On a claim of Ramanujan about certain hypergeometric series

Generalizations of Ramanujan integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications

In this paper, we obtain analytical solution of an unsolved integral $\textbf{R}_{C}(m,n)$ of Srinivasa Ramanujan [$\textit{Mess. Math}$., XLIV, 75-86, 1915], using hypergeometric approach, Mellin transforms, Infinite Fourier cosine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer's symbol. Also we have given some generalizations of the Ramanujan's integral $\textbf{R}_{C}(m,n)$ in the form of integrals $\textbf{I}^{*}_{C}(\upsilon,b,c,λ,y), \textbf{J}_C (\upsilon,b,c,λ,y), \textbf{K}_{C} (\upsilon,b,c, λ,y), \textbf{I}_{C}(\upsilon,b,λ,y)$ and sol…

Curious extensions of Ramanujan's 1-psi-1 summation formula

We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our identities, by analytic continuation, to bilateral summation formulae which contain Ramanujan's 1-psi-1 summation and a very-well-poised 4-psi-6 summation as special cases.

An Application of Hypergeometric Functions to Heat Kernels on Rectangular and Hexagonal Tori and a "Weltkonstante" -- Or -- How Ramanujan Split Temperatures

In this work we investigate the heat kernel of the Laplace--Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss' hypergeometric function $_2F_1$ and the elliptic modulus. In order to be able to do this, we employ a beautiful result of Ramanujan, connecting hypergeometric functions, the elliptic modulus and theta functions. Also, we investigate the temperature distribution of the heat kernel on hexagonal tori and use Ramanujan's corresponding theory of s…

Several transformation formulas for basic hypergeometric series

In 1981, Andrews gave a four-variable generalization of Ramanujan's ${_1ψ_1}$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${_8ϕ_7}$ series and Bailey's transformation formula for three ${_8ϕ_7}$ series. Then it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two $_3ψ_3$ summation formulas in terms of two limiting relations and Bailey's another transformation formula for three $_8ϕ_7$ series. Bas…

Ramanujan-type formulae for $1/π$: the art of translation

We outline an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for $1/π$. The principal idea is using algebraic transformations of arithmetic hypergeometric series to translate non-singular points into singular ones, where the required constants can be computed using asymptotic analysis.

拉马努金初等连分数的初等处理

An elementary treatment of Ramanujan's elementary continued fractions

本文研究拉马努金第二笔记本第12章中的初等连分数，这些连分数是其超几何与$q$-超几何连分数的前身，并给出初等处理。

Integrals and Series Representations of $q$-Polynomials and Functions: Part II Schur Polynomials and the Rogers-Ramanujan Identities

We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type identities. We also study some bilateral extensions of certain transformations in the theory of basic hypergeometric functions.

带移位指标的超级几何加速

Hypergeometric accelerations with shifted indices

本文利用 Wilf 加速方法的变体和 Pochhammer 符号的移位指标，在 Chu 与 Zhang 的超级几何变换基础上恢复并推广了多类加速级数，并构造出新的 Ramanujan 型通用常数级数。

Generalizations of Ramanujan's reciprocity formula and the Askey-Wilson integral

By using two known transformation formulas for basic hypergeometric series, we establish a direct extension of Bailey's $_6ψ_6$-series identity. Subsequently, it and Milne's identity are employed to drive multi-variable generalizations of Ramanujan's reciprocity formula. Then we utilize also Milne's identity to deduce a multi-variable generalization of the Askey-Wilson integral.

Ramanujan's cubic transformation and generalized modular equation

We study the quotient of hypergeometric functions \begin{equation*} μ_{a}^*(r)=\fracπ{2\sin{(πa)}}\frac{F(a,1-a;1;1-r^3)}{F(a,1-a;1;r^3)} \quad (r\in(0,1)) \end{equation*} in the theory of Ramanujan's generalized modular equation for $a\in(0,1/2]$, find an infinite product formula for $μ_{1/3}^*(r)$ by use of the properties of $μ_{a}^*(r)$ and Ramanujan's cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan's cubic transformation.

The Stokes phenomenon for the Ramanujan's $q$-difference equation and its higher order extension

We show connection formulae of local solutions of the Ramanujan equation between the origin and the infinity. These solutions are given by the Ramanujan function, the $q$-Airy function and the divergent basic hypergeometric series ${}_2φ_0(0,0;-;q,x)$. We use two different $q$-Borel-Laplace resummation methods to obtain our connection formulae. We also introduce the $q$-Borel-Laplace transformation of level $r-1$, which are higher order extension of these transformations. These methods are useful to obtain an asymptotic formula of a divergent series ${}_rφ_0(0,0,\dots ,0;-;q,x)$.

Certain Integrals Arising from Ramanujan's Notebooks

In his third notebook, Ramanujan claims that $$ \int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \,\mathrm{d} x + \fracπ{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d} x = 0. $$ In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if $\log x$ were replaced by $\log^2x$ in the first integral and $\log x$ were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly rela…

Identities for the Ramanujan zeta function

We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that $L(Δ, k)$ is a period in the sense of Kontsevich and Zagier when $k\ge12$. As an illustration, we reduce $L(Δ, k)$ to explicit integrals of hypergeometric and algebraic functions when $k\in\{12,13,14,15\}$.

Elliptic functions from hypergeometric integrals

As a contribution to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has shown how analogues of the Jacobian elliptic functions may be derived from incomplete hypergeometric integrals in signatures three and four. We determine precisely the signatures in which the Jacobian analogues or their squares are indeed elliptic.

Ramanujan-type formulae for $1/π$: $q$-analogues

The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of $π$, Archimedes' constant, remain an attractive object of arithmetic study. In this note we discuss some $q$-analogues of Ramanujan-type evaluations and of related supercongruences.

