Related papers: Continued Fractions for Square Series Generating F…

Legendre discovered that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. In this vein, it was recently observed that the continued fraction…

This work introduces a new functional series for expanding an analytic function in terms of an arbitrary analytic function. It is generally applicable and straightforward to use. It is also suitable for approximating the behavior of a…

Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and…

For $q \in (0, 1)$, the deformed exponential function $f(x) = \sum_{n \geq 1} x^n q^{n(n-1)/2}/n!$ is known to have infinitely many simple and negative zeros $\{x_k(q)\}_{k \geq 1}$. In this paper, we analyze the series expansions of…

Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated…

We present a theorem on taking the repeated indefinite summation of a holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying exponential bounds as the imaginary part grows. We arrive at this result using transforms…

We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P^1R, determines a…

Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and…

A century ago, P. A. MacMahon introduced two families of generating functions, $$ \sum_{1\leq n_1<n_2<\cdots<n_t}\prod_{k=1}^t\frac{q^{n_k}}{(1-q^{n_k})^2} \quad\text{ and } \sum_{\substack{1\leq n_1<n_2<\cdots<n_t\\…

We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find…

Single variable hypergeometric functions pFq arise in connection with the power series solution of the Schrodinger equation or in the summation of perturbation expansions in quantum mechanics. For these applications, it is of interest to…

We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple…

Let $\cal R$ be either the Grothendieck semiring (semiring with multiplication) of complex algebraic varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We…

In this paper we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann-Liouville fractional integral and derivative operators on a compact of the real axis.This approach has some advantages and allows us to…

The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to $p$-adic numbers where it presents many differences with…

This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an…

We express the $q$-Pochhammer symbol $(z;q)_\infty$ as an infinite product of gamma functions, analogously to how Narukawa expressed the elliptic gamma function as an infinite product of hyperbolic gamma functions. This identity is used to…

In our previous works we introduced a $q$-deformation of the generating functions for enriched $P$-partitions. We call the evaluation of this generating functions on labelled chains, the $q$-fundamental quasisymmetric functions. These…

We introduce a new generalization of Stirling numbers of the second kind and analyze their properties, including generating functions, integral representations, and recurrence relations. These numbers are used to approximate Riemann zeta…

Several authors have examined connections among restricted permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for involutions which avoid 3412. Our results…