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We extend the Faulhaber formula to the whole complex plane, obtaining an expression that fully resembles the Euler-Maclaurin summation formula, only it's exact. Thereafter, an expression for the generalized harmonic progressions valid in…

The inverse central binomial series {equation}S_k(z)=\sum_{n=1}^{\infty}\frac{n^k z^n}{\binom{2n}{n}}{equation} popularized by Ap\'ery and Lehmer is evaluated for positive integers $k$ along with the asymptotic behavior for large $k$. It is…

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…

Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite…

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, $\beta(2k+1)$, where $k$ is any nonnegative integer. Our approach relies on some properties of the Euler numbers and polynomials, and uses…

We develop ladders that reduce $\zeta(n):=\sum_{k>0}k^{-n}$, for $n=3,5,7,9,11$, and $\beta(n):=\sum_{k\ge0}(-1)^k(2k+1)^{-n}$, for $n=2,4,6$, to convergent polylogarithms and products of powers of $\pi$ and $\log2$. Rapid computability…

We introduce a rigorous arithmetic--spectral construction associating planar geometric objects with additive prime factor statistics. Let $\mathrm{sopfr}(n)$ denote the sum of prime factors of $n$, counted with multiplicity, and define the…

We derive a compact determinant formula for calculating and factorizing the hypersum polynomials S^{(L)}_k(N) \equiv \sum_{n_1=1}^N ...\sum_{n_{L+1}=1}^{n_{L}}(n_{L+1})^k expressed in the variable N(N+L+1)

For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take…

In this work we continue the investigation about the interplay between hypergeometric functions and Fourier-Legendre ($\textrm{FL}$) series expansions. In the section "Hypergeometric series related to $\pi,\pi^2$ and the lemniscate…

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series admitting meromorphic continuation to the complex plane. Assume we know the location of the poles of $A(s)$ with $|\Im s| \leq T$, and their residues, for some large constant $T$. It is…

We consider several problems at or beyond endpoint in harmonic analysis. The solutions of these problems are related to the estimates of some classes of sublinear operators. To do this, we introduce some new functions spaces…

We present a method to sum Borel- and Gevrey-summable asymptotic series by matching the series to be summed with a linear combination of asymptotic series of known functions that themselves are scaled versions of a single, appropriate, but…

Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for…

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…

Let $k$ be a positive real number, and let $M_k(q)$ be the sum of $|L(\tfrac12,\chi)|^{2k}$ over all non-principal characters to a given modulus $q$. We prove that $M_k(q)\ll_k \phi(q)(\log q)^{k^2}$ whenever $k$ is the reciprocal $n^{-1}$…

In this paper, two new series for the logarithm of the $\Gamma$-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling numbers of the first kind and have the property…

We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of…

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families…

In 2011, W. Lang derived a novel, explicit formula for the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$ involving simultaneously the Stirling numbers of the first and second kind. In this note, we first recall and then…