Related papers: The three missing terms in Ramanujan's septic thet…
Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever $$ |\lambda|\leq 2q^{\frac{1}{2}} $$ for all nontrivial eigenvalues $\lambda$ of $X$. We consider the variant…
In [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function $\theta_3(q)$ at $q=e^{-\pi}$ and encoded it in an integer sequence $(d(n))_{n\ge0}$ for which he provided a recursive procedure to…
A decomposition of identity is given as a complex integral over the coherent states associated with a class of shape-invariant self-similar potentials. There is a remarkable connection between these coherent states and Ramanujan's integral…
We introduce new analogues of the Ramanujan sums, denoted by $\widetilde{c}_q(n)$, associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
We introduce two new integer partition functions, both of which are the number of partition quadruples of $n$ with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$ by use…
In this paper, the logarithmically complete monotonicity property for a functions involving $q$-gamma function is investigated for $q\in(0,1).$ As applications of this results, some new inequalities for the $q$-gamma function are…
Ramanujan studied the analytic properties of many $q$-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious $q$-series fit into the theory of…
In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M.…
We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $\alpha _0~:=~\sqrt{3}/2\pi ~=~0.2756644477\ldots$. We…
In this work we study the Plancherel-Rotach type asymptotics for $q$-Laguerre orthogonal polynomials with complex scaling . The main term of the asymptotics contains Ramanujan function $A_{q}(z)$ for the scaling parameter on the vertical…
Let $p(n)$ be the number of partition of a positive integer $n$. We derive a new identity for complete Bell polynomials based on a generating function of $p(7n+5)$ given by Ramanujan.
In this work we study the Plancherel-Rotach type asymptotics for selected $q$-series and $q$-orthogonal polynomials with complex scalings. The $q$-series we cover are Euler's $q$-exponential, Ramanujan function, Jackson's $q$-Bessel…
We outline an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for $1/\pi$. The principal idea is using algebraic transformations of arithmetic hypergeometric series to translate…
We will use a discrete analogue of the classical Laplace method to show that the main term of the asymptotic expansions of certain entire functions, including Ramanujan's entire function $A_{q}(z)$, can be expressed in terms of…
Applying the theory of elliptic functions we establish two Jacobi theta function identities. From these identities we confirm two q-trigonometric identities conjectured by Gosper. As an application, we give a new and simple proof of a…
In this paper, for a Seifert loop (i.e., a knot in a Seifert three-manifold), first we give a family of an explicit function $\Phi(q; N)$ whose special values at roots of unity are identified with the Witten-Reshetikhin-Turaev invariants of…
In this paper, the power series and hypergeometric series representations of the beta and Ramanujan functions \begin{equation*} \mathcal{B}\left( x\right) =\frac{\Gamma \left( x\right)^{2}}{\Gamma \left( 2x\right) }\text{ and…
In the proposed work, we establish a total of six new $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 7 and 21.These equations can be regarded as modular identities in the alternate theory of…
A polynomial x^3+px^2+qx+r with the condition pr^(1/3)+ 3r^(2/3)+q=0 we call a Ramanujan cubic polynomial (RCP). We study different interest properties of RCP, in particular, an important role of a parameter pq/r. We prove some new…