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be ``continuous'' in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator…

利用Beta积分加速超几何函数以推导π和Gamma函数值的新无穷级数

Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $π$ and Values of the Gamma Function

本文通过beta积分加速超几何函数$2 F 1$，推导出$π$和Gamma函数值的新无穷级数，并基于初等反三角函数得出每项增加2.5位数字的紧凑类Ramanujan公式。

Summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1 ( a; b; q, z )$

We give summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$. We also show the $q \to 1 - 0$ limit of our summation formulae.

Semi-Finite Forms of Bilateral Basic Hypergeometric Series

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 $_1ψ_1$ summation, Bailey's $_2ψ_2$ transformations, and Bailey's $_6ψ_6$ summation.

Proof of a summation formula for an $\tilde A_n$ basic hypergeometric series conjectured by Warnaar

A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system $\tilde A_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation, Ramanujan's $_1ψ_1$-summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affine root systems $\tilde A_n$ respectively $\tilde A_m$, which generalizes at the same time the above summation formula and an identity due to Gessel and the author, is proposed as a conjecture.

Special hypergeometric motives and their $L$-functions: Asai recognition

We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai $L$-functions of Hilbert modular forms.

Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler's constant $γ$ with properties of the values of the $q$-logarithm function, with natural choice of $q$. By these means, we generalize a classical formula for $γ$ due to Ramanujan, together with Vacca's and Gosper's series for $γ$, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler's constant. The main tools are Euler-type integrals and hypergeometric series.

A new A_n extension of Ramanujan's 1-psi-1 summation with applications to multilateral A_n series

In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we pro…

theta函数与双侧基本超几何级数的变换

Theta functions and transformations of bilateral basic hypergeometric series

本文通过theta函数和双侧基本超几何级数建立新变换公式，构造伴生q级数，证明McIntosh关于q级数渐近变换的猜想，并扩展了Ramanujan和McIntosh的一些恒等式。

基本超几何级数的因式分解

Factorization of Basic Hypergeometric Series

本文研究基本超几何级数的因式分解问题，重点分析一般 $_2ψ_2$ 级数，并建立与根系上基本超几何级数理论的联系，同时给出若干经典求和与变换公式的新证明。

Wilf-Zeilberger种子与非平凡超几何恒等式

Wilf-Zeilberger seeds and non-trivial hypergeometric identities

本研究采用生成Wilf-Zeilberger对的系统方法，证明了多位学者提出的超几何恒等式猜想，涵盖Ramanujan型1/π相关公式及ζ(5)级数等重要结果。

The $\bal$\ and $\bcl$\ Bailey Transform and Lemma

We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative ``Bailey chain'' concept in the setting of basic hypergeometric series very well-poised on unitary $A_{\ell}$ or symplectic $C_{\ell}$ groups. The classical case, corresponding to $A_1$ or equivalently $\roman U(2)$, contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. In particular, our program extends much of the classical work of Rogers, Bailey, Slater, Andrews, and Bressoud.

Series acceleration formulas obtained from experimentally discovered hypergeometric recursions

In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $β$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $\frac{1}{π^2}$ due to Guillera, and a family…

Analogues of the general theta transformation formula

A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $Ξ$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives as by-products, transformation formulas of the form $F(z,α)=F(iz,β)$, where $αβ=1$. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulas of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization o…

特殊拉伸$9j$系数的$_5F_4$超几何级数表达

Expression of special stretched $9j$ coefficients in terms of $_5F_4$ hypergeometric series

本研究针对一般非平凡$9j$符号无法表示为$_5F_4$超几何级数的问题，利用Dougall-Ramanujan恒等式，证明了特殊拉伸$9j$符号可改写为$_5F_4(1)$超几何级数，拓展了角动量系数的级数表达形式。

受 Ramanujan 启发的渐近展开

An asymptotic expansion inspired by Ramanujan

本文验证了 Ramanujan 笔记本中关于一个和式渐近于 ln(x)+γ 的论断在 n=2 时正确，并讨论了误差项阶数及 n=1 情况的不同正确推广。

Applications of a class of transformations of complex sequences

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 $\frac{1}π$ due to Ramanujan. We then apply Mishev's transform to provide proofs for a number of related hypergeometric identities, including a new and simplified proof for a family of series for $\frac{1}π$ previously obtained by Levrie via Fourier--Legendre theory. We generalize this result using Mishev's transform, so as to extend a result due to Guillera on a Ramanujan-like series in…

Beta函数和Ramanujan R-函数的一些新性质

Some new properties of the beta function and Ramanujan R-function

本文研究Beta函数与Ramanujan R-函数的级数表示，证明相关函数的单调性与递减性质，验证Qiu等人的猜想，并推导出若干不等式和恒等式，对特殊函数研究有参考价值。

$q$-拉马努金和吉列拉$\pi$公式的$q$-模拟

$q$-analogues of $π$-formulas due to Ramanujan and Guillera

本文提出EKHAD-规范化方法，基于$q$-WZ对证明拉马努金$\frac{1}{\pi}$公式和Guillera $\frac{1}{\pi^2}$级数的多个新$q$-模拟，并扩展超几何展开。

Gosper 的 strange series 的进一步 q-类比

A further q-analogue of Gosper's strange series

本文基于Gosper的strange series，通过Abel型求和引理提出q-类比系列评估技术，推导出与Yamaguchi不等价的q-类比恒等式，并应用该技术构造了Yamaguchiq-类比的替代证明及多个q-类比变体。

Two General Series Identities Involving Modified Bessel Functions and a Class of Arithmetical Functions

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. Two general identities are established. The first involves the modified Bessel function $K_μ(z)$, and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_μ(z)$, the Bessel functions of imaginary argument…