Whittaker functions for generalized principal series representations of <Emphasis Type="Italic">SL</Emphasis>(3, R)

We study Whittaker functions for generalized principal series representations of the real special linear group SL(3, R) of degree 3. From the Capelli elements and Dirac-Schmid operators, we give the system of partial differential equations which is satisfied by Whittaker functions. We give six formal power series solutions of this system, which are called secondary Whittaker functions. We also give the Mellin-Barnes type integral expressions of primary Whittaker functions, i.e. the solutions having the moderate growth property.     

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.     

A unification of the Voigt functions

This paper aims to present a simple unification of the Voigt functions K(x,y) and L(x,y) through Fourier and Mellin transforms resulting in connections with the error functions, the parabolic cylinder functions and, ultimately, the Whittaker functions. Analogues of related functions are highlighted. The Voigt integrals will follow as natural consequences for analytical evaluations and uses.     

Sturm-Liouville eigenproblems with an interior double pole

The self-adjoint Sturm-Liouville eigenproblem with an interior double pole and two simple turning points is examined. Using asymptotic forms of the Whittaker functions asymptotic dispersion relations are obtained for high modes; and on employing the Titchmarsh-Weyl m-functions as well as the appropriate Whittaker functions exact solutions are obtained valid on the whole interval for the low order modes.     

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.     

Integrals involving Whittaker functions

Summary Integrals involving products of two Whittaker functions and Bessel functions are evaluated in §§ 3, 4. Also the integrals are evaluated in § 5 while in § 6 integrals involving the product of three Whittaker functions are established.     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

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.     

Whittaker Modules for Graded Lie Algebras

In this paper, we study Whittaker modules for graded Lie algebras over ℂ. We define Whittaker modules for a class of graded Lie algebras and obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra. As a consequence of this, we obtain a classification of simple Whittaker modules for such algebras. Also, we discuss some concrete algebras as examples.     

Blocks and modules for Whittaker pairs

Inspired by recent activities on Whittaker modules over various (Lie) algebras, we describe a general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case, we obtain a very general set-up for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We establish a connection between our general set-up and the general set-up of Harish-Chandra subalgebras in the sense of Drozd, Futorny and Ovsienko. For Lie algebras with triangular decomposition, we construct a family of simple Whittaker modules (roughly depending on the choice of a pair of weights in the dual of the Cartan subalgebra), describe their annihilators, and formulate several classification conjectures. In particular, we construct some new simple Whittaker modules for the Virasoro algebra. Finally, we construct a series of simple Whittaker modules for the Lie algebra of derivations of the polynomial algebra, and consider several finite-dimensional examples, where we study the category of Whittaker modules over solvable Lie algebras and their relation to Koszul algebras.     

Inequalities for Humbert functions

This paper is motivated by an open problem of Luke’s theorem. We consider the problem of developing a unified point of view on the theory of inequalities of Humbert functions and of their general ratios are obtained. Some particular cases and refinements are given. Finally, we obtain some important results involving inequalities of Bessel and Whittaker’s functions as applications.     

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.      

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).     

Whittaker modules for a Lie algebra of Block type

In this paper, we study Whittaker modules for a Lie algebra of Block type. We define Whittaker modules and under some conditions, obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules over this algebra and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra.     

Eisenstein series for SL(2)

This paper gives explicit formulas for the Fourier expansion of general Eisenstein series and local Whittaker functions over SL₂. They are used to compute both the value and derivatives of these functions at critical points.     

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.     

Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers

The classical sampling theorem has often been attributed to E.T.?Whittaker, but this attribution is not strictly valid. One must carefully distinguish, for example, between the concepts of sampling and of interpolation, and we find that Whittaker worked in interpolation theory, not sampling theory. Again, it has been said that K.?Ogura was the first to give a properly rigorous proof of the sampling theorem. We find that he only indicated where the method of proof could be found; we identify what is, in all probability, the proof he had in mind. Ogura states his sampling theorem as a ??converse of Whittaker??s theorem??, but identifies an error in Whittaker??s work. In order to study these matters in detail we find it necessary to make a complete review of the famous 1915 paper of E.T. Whittaker, and two not so well known papers of Ogura dating from 1920. Since the life and work of Ogura is practically unknown outside Japan, and there he is usually regarded only as an educationalist, we present a detailed overview together with a list of some 70 papers of his which we had to compile. K.?Ogura is presented in the setting of mathematics in Japan of the early 20th century. Finally, because many engineering textbooks refer to Whittaker as a source for the sampling theorem, we make a very brief review of some early introductions of sampling methods in the engineering context, mentioning H.?Nyquist, K.?Küpfmüller, V.?Kotel??nikov, H.?Raabe, C.E. Shannon and I.?Someya.