Related papers: Two Definite Integrals That Are Definitely (and Su…
The definite integrals $ \int_{-1}^1x[P_\nu(x)]^4\,\mathrm{d} x$ and $ \int_{0}^1x[P_\nu(x)]^2\{[P_\nu(x)]^2-[P_\nu(-x)]^2\}\,\mathrm{d} x$ are evaluated in closed form, where $ P_\nu$ stands for the Legendre function of degree $…
We give an identity which is conjectured and proved by using an implementation in Multi-WZ.
In this paper, an integral identity for twice differentiable functions is generalized. Then, by using convexity of |f''| or q-th power of |f''| and with the aid of power mean and Holder's inequalities we achieved some new results. We also…
A new inequality, $(x)^{p}+(1-x)^{\frac{1}{p}}\leq1$ for $p \geq 1$ and $\frac{1}{2} \geq x \geq 0$ is found and proved. The inequality looks elegant as it integrates two number pairs ($x$ and $1-x$, $p$ and $\frac{1}{p}$) whose summation…
In a Hom-Malcev algebra an identity, equivalent to the Hom-Malcev identity, is found.
Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of…
Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the…
In this note, we find a combinatorial identity which is closely related to the multi-dimensional integral $\gamma_{m}$ in the study of divisor functions. As an application, we determine the finite dual of the group algebra of infinite…
This is a journey through integrals of involutions and surprising consequences of the Lagrange inversion theorem. On the way, we meet unexpected logarithmic identities, hypergeometric functions with a linear regime and other mysterious…
We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.
A perfect cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The existence of such cuboids is neither proved, nor disproved. A rational perfect cuboid is a natural…
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
In this article, we explore a series of elementary yet insightful results involving integrals related to Gaussian sums. Using techniques rooted in classical calculus, we derive several identities and evaluate nontrivial definite integrals…
We describe efficient algorithms to search for cases in which binomial coefficients are equal or almost equal, give a conjecturally complete list of all cases where two binomial coefficients differ by 1, and give some identities for…
In investigating the properties of a certain class of homogeneous polynomials, we discovered an identity satisfied by their coefficients which involves simple 2F1 Gauss hypergeometric functions. This result appears to be new and we supply a…
Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of…
We give a new proof of Chan's identity involving the cubic partition function and we also give a new identity for the cubic partition function which is analogues to the Zuckerman's identity for the ordinary partition function.
In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is…
We discuss an unusual phenomenon in (integral) positive ternary quadratic forms. We also describe an interesting pairing of genera of ternary forms.
A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to new identities for binomial and q-binomial coefficients.