Related papers: Romik's Conjecture for the Jacobi Theta Function

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$…

If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+\alpha_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation…

The Weil representation discovered by Andre Weil plays an important role in the study of the tranformation properties of theta series. In this paper, we define the Weil-Schroedinger representation of the Jacobi group and prove that the…

Let $p$ be an odd prime, and let $\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ be the reduction modulo $p$ of the Artin-Hasse exponential. We obtain a polynomial expression for $a_{kp}$ in terms of those $a_{rp}$ with $r<k$, for even…

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case,…

The purpose of this paper is to formulate and study a common refinement of a version of Stark's conjecture and its $p$-adic analogue, in terms of Fontaine's $p$-adic period ring and $p$-adic Hodge theory. We construct period-ring-valued…

We give a new proof for a product formula of Jacobi which turns out to be equivalent to a $q$-trigonometric product which was stated without proof by Gosper. We apply this formula to derive a $q$-analogue for the Gauss multiplication…

In 1997, B\'aez-Duarte gave a probabilistic proof of the asymptotic formula for the partition function, which had originally been proved by Hardy-Ramanujan. Based on the probabilistic approach, this paper proves an asymptotic formula for…

We prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of…

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv1\pmod{2m}$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product $\prod_{k\in…

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a…

In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of…

For a given $\theta\in (-1,1)$, we find out all parameters $\alpha,\beta\in \{0,1\}$ such that, there exists a linear combination of Jacobi polynomials $J_{n+1}^{(\alpha,\beta)}(x)-C J_{n}^{(\alpha,\beta)}(x)$ which generates a Lobatto…

We form the Jacobi theta distribution through discrete integration of exponential random variables over an infinite inverse square law surface. It is continuous, supported on the positive reals, has a single positive parameter, is unimodal,…

All the periodic points of a certain algebraic function related to the Rogers-Ramanujan continued fraction $r(\tau)$ are determined. They turn out to be $0, \frac{-1 \pm \sqrt{5}}{2}$, and the conjugates over $\mathbb{Q}$ of the values…

A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to…

It is widely believed that the parity of the partition function $p(n)$ is ``random.'' Contrary to this expectation, in this note we prove the existence of infinitely many congruence relations modulo 4 among its values. For each square-free…

Okazaki and Smith discovered many elegant formulas expressing some matrix integrals as some celebrated $q$-series such as the Rogers--Ramanujan functions or Jacobi theta functions. These integrals arise as Wilson line defect half-indices of…

Assume that the Riemann hypothesis holds for Dedekind zeta functions. Under this assumption, we prove that a degree $d$ polynomial with random multiplicative $\pm1$ coefficients is irreducible in $\mathbb{Z}[x]$ with probability…

We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special…