Related papers: Heine's method and $A_n$ to $A_m$ transformation f…
We demonstrate how formulas that express Hecke-type double-sums in terms of theta functions and Appell--Lerch functions -- the building blocks of Ramanujan's mock theta functions -- can be used to give general string function formulas for…
We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in…
We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to a suitable loop algebra we recover a generalized Drinfeld-Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.
A result of Helson on general Dirichlet series $\sum a_{n} e^{-\lambda_{n}s}$ states that, whenever $(a_{n})$ is $2$-summable and $\lambda=(\lambda_{n})$ satisfies a certain condition introduced by Bohr, then for almost all homomorphism…
In the paper we consider the Heun functions, which are solutions of the equation introduced by Karl Heun in 1889. The Heun functions generalize many known special functions and appear in many fields of modern physics. Evaluation of the…
In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class…
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:…
The history of linear differential equations is over 350 years. By using Frobenius method and putting the power series expansion into linear differential equations, the recursive relation of coefficients starts to appear. There can be…
Through an application of a remarkable result due to Mishev in 2018 concerning the inverses for a class of transformations of sequences of complex numbers, we obtain a very simple proof for a famous series for $\frac{1}{\pi}$ due to…
We shall make use of the method of partial fractions to generalize some of Ramanujan's infinite series identities, including Ramanujan's famous formula for $\zeta(2n+1)$, and we shall also give a generalization of the transformation formula…
We generalize an algorithm of Leclerc describing explicitly the bijection of Lalonde-Ram from finite to affine Lie algebras. In type $A_n^{(1)}$, we compute all affine standard Lyndon words for any order of the simple roots, and establish…
The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric_2F_1,_1F_1 and_0F_1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice…
General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new…
In this paper, we introduced the theory of the sieve function transformation. Using the principle of sieve function transformation, we improved sieve method, and obtained the difference range of similar sieve function values. For this, we…
Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system An, with different dimensions n. We give, with a new, elementary, proof, an elliptic analogue of this transformation. We also…
I consider the power series expansion of Lame function in the Weierstrass's form and its integral forms applying three term recurrence formula[1]. I investigate asymptotic expansions of Lame function for the cases of infinite series and…
The theory of Mellin transform is an incredibly useful tool in evaluating some of the well known results for the zeta function. Ramanujan in his quarterly reports \cite{1} gave a theorem for Mellin transform which is now known as…
Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several…
We define a finite-field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier, et. al in the single variable case. We develop geometric connections…
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…