On A Limiting Relation Between Ramanujan's Entire Function $A_{q}(z)$ And $θ$-Function

On The Chaotic Asymptotics of Ramanujan's Entire Function $A_q(z)$

We will use a discrete analogue of the classical \emph{Laplace} method to show that for infinitely many positive integers $n$, the main term of the asymptotic expansions of scaled confluent basic hypergeometric functions, including the Ramanujan's entire function $A_{q}(z)$, could be expressed in $θ$-functions, and the error term depends on the ergodic property of certain real scaling parameter.

Zeros of Ramanujan Type Entire Functions

In this work we establish some polynomials and entire functions have only real zeros. These polynomials generalize q-Laguerre polynomials $L_{n}^{(α)}(x;q)$, while the entire functions are generalizations of Ramanujan's entire function $A_{q}(z)$, q-Bessel functions $J_ν^{(2)}(z;q)$, $J_ν^{(3)}(z;q)$ and confluent basic hypergeometric series.

广义拉马努金整函数的连接公式

Connection Formulae for a Generalised Ramanujan Entire Function

本文推广拉马努金-$q$-艾里连接公式至单参数变形，通过$q$-波雷尔反演和离散求和，给出整函数无穷远处的发散渐近展开及连接系数，并证明余项估计。

拉马努金整函数 $A_q(z)$ 的艾里转折点渐近分析

Airy Turning-Point Asymptotics for Ramanujan's Entire Function $A_q(z)$

本文对拉马努金整函数 $A_q(z)$ 在特定转折点标度下进行渐近分析，得到其艾里函数展开式及一阶修正，确定了其正零点与艾里零点的对应关系，验证了相关差分方程与微分方程的联系，补充了森田反对称伴随函数的归一化极限结果。

On Certain Generalizations of Rogers-Ramanujan Type Identities

We state and prove a number of unilateral and bilateral $q$-series identities and explore some of their consequences. Those include certain generalizations of the $q$-binomial sum which also generalize the $q$-Airy function introduced by Ramanujan, as well as certain identities with an interesting variable-parameter symmetry based on limiting cases of Heine's transformation of basic hypergeomteric functions.

Ramanujan's Beautiful Integrals

Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the \emph{Journal of the Indian Mathematical Society}, notebooks, Quarterly Reports to the University of Madras, letters to Hardy, published papers and the Lost Notebook. His evaluations are often surprising, beautiful, elegant, and useful in other mathematical contexts. He also discovered general methods for evaluating and approximating integrals. A survey of Ramanujan's contributions to the evaluation of integrals is given, with examples provided from each o…