A generating function for a class of generalized Bernoulli polynomials

First we derive a generating function and a Fourier expansion for a class of generalized Bernoulli polynomials. Then we derive formulas that allow certain Dirichlet series to be evaluated in terms of these generalized Bernoulli polynomials.      

The Generalized Segal–Bargmann Transform and Special Functions

Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner–Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions.     

The correlation functions of vertex operators and Macdonald polynomials

The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n = 1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.     

De Moivre and Bell polynomials

We survey a family of polynomials that are very useful in all kinds of power series manipulations, and appearing more frequently in the literature. Applications to formal power series, generating functions and asymptotic expansions are described, and we discuss the related work of De Moivre, Arbogast and Bell.     

Laguerre–Freud equations for three families of hypergeometric discrete orthogonal polynomials

The Cholesky factorization of the moment matrix is considered for discrete orthogonal polynomials of hypergeometric type. We derive the Laguerre–Freud equations when the first moments of the weights are given by the ₁F₂, ₂F₂, and ₃F₂ generalized hypergeometric series.     

A generalization of the class laguerre polynomials: asymptotic properties and zeros

We consider a modification of the gamma distribution by adding a discrete measure Support in the point x = 0. We study some properties of the polynomials orthogonal with respect to such measures [1]. In particular, we deduce the second order differential to'1ttatiolt and the three term recurrence relation which such polynomials satisfy as well as, for large n. the behaviour of their zeros.     

Polynomial Interpolation of the Discrete Logarithm

Recently, Coppersmith and Shparlinski proved several results on the interpolation of the discrete logarithm in the finite prime field by polynomials modulo p and modulo p-1, respectively. In this paper most of these results are extended to arbitrary .     

Orthogonal polynomials and Gaussian quadrature for refinable weight functions

We show how to compute the modified moments of a refinable weight function directly from its mask in O(N²n) rational operations, where N is the desired number of moments and n the length of the mask. Three immediate applications of such moments are:

• the expansion of a refinable weight function as a Legendre series;

• the generation of the polynomials orthogonal with respect to a refinable weight function;

• the calculation of Gaussian quadrature formulas for refinable weight functions.

In the first two cases, all operations are rational and can in principle be performed exactly.

A Formula for <Emphasis Type="Italic">N</Emphasis>-Row Macdonald Polynomials

We derive a formula for the n-row Macdonald polynomials with the coefficients presented both combinatorically and in terms of very-well-poised hypergeometric series.     

Computation of certain infinite series of the form for arbitrary real-valued k

We provide a general method from which one may obtain more arbitrary closed form solutions for infinite series of the form ∑f(n)n^k, for and sufficiently well-behaved f(n). We illustrate the methods for the special case of f(n)=aⁿ/n!, obtaining generalizations to the Bell numbers and polynomials.     

Approximation of nonnegative functions by means of exponentiated trigonometric polynomials

We consider the problem of approximating a nonnegative function from the knowledge of its first Fourier coefficients. Here, we analyze a method introduced heuristically in a paper by Borwein and Huang (SIAM J. Opt. 5 (1995) 68–99), where it is shown how to construct cheaply a trigonometric or algebraic polynomial whose exponential is close in some sense to the considered function. In this note, we prove that approximations given by Borwein and Huang's method, in the trigonometric case, can be related to a nonlinear constrained optimization problem, and their convergence can be easily proved under mild hypotheses as a consequence of known results in approximation theory and spectral properties of Toeplitz matrices. Moreover, they allow to obtain an improved convergence theorem for best entropy approximations.     

Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series

We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p-adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p-adic type.     

Two finite hypergeometric sequences of discrete orthogonal polynomials

Two finite hypergeometric sequences of symmetric orthogonal polynomials of a discrete variable are introduced and their standard properties, such as second-order difference equations, explicit forms of the polynomials and three term recurrence relations are obtained. As a consequence of two specific Sturm–Liouville problems, it is proved that these polynomials are finitely orthogonal with respect to two symmetric weight functions.     

Functional Non-Central and Central Limit Theorems for Bivariate Appell Polynomials

We consider quadratic forms in bivariate Appell polynomials involving strongly dependent time series. Both the spectral density of these time series and the Fourier transform of the kernel of the quadratic forms are regularly varying at the origin and hence may diverge, for example, like a power function. We obtain functional limit theorems for these quadratic forms by extending the recent results on the convergence of their finite-dimensional distributions. Some of these are functional central limit theorems where the limiting process is Brownian motion. Others are functional non-central limit theorems where the limiting processes are typically not Gaussian or, if they are Gaussian, then they are not Brownian motion.     

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.     

A Rigorous Proof of the Waterloo Algorithm for the Discrete Logarithm Problem

In this paper we are concerned with the Waterloo variant of the index calculus method for the discrete logarithm problem in . We provide a rigorous proof for the heuristic arguments for the running time of the Waterloo algorithm. This implies in studying the behavior of pairs of coprime smooth polynomials over finite fields. Our proof involves a double saddle point method, and it is in nature similar to the one of Odlyzko for the rigorous analysis of the basic index calculus.     

A General Polynomial Sieve

An important component of the index calculus methods for finding discrete logarithms is the acquisition of smooth polynomial relations. Gordon and McCurley (1992) developed a sieve to aid in finding smooth Coppersmith polynomials for use in the index calculus method. We discuss their approach and some of the difficulties they found with their sieve. We present a new sieving method that can be applied to any affine subspace of polynomials over a finite field.     

A generating function for Blattner's formula

Let be a connected, semisimple Lie group with finite center and let be a maximal compact subgroup. We investigate a method to compute multiplicities of -types in the discrete series using a rational expression for a generating function obtained from Blattner's formula. This expression involves a product with a character of an irreducible finite-dimensional representation of and is valid for any discrete series system. Other results include a new proof of a symmetry of Blattner's formula, and a positivity result for certain low rank examples. We consider in detail the situation for of type split . The motivation for this work came from an attempt to understand pictures coming from Blattner's formula, some of which we include in the paper.