectively computable results locally at each place of a number field in particular at archimedean places we need some detailed calculations on the harmonic analysis of real lie groups and have to face certain combinatorial complexity in l1 l2 the author works out the local archimedean theory for cohomological rep resentations and the connection to special values of l functions in the stable range the local theory of theta correspondence for unitary groups is developed in hks for p adic fields and in p for field of real numbers it is made more precise in h1 h2 and applied to obtain results about special values of l functions the article hls outlines the general global theory of theta correspondence for cuspidal automorphic representations of unitary groups in which the authors obtain a general formula for the local zeta integral on unitary groups and show that the central l value is non negative we shall explain the motivation and background of this paper by a brief review for above context following hls let f be a totally real number field f a totally imaginary quadratic extension of f a a f the adele ring of f let v resp v be a hermitian resp skew hermitian vector space of dimension n over f and w v f v a symplectic space over f fixing an additive character and a complete polarization w x y we have the schrodinger model of the oscillator representation of f sp w a realized on the space s x a of schwartz bruhat functions on x a let g u v g u v by choosing a global splitting character of a f cf hls then defines a weil representation v v of g a g a on s x a as usual for s x a we have the theta kernel g g x x f

explicit induction principle and symplectic orthogonal theta lifts abstract during the last two decades great efforts have been devoted to the calculation of the local theta correspondence for reductive dual pairs however uniform formulas remain elusive for real dual pairs of type i the purpose of this paper is twofold to formulate an explicit version of induction principle for dual pairs o p q s p 2 n r with p q even and to apply it to obtain a complete and explicit description of the local theta correspondence when p q 4 our approach is very elementary by analysis on the infinitesimal characters and k types under the theta correspondence

inversion de l espace et du temps des processus de levy stables we proove a new duality relation between stable levy processes with index open image in new window and those with index open image in new window this duality appears to be the trajectorial version of the duality of zolotarev which concerns one dimensional stable laws we give an application of this result to the behaviour of the paths at small and large times of the process conditioned to stay positive

characters dual pairs and unipotent representations abstract we lift distribution characters of irreducible unitary representations of classical groups from the group to the lie algebra via the cayley transform then a specific class of these characters admits fourier transform supported on the closure of a single nilpotent coadjoint orbit we calculate also the wave front set of the most singular low rank representations

on the law of the supremum of levy processes we show that the law of the overall supremum overline x t sup s let x s undefined of a levy process x before the deterministic time t is equivalent to the average occupation measure t dx t0p xs dx ds whenever 0 is regular for both open halflines 0 and 0 in this case p x t dx is absolutely continuous for some and hence for all t 0 if and only if the resolvent measure of x is absolutely continuous we also study the cases where 0 is not regular for both halflines then we give absolute continuity criterions for the laws of gt x t and gt x t xt where gt is the time at which the supremum occurs before t the proofs of these results use an expression of the joint law p gt ds xt dx x t dy in terms of the entrance law of the excursion measure of the reflected process at the supremum and that of the reflected process at the infimum as an application this law is made partly explicit in some particular instances

on wiener hopf factorisation and the distribution of extrema for certain stable processes it is shown that when the index 0 0 l alpha 1 k where l and k are integers darling s integral can be evaluated this leads to explicit formulas for a transform of the laplace transform of sup 0 leq t leq 1 x t and the wiener hopf factors of x t t geq 0

asymptotique des nombres de betti des varietes arithmetiques we study the question of the growth of betti numbers of certain arithmetic varieties in tower of congruence coverings in fact our results are about siegel varieties and varieties associated to orthogonal groups we explain how a theorem of waldspurger can be used to obtain lower and upper bound our results are in the direction of conjectures made by sarnak and xue

compositions of theta correspondences theta correspondence y over r is established by howe j amer math soc 2 1989 535 in he j funct anal 199 2003 92 we prove that y preserves unitarity under certain restrictions generalizing the result of li invent math 97 1989 237 the goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences we show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method j algebra 190 1997 518 representation theory of lie groups park city 1998 pp 179 238 the orbit correspondence for real and complex reductive dual pairs preprint 2001 one can compose theta correspondences to obtain unitary representations we call this process quantum induction r 2004 published by elsevier inc msc primary 22e45 22e46

