In this paper we will study the number-theoretical properties of the expression v1 nkal rcka,, d(p; a I . . . . . an) = ( 1) n/2 ~ cot cot (1) k=l P P and of related finite trigonometric sums. In Eq. (I), p is a positive integer, a~ . . . . . a, are integers prime to p, and n is even (for n odd the sum is clearly equal to zero). There are two reasons for being interested in sums of this type. First of all, the case n = 2 is, up to a factor, the classical Dedekind sum:

/pdf/higher-dimensional-dedekind-sums-2g7ksjkkpf.pdf

Higher Dimensional Dedekind Sums

In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q=1-p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the limit as N goes to infinity. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schuetz for the case N=2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m'th left-most particle is at x at time t. In one of these formulas we can take the limit as N goes to infinity, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.

Integral Formulas for the Asymmetric Simple Exclusion Process

pdf/explicit-evaluations-and-reciprocity-theorems-for-finite-3nzou0t6yf.pdf

Explicit evaluations and reciprocity theorems for finite trigonometric sums

Introduction. This paper is the outgrowth of the author's Ph.D. thesis written under the supervision of Professor A. Peyerimhoff, whose paper on zeros of power series [4] has been the starting point. §1 contains some lemmas due to A. Peyerimhoff, which are needed in §2. In all theorems (with the exception of Theorem 4 only) of the remaining part of this paper, real entire functions flz) of finite order with infinitely many real and finitely many complex^) zeros are taken as starting point. Then in §2 functions of the form g(z) = A(z)f(z) + B(z)f'(z) are considered where A(z) and B(z) are real polynomials. In Theorem 1 upper bounds for the number of zeros of g(z) in the complex plane with the exception of certain real intervals are obtained. In certain instances the exact number of zeros is obtained. At the end of §2 examples are given which show that the results of Theorem 1 are best possible. Questions of this kind have been considered by Laguerre and Borel especially [1], [2] for the particular case g(z) =f'(z). One of the results of [4] deals with functions g(z) = af(z) + zf'(z) where a is real and/(z) of order < 1. The method of proof of this result has been generalized in the present paper. In §3 a couple of theorems (2, 3, 5, 6) which are derived from Theorem 1 are partly slight generalizations of known theorems, especially of Laguerre [2], [3], [5], [6]. These theorems deal with questions of the following kind. If one has some information on the zeros of the entire function flz) = 2 anzn, one wants to gain information on the zeros of F(z) = 2 anG(n)zn, where G(z) is an entire function of a certain type. For Theorem 4, which is known already, a short proof is given here which is independent of Theorem 1. From this theorem follows immediately the wellknown fact that the Besselfunctions of real order > — 1 have real zeros only. From Theorem 6 Hurwitz's Theorem on the complex zeros of the Besselfunctions of real order < — 1 follows as a special case. In Theorem 9 functions g(z) = af(z) + zlf'(z) are considered, where a is real, / odd, and flz) is a real canonical product of finite genus. Then the exact number of complex zeros of g(z) is obtained. Especially we find that for a>0 and / odd the functions a sin z-l-z' cos z and a cos z—z' sin z have exactly I— 1 complex zeros.

/pdf/zeros-of-entire-functions-3vsvgunjos.pdf

Zeros of entire functions

The radius of uniform convexity of Bessel functions

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