Alin Bostan, Fernando Chamizo, Mikael P. Sundqvist
We give three elementary proofs of a nice equality of definite integrals, which arises from the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a generalization together with an equally elementary proof and discuss some consequences.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give a new proof of a theorem of Zudilin that equates a very-well-poised hypergeometric series and a particular multiple integral. This integral generalizes integrals of Vasilenko and Vasilyev which were proposed as tools in the study of the arithmetic behaviour of values of the Riemann zeta function at integers. Our proof is based on limiting cases of a basic hypergeometric identity of Andrews.
Among the several proofs known for $\sum_{n=1}^\infty{1/n^2} = {π^2/6}$, the one by Beukers, Calabi, and Kolk involves the evaluation of $\,\int_0^1 {\int_0^1{1/(1-x^2 y^2) \, dx} \, dy}$. It starts by showing that this double integral is equivalent to $\frac34 \sum_{n=1}^\infty{1/n^2}$, and then a non-trivial \emph{trigonometric} change of variables is applied which transforms that integral into $\,{\int \int}_T \: 1 \; du \, dv$, where $T$ is a triangular domain whose area is simply ${π^2/8}$. Here in this note, I introduce a hyperbolic version of this change of variables and, by applying it…
A family of general integral identities is derived and several applications of physical interest are presented
The Hilbert transform $H$ satisfies the Bedrosian identity $H(fg)=fHg$ whenever the supports of the Fourier transforms of $f,g\in L^2(R)$ are respectively contained in $A=[-a,b]$ and $B=R\setminus(-b,a)$, $0\le a,b\le+\infty$. Attracted by this interesting result arising from the time-frequency analysis, we investigate the existence of such an identity for a general bounded singular integral operator on $L^2(R^d)$ and for general support sets $A$ and $B$. A geometric characterization of the support sets for the existence of the Bedrosian identity is established. Moreover, the support sets for …
In this article we prove a new elliptic hypergeometric integral identity. It previously appeared (as a conjecture) in articles by Rains, and Spiridonov and Vartanov. Moreover it gives a different proof of an identity in another article by Rains. We also give some basic hypergeometric and classical limits of this identity. The classical limit gives identities (some known, some new) between generalizations of the Selberg integral.
In this paper, a general integral identity for twice differentiable functions is derived. By using of this identity, the author establish some new estimates on Hermite-Hadamard type and Simpson type inequalities for s-convex via Riemann Liouville fractional 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.
In this paper, a general integral identity for a twice differentiable functions is derived. By using of this identity, the author establishes some new Hermite-Hadamard type and Simpson type inequalities for differentiable (α,m)-convex functions via Riemann Liouville fractional integral.
New identity similar to an identity of [13] for fractional integrals have been defined. Then making use of this identity, some new Ostrowski type inequalities for Riemann-Liouville fractional integral have been developed. Our results have some relationships with the results of Alomari et. al., proved in [13] [published in. Appl. Math. Lett. 23 (2010) 1071-1076] and the analysis used in the proofs is simple.
In this paper, a new identity for fractional integrals is established. Then by making use of the established identity, some new Ostrowski type inequalities for harmonically s-convex functions via Riemann--Liouville fractional integral are established.
New identity for fractional integrals have been defined. By using of this identity, we obtained new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for s-convex, quasi-convex, m-convex functions via Riemann Liouville fractional integral.
In this paper, the authors gives a new identity for Hadamard fractional integrals. By using of this identity, the authors obtains new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for (alpha?;m)-GA-convex function via Hadamard fractional integral.
A singular integral identity for surface measure
本文证明了一类 Riesz 型核在可整流集合上的积分为常数，由此导出曲面测度公式，并给出几何解释及凸区域刻画，加强了 Steinerberger 的近期不等式。
In this paper, it is a fuction that is a GA-convex differentiable for a new identity. As a result of this identity, some new and general integral inequalities for differentiable GA-convex functions are obtained.
In this paper, the author introduces the concept of the quasi-geometrically convex and defines a new identity for fractional integrals. By using of this identity, author obtains new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for GA-s-convex, quasi-geometrically convex and (s,m)-GA-convex functions via Riemann Liouville fractional integral.
We consider a new identity involving integrals and sums of Bessel functions. The identity provides new ways to evaluate integrals of products of two Bessel functions. The identity is remarkably simple and powerful since the summand and integrand are of exactly the same form and the sum converges to the integral relatively fast for most cases. A proof and numerical examples of the identity are discussed.
In this paper, new identity for fractional integrals have been defined. By using of this identity, we obtained new general inequalities containing all of Hadamard, Ostrowski and Simpson type inequalities for for functions whose derivatives in absolute value at certain power are (α,m)-convex via Riemann Liouville fractional integral.
In this paper, it is a fuction that is a harmonically-convex differentiable for a new identity. As a result of this identity some new and general inequalities for differentiable harmonically-convex functions are obtained.