traffic grooming on the path in a wdm network routing a request consists in assigning it a route in the physical network and a wavelength if each request uses at most 1 c of the bandwidth of the wavelength we will say that the grooming factor is c that means that on a given edge of the network we can groom group at most c requests on the same wavelength with this constraint the objective can be either to minimize the number of wavelengths related to the transmission cost or minimize the number of add drop multiplexers shortly adm used in the network related to the cost of the nodes we consider here the case where the network is a path on n nodes pn thus the routing is unique for a given grooming factor c minimizing the number of wavelengths is an easy problem well known and related to the load problem but minimizing the number of adms is np complete for a general set of requests and no results are known here we show how to model the problem as a graph partition problem and using tools of design theory we completely solve the case where c 2 and where we have a static uniform all to all traffic one request for each pair of vertices

genome rearrangements and sorting by reversals sequence comparison in molecular biology is in the beginning of a major paradigm shift a shift from gene comparison based on local mutations i e insertions deletions and substitutions of nucleotides to chromosome comparison based on global rearrangements i e inversions and transpositions of fragments the classical methods of sequence comparison do not work for global rearrangements and little is known in computer science about the edit distance between sequences if global rearrangements are allowed in the simplest form the problem of gene rearrangements corresponds to sorting by reversals i e sorting of an array using reversals of arbitrary fragments recently kececioglu and sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements one of these problems is gollan s conjecture on the reversal diameter of the symmetric group this paper proves the conjecture further the problem of expected reversal distance between two random permutations is investigated the reversal distance between two random permutations is shown to be very close to the reversal diameter thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies the gene rearrangement problem forces us to consider reversals of signed permutations as the genes in dna could be positively or negatively oriented an approximation algorithm for signed permutation is presented which provides a performance guarantee of 3 over 2 finally using the signed permutations approach an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of 7 over 4

computational strategies for the riemann zeta function we provide a compendium of evaluation methods for the riemann zeta function presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new we concentrate primarily on practical computational issues such issues depending on the domain of the argument the desired speed of computation and the incidence of what we call value recycling c 2000 elsevier science b v all rights reserved

geometric realization of discrete series for semisimple symmetric spaces let g be a connected semisimple lie group a an involution of g and h the connected component of the fixed point set of a containing the identity then g h is a semisimple symmetric space we assume g is a real form of a complex lie group and rank g h rank k k c h where k is a a stable maximal compact subgroup of g flensted jensen using his duality principle and poisson transform constructed countably many discrete series in u g h f j i then using deep results of hyperfunction theory and microlocal analysis oshima and matsuki gave a detailed characterization of these discrete series including a proof of flensted jensen s conjecture c 0 om based on these results we give the discrete series a geometric interpretation in locally symmetric spaces f c where g k and f cg is a cocompact discrete group such that f c h c h is cocompact our geometric interpretation is the following first of all to a flensted jensen function which is the generating function of a discrete series v we associate a harmonic form cb z on g with values in a finite dimensional g module v right g translates of coefficients of 5z realizes the contragredient t of r and tba is left invariant under h and under right k action it transforms according to the minimal k type with highest weight of 3 is the pull back of a v valued from coz on and v determines a locally constant bundle e on f so that if convergence holds

on the discrete series of generalized stiefel manifolds a study of the discrete series of generalized stiefel manifolds is made using the oscillator representation new infinite families of such discrete series are constructed

approximation algorithms for grooming in optical network design we study traffic grooming in optical network design where the goal is to aggregate low bandwidth traffic streams to utilize efficiently high bandwidth media such as wavelength channels more precisely given traffic demands to be routed in a network the design problem is to define a collection of light paths such that each demand can follow a sequence of consecutive light paths each light path has a unit wavelength bandwidth and multiple sub wavelength demands may share a common light path traffic must enter and depart from a light path at its two endpoints only most previous work on grooming focused on the ring topology and typically involved only uniform bandwidth demands whereas we deal with more general settings two objectives are considered one is to minimize the total cost of equipment necessary to support the light paths the other is simply to minimize the number of light paths even for the extremely restricted special case of a line topology and traffic demands that request half wavelength bandwidth we show that both objectives are apx hard to optimize which means we cannot approximate the optimum arbitrarily closely on the other extreme of generality for arbitrary network topologies and traffic demands that request arbitrary amounts of bandwidth we show a logarithmic approximation for cost minimization and a 2 approximation for minimizing light path counts furthermore we discover that the special case of half wavelength demands has rich combinatorial properties closely related to graph problems such as cycle packing and pattern matching problems such as interchange distance in strings we show how to approximate both objectives up to small constant factors in this case while similarly improving the approximation and hardness of the interchange distance problem as well

