Related papers: More Infinite Products: Thue-Morse and the Gamma f…

The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In…

In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is…

We derive hypergeometric formulas for Euler's constant, gamma. A "by-product" of Thomae's transformation is an infinite product for e^gamma involving the binomial coefficients. Alternate, non-hypergeometric proofs use a double integral for…

We give an exact coefficients formula of any infinite product of power series with constant term equal to $1$, by using structures from partitions of integers and permutation groups. This is an universal theorem for various of Binomial-type…

Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations…

We study a binary Thue--Morse-type sequence arising from the base-$3/2$ expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms…

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vign\'{e}ras multiple gamma function by considering the classical limit of the multiple q-gamma…

A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse…

We say a power series $a_0+a_1q+a_2q^2+\cdots$ is \emph{multiplicative} if $n\mapsto a_n/a_1$ for positive integers $n$ is a multiplicative function. Given the Eisenstein series $E_{2k}(q)$, we consider formal multiplicative power series…

One of the earliest examples of analytic representations for $\pi$ is given by an infinite product provided by Wallis in 1655. The modern literature often presents this evaluation based on the integral formula $$ \frac{2}{\pi} \int_0^\infty…

Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that $$\liminf_{n\to \infty} (q_{n+1}-q_n) \le 6.$$ This sharpens an earlier result of the authors (arXivMath NT/0506067), which had 26 in place…

Let $(m_1,\ldots,m_J)$ and $(r_1,\ldots,r_J)$ be two sequences of $J$ positive integers satisfying $1\le r_j< m_j$ for all $j=1,\ldots,J$. Let $(\delta_1,\ldots,\delta_J)$ be a sequence of $J$ nonzero integers. In this paper, we study the…

Two results concerning the number of threshold functions $P(2, n)$ and the probability ${\mathbb P}_n$ that a random $n\times n$ Bernoulli matrix is singular are established. We introduce a supermodular function $\eta^{\bigstar}_n : 2^{{\bf…

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{\alpha_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty…

Let $h_n(v)$ be the sequence of rational functions with $$ \frac{h_n(v)}{v}-nh_n(v)+(n-1)h_{n-1}(v)-vh_{n-1}'(v)+\frac{v(v(vh_{n-1}(v))')'}{4}=0 $$ for $n>0$ and $h_0(v)=1$. We prove that $h_n(v)$ has a pole at $v=\frac{1}{n}$ if and only…

We obtain infinite product expansions in the sense of Borcherds for theta functions associated with certain positive-definite binary quadratic and binary hermitian forms. Among other things, we show that every weight 1 binary theta function…

The main goal of this paper is to give a modular type representation for the infinite product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$. It is shown that this representation essentially contains the well-known modular formulae either for Dedekind's…

We show that the infinite symmetric product of a connected graded-commutative algebra over the rationals is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular,…

We investigate the continued fraction expansion of the infinite products $g(x) = x^{-1}\prod_{t=0}^\infty P(x^{-d^t})$ where polynomials $P(x)$ satisfy $P(0)=1$ and $\deg(P)<d$. We construct relations between partial quotients of $g(x)$…

We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers-Ramanujan continued fraction. For example, if $$\sum_{n=0}^{\infty}A(n)q^{n}:=…