For $\boldsymbol{\large λ} = (λ_1,\ldots,λ_d)$ with $λ_i > 0$, it is proved that \begin{equation*} \prod_{i=1}^d \frac{ 1}{(1- r x_i)^{λ_i}} = \frac{Γ(|\boldsymbol{\large λ}|)}{\prod_{i=1}^{d} Γ(λ_i)} \int_{\mathcal{T}^d} \frac{1}{ (1- r \langle x, u \rangle)^{|\boldsymbol{\large λ}|}} \prod_{i=1}^d u_i^{λ_i-1} du, \end{equation*} where $\mathcal{T}^d$ is the simplex in homogeneous coordinates of $\mathbb{R}^d$, from which a new integral relation for Gegenbuer polynomials of different indexes is deduced. The latter result is used to derive closed formulas for reproducing kernels of orthogonal …
In this paper, a new identity for convex functions is derived. A consequence of the identity is that we can derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are convex. Some applications to special means of real numbers are also given.
While investigating the properties of a galaxy model used in Stellar Dynamics, a curious integral identity was discovered. For a special value of a parameter, the identity reduces to a definite integral with a very simple symbolic value; but, quite surprisingly, all the consulted tables of integrals, and computer algebra systems, do not seem aware of this result. Here I show that this result is a special case ($n=0$ and $z=1$) of the following identity (established by elementary methods): $$ I_n(z)\equiv\int_0^1{{\rm K}(k) k\over (z+k^2)^{n+3/2}}dk = {(-2)^n\over (2n+1)!!} {d^n\over dz^n} {{\r…
In this paper, we define a new identity for twice differentiable mappings and obtained some new estimates on the generalization of Hadamard's and Simson's type inequalities for quasi-geometrically convex mappings using of this identity.
A Different Demonstration for Integral Identity Across Distinct Time Scales
针对时间尺度积分中Δ-积分转实数域经典积分能否仅用黎曼和实现的问题，本文给出肯定证明，即可积函数下该转化无需勒贝格积分理论支撑。
In this paper we provide some relationships between Catalan's constant and the $_3{\rm F}_2$ and $_4{\rm F}_3$ hypergeometric functions, deriving them from some parametric integrals. In particular, using the complete elliptic integral of the first kind, we found an alternative proof of a result of Ramanujan for $_3{\rm F}_2$, a second identity related to $_4{\rm F}_3$ and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.
We generalize aspects of Fourier Analysis from intervals on $\mathbb{R}$ to bounded and measurable subsets of $\mathbb{R}^n$. In doing so, we obtain a few interesting results. The first is a new proof of the famous Integral Cauchy-Schwarz Inequality. The second is a restatement of Parseval's Identity that doubles as a representation of integrating bounded and measurable functions over bounded and measurable subsets of $\mathbb{R}^n$. Finally, we apply these first two results to develop some sufficient criteria for additional integral inequalities that are elementary in nature.
We prove an integral representation and a power series expansion for the function $\det(A)^{-1}$ in a small neighborhood of the identity matrix. Both results are closely linked to the formula for the change of coordinates of the Dirac delta distribution in $\mathbb{R}^m$.
In this paper, we prove two related central binomial series identities: $B(4)=\sum_{n \geq 0} \frac{\binom{2n}n}{2^{4n}(2n+1)^3}=\frac{7 π^3}{216}$ and $C(4)=\sum_{n \in \mathbb{N}} \frac{1}{n^4 \binom{2n}n}=\frac{17 π^4}{3240}.$ Both series resist all the standard approaches used to evaluate other well-known series. To prove the first series identity, we will evaluate a log-sine integral that is equal to $B(4).$ Evaluating this log-sine integral will lead us to computing closed forms of polylogarithms evaluated at certain complex exponentials. To prove the second identity, we will evaluate a …
Legendre's relation for the complete elliptic integrals of the first and second kinds is generalized. The proof depends on an application of the generalized trigonometric functions and is alternative to the proof for Elliott's identity.
A determinant identity for the sum of contour integral matrices
本文推导了两个由轮廓积分定义的n×n矩阵A和B之和的行列式恒等式，将其表示为Fredholm行列式，扩展了最近在文献中的结果。
We give a direct evaluation of a curious integral identity, which follows from the work of Ismail and Valent on the Nevanlinna parametrization of solutions to a certain indeterminate moment problem.
By making use of the identity obtained by Sarikaya, some new Hermite-Hadamard type inequalities for h-convex functions on the co-ordinates via fractional integrals are established. Our results have some relationships with the results of Sarikaya([16])
We prove an elementary multilinear identity for the Fourier extension operator on $\mathbb{R}^n$, generalising to higher dimensions the classical bilinear extension identity in the plane. In the particular case of the extension operator associated with the paraboloid, this provides a higher dimensional extension of a well-known identity of Ozawa and Tsutsumi for solutions to the free time-dependent Schrödinger equation. We conclude with a similar treatment of more general oscillatory integral operators whose phase functions collectively satisfy a natural multilinear transversality condition. T…
We give an alternative, combinatorial/geometrical evaluation of a class of improper sinc integrals studied by the Borweins. A probabilistic interpretation is also noted and used to shed light on a related combinatorial identity.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。