Contents 1. Introduction 1 2. Some preliminaries 5 3. The point spectrum of U p 7 4. A structure theorem for eigenfunctions 14 5. A decomposition into finite dimensional eigenspaces 21 6. Simultaneous eigenfunctions 23 7. A first application: tensor products of Hecke operatorsand the Riemann zeta function 27 8. A second application: completely multiplicative coefficients 30 9. Appendix: Explici...

We investigate a transformation on the space of rational functions.

The denseness of rational functions with prescribed poles in the Hardy space and disk algebra is considered. Notations. C complex plane D unit disk fz : jzj < 1g Tunit circle fz : jzj = 1g H p Hardy space of analytic functions on D kfk 1 := supfjf(z)j : z 2 D g, the H 1 norm A(D) disk algebra of functions analytic on D and continuous on D P n set of polynomials of degree at most n

In his book on the theory of meromorphic functions, R. Nevanlinna proved a number of "uniqueness theorems." The most important of them states that if two functions w=f(x) and w = g(x), meromorphic in the whole x-plane, assume five values of w (finite or infinite) at the same points x they must be identical. If we understand by the distribution of a function w = (x) with respect to a given va...

We study the following questions: (1) What are all solutions to f ◦ f̂ = g ◦ ĝ in complex rational functions f, g ∈ C(X) and meromorphic functions f̂ , ĝ on the complex plane? (2) For which rational functions f(X) and g(X) with coefficients in an algebraic number field K does the equation f(a) = g(b) have infinitely many solutions with a, b ∈ K? We utilize various algebraic, geometric and analyti...

This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present a certi cate that consists of a pair of functions of two integer variables. To prove the identity, take the two functions that are given, check that condition (1) below is satis ed (a simple mechanical task), and check the equally simple fact that the boundary conditio...

Stony Brook IMS Preprint #2008/4 September 2008 Abstract. Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston–Teichmüller theory. We will discuss a topological theory of regluing, and trace a direction in...

We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input variables of the (quasi-polynomial) function are fixed, there is a polynomial time algorithm which converts between the two representations. Examples of such countin...

Our final integration technique deals with the class of functions known as rational functions. Recall from Calculus I that DEFINITION 7.1. A rational function 1 is a function that is the ratio of two polynomials 1 Here 'rational' means 'ratio', as in 'the ratio of two polynomials.' r(x) = p(x) q(x) , where p(x) and q(x) are polynomials. (Remember a polynomial has the form p(x) = a n x n + a n−1...

The rational functions are shown to coincide with the compositions of endmarkings, morphisms and inverses of injective morphisms. To represent a rational function x we need one ednmarking \xm, two morphisms au <x3 and one inverse of an injective morphism <x2 and then Resumé. — On montre que les fonctions rationnelles coïncident avec les compositions de marquages \im est un marquage terminal, <x...