Related papers: Jacobian Elliptic Functions, Continued Fractions a…
We investigate the quantitative unique continuation properties of solutions to second-order elliptic equations with lower-order terms. In particular, we establish quantitative forms of the strong unique continuation property for solutions…
In the present work, we established continued fractions of level eighteen, twenty six and thirty. Further, we obtained vanishing coefficients and many algebraic relations. To validate our result colored partitions are also obtained.
By applying the MC algorithm and the Bauer-Muir transformation for continued fractions, in this paper we shall give six examples to show how to establish an infinite set of continued fraction formulas for certain Ramanujan-type series, such…
In this article with the help of the inverse function of the singular moduli we evaluate the Rogers Ranmanujan continued fraction and his first derivative.
Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.
A contiguous relation for complementry pairs of very well poised balanced ${}_{10}\phi_9$ basic hypergeometric functions is used to derive an explict expression for the associated continued fraction. This generalizes the continued fraction…
We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…
By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive…
We use the method of generating functions to find the limit of a $q$-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities,…
The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$,…
Jacobi elliptic functions and complete elliptic integrals are generalized using three parameters. These generalized functions and integrals are closely related to ordinary differential equations involving $p$-Laplacian. In this paper,…
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 prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
Finite trigonometric sums appear in various branches of Physics, Mathematics and their applications. For p; q to coprime positive integers and r we consider the finite trigonometric sums involving the product of three trigonometric…
We begin with an improvement to an extension result for subharmonic functions of Blanchet et al. With the aid of this improvement we then give extension results for subharmonic functions, for separately subharmonic functions, for harmonic…
In this paper, we apply high level versions of Jacobi's derivative formula to number theory such as quarternary quadratic forms and convolution sums of some arithmetical functions.
We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction.…
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…
Jacobi sums and cyclotomic numbers are the important objects in number theory. The determination of all the Jacobi sums and cyclotomic numbers of order $e$ are merely intricate to compute. This paper presents the lesser numbers of Jacobi…
We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in…