howe correspondence for real unitary groups roger howe proved that for any reductive dual pair g g in the symplectic groupsp 2n r there is a one to one correspondence of irreducible admissible representations of some two fold covers ofgandg we determine this correspondence explicitly for dual pairs of the form u p q u r s withr s p q and describe it in terms of langlands parameters in this case the correspondence may be understood in a natural way as a correspondence of representations of the linear groups rather than the appropriate covers we show that every irreducible admissible representation ofu p q occurs in the correspondence with precisely one unitary group of equal rank this result verifies a conjecture of harris kudla and sweet who investigated the correspondence for unitary groups of equal size overp adic fields the correspondence of discrete series representations was determined by j s li for induced representations the correspondence is obtained in a natural way from the corresponding discrete series on the respective levi factors of the parabolic subgroups ofu p q andu r s generalizing a result of li we show that under the correspondence representations with nonzero cohomology are matched in an interesting way with unitarity not necessarily preserved the proof uses the induction principle which is due to kudla and an argument involvingk types and the space of joint harmonics howe

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

theta dichotomy for unitary groups some recent work of gross and prasad 14 suggests that the root numbers attached to certain symplectic representations of the weil deligne group of a local field f control certain branching rules for representations of orthogonal groups over f on a global level this local phenomenon should have implications for the structure of the value at the center of symmetry for certain l functions of arithmetic interest 12 this conjectural picture is based to some extent on the now classic work of tunnell 47 and waldspurger 49 as well as on the case of the triple product l function 13 34 17 in all of these examples the local root number detects the existence of a certain type of invariant linear functional it is possible to set up analogous conjectures for the isometry groups of hermitian or quaternion hermitian forms it turns out that root numbers also play a role in the local and global theta correspondence roughly speaking certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the same size e g for dual pairs of the form sp n o 2n 1 and u n u n as will be seen the theta correspondence for such pairs is connected with a certain induced representation in s at the point s 0 on the unitary axis by contrast the correspondence for the pairs sp n o 2n and sp n o 2n 2 discussed by prasad 35 is connected to the behavior of a similar induced representation at the points 12 in which case no epsilon factor arises in this paper we consider the local theta correspondence for unitary groups in the non archimedean case let f be a non archimedean local field of characteristic not equal to 2 and let e be a quadratic extension of f for further notation see the notation section at the end of this introduction let v and w be e vector spaces of dimensions m and n equipped respectively with a hermitian form v v e and a skew hermitian form w w e then the isometry groups g v and g w of the spaces v and w form a dual reductive pair in the symplectic group sp w where w v e w viewed as an f vector space of dimension 2mn and equipped with the symplectic form

l functions and theta correspondence for classical groups using the doubling method of piatetski shapiro and rallis we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the l functions and local theta correspondence

on barnes beta distributions and applications to the maximum distribution of the 2d gaussian free field a new family of barnes beta distributions on 0 infty is introduced and its infinite divisibility moment determinacy scaling and factorization properties are established the morris integral probability distribution is constructed from barnes beta distributions of types 1 0 and 2 2 and its moment determinacy and involution invariance properties are established for application the maximum distributions of the 2d gaussian free field on the unit interval and circle with a non random logarithmic potential are conjecturally related to the critical selberg and morris integral probability distributions respectively and expressed in terms of sums of barnes beta distributions of types 1 0 and 2 2

