Related papers: On Ramanujan's cubic continued fraction

Let K be a number field, let f: P_1 --> P_1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w in P_1(K) are not preperiodic under f. We prove that the set of (m,n) in N^2…

This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered…

In a paper published in 2023, Wagner introduced and studied Jacobi forms with complex multiplication, and gave several applications. One such application was in constructing a new doubly-infinite family of partition-theoretic objects,…

We consider the polynomials $\displaystyle f(x)=x^d+c$, where $d\ge 2$ and $c\in\mathbb Q$. It is conjectured that if $d=2$, then $f$ has no rational periodic point of exact period $N\ge 4$. In this note, fixing some integer $d\ge 2$, we…

We study and partially classify cubic rational expressions $g(x)/h(x)$ over a finite field $\mathbb{F}_q$, up to pre- and post-composition with independent M\"obius transformations. In particular, we obtain a full classification when $q$ is…

It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function ${}_2F_1$ arises from a relation between modular curves, namely the covering of $X_0(3)$ by $X_0(9)$. In general, when $2\le N\le 7$ the N-fold cover of…

We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the `normal' statistics given by the Gauss-Kuzmin measure. As a by-product,…

In this note we consider functions with Moebius-periodic rational coefficients. These functions under some conditions take algebraic values and can be recovered by theta functions and the Dedekind eta function. Special cases are the…

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest,…

We derived $q$-continued fractions $X_i(q)$ of order thirty-four and continued fractions $Y_i(q)$ of order sixty-eight from a general continued fraction identity of Ramanujan, where $i=1,2,3,4,5,6,7$ and $8$. We established some…

In this work, we present continued fractions for the arithmetic, geometric, harmonic and cotangent means of $[a_0,a_1,\dots,a_k]$ and $[a_0,a_1,\dots,a_k,a_{k+1}]$, and some of their applications.

In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any…

Euler gives a continued fraction representation of (1+x)^n involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for ``vanishing'' n, and for infinite…

We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational…

We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula…

Let $R$ be a complete equicharacteristic noetherian local domain and $\nu$ a valuation of its field of fractions whose valuation ring dominates $R$ with trivial residue field extension. The semigroup of values of $\nu$ on $R\setminus \{0\}$…

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field $\mathbb{F}_q$. We count the irreducible polynomials in $\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\, f}\cdot…

Recently the partial wave cutoff method was developed as a new calculational scheme for a functional determinant of quantum field theory in radial backgrounds. For the contribution given by an infinite sum of large partial waves, we derive…

For a given point P in the group of K-rational points E(K) of an elliptic curve, we consider the sequence of values (F_1(P),F_2(P),F_3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is…

We obtained the $C^\alpha$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form $${\partial_t u} = \partial_x(a(x,y,t)\partial_x u) +b_0(x,y,t)\partial_x u+b(x,y,t)\partial_y u,$$…