Related papers: Identities for the shifted harmonic numbers and bi…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
We construct the generating function for products of inverse central binomial coefficients with harmonic numbers.
The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin…
We explore new types of binomial sums with Fibonacci and Lucas numbers. The binomial coefficients under consideration are $\frac{n}{n+k}\binom{n+k}{n-k}$ and $\frac{k}{n+k}\binom{n+k}{n-k}$. The identities are derived by relating the…
We evaluate some new three parameter families of finite reciprocal sums involving Horadam numbers. We will also be able to state the results for the infinite sums. Some Fibonacci and Lucas sums will be presented as examples.
A systematic study is performed on the finite harmonic sums up to level four. These sums form the general basis for the Mellin transforms of all individual functions $f_i(x)$ of the momentum fraction $x$ emerging in the quantities of…
We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate…
When $a_n$ is the $n$'th coefficient of some holomorphic cusp form, we prove a variety of Omega results for the twisted sum $\sum_{n\le x}a_ne^{2\pi in\alpha}$ and discuss their applications to the Ramanujan $\tau$-function and sums of…
Series representations consisting of spherical harmonics are obtained for characteristic exponents and probability density functions of multivariate stable distributions under various conditions. A esult potentially applicable in a…
In this paper, we give a purely cohomological interpretation of the extension problem for associative algebras; that is the problem of extending an associative algebra by another associative algebra. We then give a similar interpretation of…
In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta…
We define reflective numbers and their iterative summations. We provide classification of reflective numbers based on their iterative cyclical limits.
We develop closed form expressions for various finite binomial Fibonacci and Lucas sums depending on the modulo 5 nature of the upper summation limit. Our expressions are inferred from some trigonometric identities.
Infinite series sum_{n=1}^infty {(alpha/2)_n / (n n!)}_1F_1(-n, gamma, x^2), where_1F_1(-n, gamma, x^2)={n!_(gamma)_n}L_n^(gamma-1)(x^2), appear in the first-order perturbation correction for the wavefunction of the generalized spiked…
We obtain several estimates for bilinear form with exponential sums with binomials $mx^k + nx^\ell$. In particular we show the existence of nontrivial cancellations between such sums when the coefficients $m$ and $n$ vary over rather sparse…
Using a property of the q-shifted factorial, an identity for q-binomial coefficients is proved, which is used to derive the formulas for the q-binomial coefficient for negative arguments. The result is in agreement with an earlier paper…
We refine and extend quantitative bounds, on the fraction of nonnegative polynomials that are sums of squares, to the multihomogenous case.
We give a new identity involving Bernoulli polynomials and combinatorial numbers. This provides, in particular, a Faulhaber-like formula for sums of the form $1^m (n-1)^m + 2^m (n-2)^m + \cdots + (n-1)^m 1^m$ for positive integers $m$ and…
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as…
We briefly describe some well-known means and their properties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, deriving from the harmonic mean, motivate the definition of a new kind of mean that we…