Related papers: A simple mnemonic to compute sums of powers

We compute the sum and the alternating sum of the reciprocals of triangular numbers using two standard methods from calculus: a telescoping series approach and a power series approach. We then extend these results to generalized…

This paper gives new explicit formulas for sums of powers of integers and their reciprocals.

Faulhaber's formula expresses the sum of the first $n$ positive integers, each raised to an integer power $p\geq 0$, as a polynomial in $n$ of degree $p+1$. Ramanujan expressed this sum for $p\in\{\frac12,\frac32,\frac52,\frac72\}$ as the…

We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers.

The Euler-Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler-Maclaurin summation formula and the Poission…

In modern usage the Bernoulli numbers and Bernoulli polynomials follow Euler's approach and are defined using generating functions. We consider the functional equation $f(x)+x^k=f(x+1)$ and show that a solution can be derived from…

Sums of the form $\sum_{N_m=q}^{n}{\cdots \sum_{N_1=q}^{N_2}{a_{(m);N_m}\cdots a_{(1);N_1}}}$ where the $a_{(k);N_k}$'s are same or distinct sequences appear quite often in mathematics. We will refer to them as recurrent sums. In this…

This paper sets the groundwork for the consideration of families of recursively defined polynomials and rational functions capable of describing the Bernoulli numbers. These families of functions arise from various recursive definitions of…

We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.

In this note, we derive an alternative recursive formula for the sums of powers of integers involving the Stirling numbers of the first kind. As a remarkable by-product, we provide a non-recursive definition of the Catalan numbers.

We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of…

Gerard and Washington proved that, for $k > -1$, the number of primes less than $x^{k+1}$ can be well approximated by summing the $k$-th powers of all primes up to $x$. We extend this result to primes in arithmetic progressions: we prove…

The distribution of the sum of r-th power of standard normal random variables is a generalization of the chi-squared distribution. In this paper, we represent the probability density function of the random variable by an one-dimensional…

We show that explicit forms for certain polynomials~$\psi^{(a)}_m(n)$ with the property \[ \psi^{(a+1)}_m(n) = \sum_{\nu=1}^n \psi_m^{(a)}(\nu) \] can be found (here, $a,m,n\in\mathbb{N}_0$). We use these polynomials as a basis to express…

A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…

In this note, we present a simple summation formula for $k$-bonacci numbers. The derivation consists in obtaining the generating function of such numbers, and noting that its evaluation at a particular value yields a formula generalizing a…

Recently, E. Samsonadze (arXiv:2411.11859v1) has given an explicit formula for the sums of powers of integers $S_k(n) = 1^k +2^k +\cdots + n^k$. In this short note, we show that Samsonadze's formula corresponds to a well-known formula for…

Due to Girard's (sometimes called Waring's) formula the sum of the $r-$th power of the zeros of every one variable polynomial of degree $N$, $P_{N}(x)$, can be given explicitly in terms of the coefficients of the monic ${\tilde P}_{N}(x)$…

We observe that the classical Faulhaber's theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression $a+b, a+2b, ..., a+nb$ is a polynomial in $na+n(n+1)b/2$.…

The aim of this paper is to prove wordlessly the sum formula of $1^{k}+2^{k}+\ldots +n^{k}$, $k\in\{1,2,3\}$.