on the gauge orbit space stratification a review first we review the basic mathematical structures and results concerning the gauge orbit space stratification this includes general properties of the gauge group action fibre bundle structures induced by this action basic properties of the stratification and the natural riemannian structures of the strata in the second part we study the stratification for theories with gauge group su n in spacetime dimension 4 we develop a general method for determining the orbit types and their partial ordering based on the 1 1 correspondence between orbit types and holonomy induced howe subbundles of the underlying principal su n bundle we show that the orbit types are classified by certain cohomology elements of spacetime satisfying two relations and that the partial ordering is characterized by a system of algebraic equations moreover operations for generating direct successors and direct predecessors are formulated which allow one to construct the set of orbit types starting from the principal type finally we discuss an application to nodal configurations in yang mills chern simons theory

on the howe correspondence for symplectic orthogonal dual pairs we reformulate some of moeglin s results on the correspondence for the dual pairs sp 2n r o p q with p and q even and fill in the cases where p and q are both odd we arrive at a complete and detailed description in terms of langlands parameters of the dual pair correspondence for the cases p q 2n and p q 2n 2 in addition we point out and suggest a way to correct an error in moeglin s paper

on the gan gross prasad conjecture for u p q in this paper we give a proof of the gan gross prasad conjecture for the discrete series of u p q there are three themes in this paper branching laws of a small a mathfrak q lambda branching laws of discrete series and inductive construction of discrete series these themes are linked together by a reciprocity law and the notion of invariant tensor product

on the non vanishing of theta liftings of tempered representations of u p q abstract in this paper we give an explicit determination of the non vanishing of the theta liftings of tempered representations for unitary dual pairs u p q u r s for arbitrary non negative integers p q r s for discrete series representations in terms of harish chandra parameters we give a complete criterion when the theta lifts are nonzero for tempered representations we determine the non vanishing in terms of the local langlands correspondence assuming the local gan gross prasad conjecture

the explicit theta correspondence for reductive dual pairs sp p q o 4 abstract for every irreducible harish chandra module of o 4 we determine its theta lifts to s p p q in terms of the langlands parameters for all non negative integers p and q our strategy is to determine the desired theta lifts by their infinitesimal characters and lowest k types

mvw extensions of quaternionic classical groups we show that inside the real quaternionic symplectic group mathrm sp p q every element is conjugate to its inverse similar result is proved for the real quaternionic orthogonal group mathrm o 2n we also show that no similar result is possible in the p adic case

on extrema of stable processes we study the wiener hopf factorization and the distribution of extrema for general stable processes by connecting the wiener hopf factors with a certain elliptic like function we are able to obtain many explicit and general results such as infinite series representations and asymptotic expansions for the density of supremum explicit expressions for the wiener hopf factors and the mellin transform of the supremum quasi periodicity and functional identities for these functions finite product representations in some special cases and identities in distribution satisfied by the supremum functional

density behaviour related to levy processes let p t x f t x and q t x be the densities at time t of a real levy process its running supremum and the entrance law of the reflected excursions at the infimum we provide relationships between the asymptotic behaviour of p t x f t x and q t x when t is small and x is large then for large x these asymptotic behaviours are compared to this of the density of the levy measure we show in particular that under mild conditions if p t x is comparable to t x as t 0 and x then so is f t x

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

the explicit duality correspondence of sp p q o 2n abstract we investigate the type i dual pairs over the quaternion algebra h namely the family of dual pairs sp p q o 2n we give a complete and explicit description of duality correspondence for p q n as well as some of the cases for p q n in terms of the langlands parameters

ruling out ptas for graph min bisection dense k subgraph and bipartite clique assuming that np not subseteq cap epsilon 0 bptime 2 n epsilon we show that graph min bisection dense k subgraph and bipartite clique have no polynomial time approximation scheme ptas we give a reduction from the minimum distance of code mdc problem starting with an instance of mdc we build a quasi random probabilistically checkable proof pcp that suffices to prove the desired inapproximability results in a quasi random pcp the query pattern of the verifier looks random in a certain precise sense among the several new techniques we introduce the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials as is important for our purpose the certificate itself happens to be another polynomial and it can be checked probabilistically by reading a constant number of its values

uniform asymptotic expansions for the barnes double gamma function we develop uniform asymptotic approximations for the barnes double gamma function for various parameter values this involves the use of the method of matched asymptotic expansions to solve the defining functional difference equation in various asymptotic regions the resulting solutions provide extremely accurate approximations to the double gamma function we also give applications of our results to functional difference equations that arise in the theory of moving contact lines and acoustic wave diffraction