Principal series and generalized principal series Whittaker functions with peripheral K-types on the real symplectic group of rank 2

First, we give some explicit formulas of principal series Whittaker functions on the real symplectic group of rank 2 with arbitrary one-dimensional K-types. These formulas are extension of Ishii??s formulas for Whittaker functions with minimal K-types. Secondly, we compute explicit formulas of the holonomic system for the radial part of Whittaker functions with peripheral K-types belonging to the generalized principal series representations induced from the Siegel maximal parabolic subgroup (i.e., P _S-series). Thirdly, we derive eight power series solutions for our holonomic system utilizing the embedding of the P _S-series into various principal series, from the power series Whittaker functions belonging to the principal series.     

On the Nevanlinna characteristic of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">z</Emphasis>+<Emphasis Type="Italic">η</Emphasis>) and difference equations in the complex plane

We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations. This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (HKUST6135/01P). The second author was also partially supported by the National Natural Science Foundation of China (Grant No. 10501044) and the HKUST PDF Matching Fund.     

Double <Emphasis Type="Italic">SO</Emphasis>(2, 1)-integrals and formulas for Whittaker functions

With the help of some double integral bilinear functionals with homogeneous kernels defined on a pair of representation spaces of the group SO(2, 1) we obtain some functional relations for Whittaker functions and calculate the sum of one series of Gauss hypergeometric functions converging to a Whittaker function.     

Fourier–Jacobi Type Spherical Functions for Discrete Series Representations of Sp(2, R)

In this paper we define a kind of generalized spherical functions on Sp(2, R). We call it Fourier–Jacobi type, since it can be considered as a generalized Whittaker model associated with the Jacobi maximal parabolic subgroup. Also we give the multiplicity theorem and an explicit formula of these functions for discrete series representations of Sp(2, R).     

LU factorizations, q = 0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q = 0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas. Dedicated to Dick Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—33D45, 33D80 Work done at KdV Institute, Amsterdam and supported by NWO, project number 613.006.573.     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

Interlacing of Hurwitz series

In this paper, we introduce the notion of interlacing of Hurwitz series. We begin by reviewing some important properties of the ring of Hurwitz series over a commutative ring A of arbitrary characteristic, and we introduce and investigate properties of the maps exp and log. We show that solutions of linear homogeneous differential equations with constant coefficients from the ring A can be described simply as interlacings of solutions of a first order system of differential equations. We give several examples to illustrate this result, and we conclude by defining and investigating properties of trigonometric functions using interlacings of Hurwitz series.     

Differential equations for principal series whittaker functions on SU(2,2)

Let SU(2,2) be the special unitary group of index (2,2). In this paper, two explicit linear partial differential equations are obtained: one from the Casimir operator and the other from the Schmid operator by taking their radial parts. Whittaker functions belonging to an irreducible principal series representation of SU(2,2) satisfy the system of differential equations and furthermore this system becomes holonomic when the dimension of a minimal K-type of the representation is one or two.     

The local period integrals and essential vectors

By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of GL_n over p-adic fields. In each case, we show that the integrals achieve local formal L-functions defined by Langlands parameters, when the test vector is associated to the new form. We give the relation between local periods involving essential Whittaker functions and special values of formal L-factors at $s=1$ for certain distinguished or unitary representations. The period integrals are also served as standard nonzero distinguished forms.     

Existence of the best <Emphasis Type="Italic">n</Emphasis>-term approximants for structured dictionaries

We extend the Pizzetti formulas, i.e., expansions of the solid and spherical means of a function in terms of the radius of the ball or sphere, to the case of real analytic functions and to functions of Laplacian growth. We also give characterizations of these functions. As an application we give a characterization of solutions analytic in time of the initial value problem for the heat equation ∂ _t u = Δu in terms of holomorphic properties of the solid and/or spherical means of the initial data.     

A new proof of regularity of weak solutions of the <Emphasis Type="Italic">H</Emphasis>-surface equation

Abstract. We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions of the H-surface system are locally H?lder continuous provided that H is a bounded Lipschitz function. Contrary to Bethuel's, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Hélein's theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in , this time assuming in addition that decays at infinity like . Received: 10 May 2001 / Accepted: 7 June 2001 / Published online: 18 January 2002 The author gratefully acknowledges the generous support of Alexander von Humboldt Foundation, and the hospitality of Mathematisches Institut der Universit?t Bonn, where this research has been carried out. In particular, many thanks are due to Professor Stefan Hildebrandt.     

