Related papers: On Catalan Constant Continued Fractions
We show that logarithmic derivative of the Zeta function of any regular graph is given by a power series about infinity whose coefficients are given in terms of the traces of powers of the graph's Hashimoto matrix. We then consider the…
We prove a number of results regarding odd values of the Ramanujan $\tau$-function. For example, we prove the existence of an effectively computable positive constant $\kappa$ such that if $\tau(n)$ is odd and $n \ge 25$ then either \[…
We continue our investigation of the distribution of the fractional parts of $a \gamma$, where $a$ is a fixed non-zero real number and $\gamma$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We…
Easily computable lower and upper bounds are found for the sum of Catalan numbers. The lower bound is proven to be tighter than the upper bound, which previously was declared to be only an asymptotic. The average of these bounds is proven…
Ramanujan's $q$-continued fractions are a central part of Ramanujan's development of basic hypergeometric series. They appear in Chapter 16 of Part III and Chapter 32 of Part V of {\em Ramanujan's Notebooks} edited by Berndt, and in Volume…
We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients…
The generating functions of some sequences of Catalan numbers and Narayana polynomials have simple expansions as continued fractions of Jacobi type. We give an overview of these facts and prove analogous results for q-Narayana polynomials…
We define a new generalization of Catalan numbers to multinomial coefficients. With arithmetic methods, we study their integrality and the integrality of their Lucasnomial generalization. We give a complete characterization of regular Lucas…
In this paper, Riemann's Zeta function with odd positive integer argument is represented as an infinite summation of integer powers of $\pi$ with rational coefficients. Specific values for Apery's Constant and Catalan's Constant are then…
Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In…
In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a…
We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More…
Let $n\geq2$ be an integer. In this paper, we study the convexity of the so-called MacMahon's $q$-Catalan polynomials $C_n(q)=\frac1{[n+1]_q}\left[ 2n \atop n \right]_q$ as functions of $q$. Along the way, several intermediate results on…
The paper describes a prime factorization of the Catalan numbers. Odd prime factors are distributed in layers in accordance with Legendre's formula. The content of each layer is a network of the intervals, Chebyshev's Segments. The primes…
In this paper, we study the combinatorial structures of straight and ordinary m\'enage permutations. Based on these structures, we prove four formulas. The first two formulas define a relationship between the m\'enage numbers and the…
In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and…
We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared…
We present results for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-dimensional and two-dimensional series. Most of these series can be expressed in…
The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the…
We prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then…