Related papers: Lost in translation
By applying the partial derivative operator to several summation formulas for hypergeometric series, we prove several double series for $\pi$ in this paper. Similarly, we also establish several $q$-analogues of them.
Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the…
By using two known transformation formulas for basic hypergeometric series, we establish a direct extension of Bailey's $_6\psi_6$-series identity. Subsequently, it and Milne's identity are employed to drive multi-variable generalizations…
Ramanujan sums have attracted significant attention in both mathematical and engineering disciplines due to their diverse applications. In this paper, we introduce an algebraic generalization of Ramanujan sums, derived through polynomial…
We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental…
We provide a simple way to add, multiply, invert, and take traces and norms of algebraic integers of a number field using integral matrices. With formulas for the integral bases of the ring of integers of at least a significant proportion…
In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $\pi$. Among these, one of the most celebrated is the following series:…
We present a new method for producing series for $1/\pi$ and other constants using Legendre's relation, starting from a generation function that can be factorised into two elliptic $K$'s; this way we avoid much of modular theory or creative…
Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a…
We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic…
We show that a binomial identity arising in the context of the study of series expansions of $1/\pi$ can be seen as an incarnation of Whipples second theorem for hypergeometric series.
We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…
We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's…
In this paper we describe all rotation $H$-hypersurfaces in $H^n \times R$ and use them as barriers to prove existence and characterization of certain vertical $H$-graphs and to give symmetry and uniqueness results for compact…
Each of Ramanujan's series for $\frac{1}{\pi}$ is of the form $$ \sum_{n=0}^{\infty} z^n \frac{ (a_{1})_{n} (a_{2})_{n} (a_{3})_{n} }{ (b_{1})_{n} (b_{2})_{n} (b_{3})_{n} } (c_{1} n + c_2) $$ for rational parameters such that the difference…
We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.
This article gives a classification scheme of algebraic transformations of Gauss hypergeometric functions, or pull-back transformations between hypergeometric differential equations. The classification recovers the classical transformations…
Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define…
We apply the method established in our previous work to derive a Chudnovsky-Ramanujan type formula for the Legendre family of elliptic curves. As a result, we prove two identities for $1/\pi$ in terms of hypergeometric functions.
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.