On the gamma factor of the triple product L-functions

Let π₁ and π₂ be essentially (limit of) discrete series representations of GL₂(R), and π₃ be a principal series representation of GL₂(R). We calculated the gamma factor of the triple product L-function L(s,π₁×π₂×π₃) by constructing the normalized good sections and Whittaker functions for πi explicitly and showed that they coincide the functions which have been predicted by Langlands philosophy.     

Linear Independence of the Values of <Emphasis Type="Italic">q</Emphasis>-Hypergeometric Series and Related Functions

We give linear independence results for the values of certain entire series and of functions satisfying certain first order q-difference equations. The former generalizes a result of Bézivin, while the latter refines that of the second named author in qualitative form. These results imply linear independence of the values of q-hypergeometric series.Research supported in part by Grant-in-Aid for Scientic Research (No. 13640007), the Ministry of Education, Science, Sports and Culture of Japan.2000 Mathematics Subject Classification: Primary—11J72     

Extrinsic diameter of immersed flat tori in <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S ³. In the proof of that result, an estimate of extrinsic diameter of flat tori plays a crucial role. It is reasonable to expect that the same rigidity holds in the class of immersed flat tori in S ³. In this paper, we give a new method for characterizing immersed flat tori in S ³ with extrinsic diameter π, which is a somewhat similar technique to the proof of the 6-vertex theorem for certain closed plane curves given by the second author. As an application, we show that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Recently, the global behaviour of flat surfaces in H ³ and R ³ regarded as wave fronts has been studied. We also give here a formulation of flat tori in S ³ as wave fronts. As an application, we shall exhibit a flat torus as a wave front whose extrinsic diameter is less than π.     

An Excursion-Theoretic Approach to Stability of Discrete-Time Stochastic Hybrid Systems

We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for L ₁-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global asymptotic stability almost surely and in L ₁. The systems need not be time-homogeneous, and our results apply to certain systems for which functional-analytic or martingale-based estimates are difficult or impossible to get.     

On Whittaker modules over a class of algebras similar to U(sl 2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A ₁ and finding interesting families of new noetherian rings, a class of algebras similar to U(sl ₂) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.      

Telescoping partial fractions decompositions, the little q-Jacobi functions of complex order, and the nonterminating q-Saalschütz sum

We use telescoping partial fractions decompositions to give new proofs of the orthogonality property and the normalization relation for the little q-Jacobi polynomials, and the q-Saalschütz sum. In [20], we followed the development [19] of Schur functions for partitions with complex parts, and we showed that there exist natural little q-Jacobi functions of complex order which satisfy extensions of the orthogonality property and normalization relation of the little q-Jacobi polynomials, and that these two results follow from and together imply the nonterminating form of the q-Saalschütz sum. Writing the q-Pochhammer symbol of complex order as a ratio of infinite products in the usual way, we obtain new telescoping partial fractions decomposition proofs of our results [20] for the little q-Jacobi functions of complex order. We give several new proofs of the q-Saalschütz sum and its nonterminating form. For our friends Dick and Liz 2000 Mathematics Subject Classification Primary—42C05; Secondary—33C45, 33C47     

New lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-Hamiltonian structure

We study restricted multiple three wave interaction system by the inverse scattering method. We develop the algebraic approach in terms of classical r-matrix and give an interpretation of the Poisson brackets as linear r-matrix algebra. The solutions are expressed in terms of polynomials of theta functions. In particular case for n = 1 in terms of Weierstrass functions.      

Generating functions of the (<Emphasis Type="Italic">h,q</Emphasis>) extension of twisted Euler polynomials and numbers

By using p-adic q-deformed fermionic integral on ℤ _p , we construct new generating functions of the twisted (h, q)-Euler numbers and polynomials attached to a Dirichlet character χ. By applying Mellin transformation and derivative operator to these functions, we define twisted (h, q)-extension of zeta functions and l-functions, which interpolate the twisted (h, q)-extension of Euler numbers at negative integers. Moreover, we construct the partially twisted (h, q)-zeta function. We give some relations between the partially twisted (h, q)-zeta function and twisted (h, q)-extension of Euler numbers.