arXiv subjects

José L. Cereceda

Publications and source records attributed to José L. Cereceda.

At least 19 recordsLinked to original sources

arXiv2026

The most concise recurrence formula for the sums of integer powers

For integers $n,k \geq 1$, let $S_k(n)$ denote the power sum $1^k +2^k + \cdots + n^k$. In this note, we first recall the minimal recurrence relation connecting $S_k(n)$ and $S_{k-1}(n)$ established by Abramovich (1973). We then discuss an old algorithm to determine the coefficients of the power sum polynomial $S_k(n)$ in terms of the coefficients of $S_{k-1}(n)$ (see, e.g., Bloom (1993) and Owens (1992)). Moreover, we bring to light an explicit relationship between $S_k(n)$ and $S_{k+1}(n)$ put forward by Budin and Cantor (1972). We conclude that these procedures (including the integration formula expressing $S_k(n)$ in terms of $S_{k-1}(n)$) all constitute equivalent methods to determine $S_k(n)$ starting from $S_{k-1}(n)$. In addition, as a by-product, we provide a determinantal formula for the Bernoulli numbers involving the binomial coefficients.

math.HO↗

arXiv2025

Formulas involving Cauchy polynomials, Bernoulli polynomials, and generalized Stirling numbers

In this paper, we derive novel formulas and identities connecting Cauchy numbers and polynomials with both ordinary and generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler polynomials, $r$-Whitney numbers, and hyperharmonic polynomials, as well as Bernoulli numbers and polynomials. We also provide formulas for the higher-order derivatives of Cauchy polynomials and obtain corresponding formulas and identities for poly-Cauchy polynomials. Furthermore, we introduce a multiparameter framework for poly-Cauchy polynomials, unifying earlier generalizations like shifted poly-Cauchy numbers and polynomials with a $q$ parameter.

math.CO↗

arXiv2025

A remark on an explicit formula for the sums of powers of integers

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 $S_k(n)$ involving the Stirling numbers of the second kind.

math.GM↗

arXiv2025

Sums of powers of integers and the sequence A304330

For integer $k \geq 1$, let $S_k(n)$ denote the sum of the $k$th powers of the first $n$ positive integers. In this paper, we derive a new formula expressing $2^{2k}$ times $S_{2k}(n)$ as a sum of $k$ terms involving the numbers in the $k$th row of the integer sequence A304330, which is closely related to the central factorial numbers with even indices of the second kind. Furthermore, we provide an alternative proof of Knuth's formula for $S_{2k}(n)$ and show that it can equally be expressed in terms of A304330. Moreover, we obtain corresponding formulas for $2^{2k-1}S_{2k-1}(n)$ and determine the Faulhaber form of both $S_{2k}(n)$ and $S_{2k+1}(n)$ in terms of A304330 and the Legendre-Stirling numbers of the first kind.

math.GM↗

arXiv2023

A refinement of Lang's formula for the sum of powers of integers

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 slightly refine Lang's formula for $S_k(n)$. As it turns out, the refined Lang's formula constitutes a special case of a well-known relationship between the power sums, the elementary symmetric functions, and the complete homogeneous symmetric functions. In addition, we provide several applications of this general relationship.

math.NT↗

arXiv2023

Sums of powers of integers via differentiation

For integer $k \geq 0$, let $S_k$ denote the sum of the $k$th powers of the first $n$ positive integers $1^k + 2^k + \cdots + n^k$. For any given $k$, the power sum $S_k$ can in principle be determined by differentiating $k$ times (with respect to $x$) the associated exponential generating function $\sum_{k=0}^{\infty}S_k x^k/k!$, and then taking the limit of the resulting differentiated function as $x$ approaches $0$. In this paper, we exploit this method to establish a couple of seemingly novel recurrence relations, one of them involving the even-indexed power sums $S_2, S_4,\ldots, S_{2k}$, and the other the odd-indexed power sums $S_{1}, S_3, \ldots, S_{2k-1}$, with both recurrence relations depending explicitly on the parameter $N = n + \frac{1}{2}$. From this, we obtain a determinantal formula of order $k$ which yields $S_{2k}$ [$S_{2k-1}$] in the Faulhaber form, that is, as an odd [even] polynomial in $N$. As a byproduct, we discover a new determinantal formula for the Bernoulli number $B_{2k}$. Furthermore, we show that $S_{2k}$ and $S_{2k-1}$ can be obtained by taking the corresponding higher order derivatives of the Chebyshev polynomials of the second kind.

math.NT↗

arXiv2022

Sums of powers of integers and generalized Stirling numbers of the second kind

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 involves the $r$-Stirling numbers of the second kind $\genfrac{\{}{\}}{0pt}{}{k}{j}_r$, $j=0,1,\ldots,k$, while the other involves their duals $\genfrac{\{}{\}}{0pt}{}{k}{j}_{-r}$, with both families of formulas being indexed by the non-negative integer $r$. As a by-product, we obtain three additional formulas for $S_k(n)$ involving the numbers $\genfrac{\{}{\}}{0pt}{}{k}{j}_{n+m}$, $\genfrac{\{}{\}}{0pt}{}{k}{j}_{n-m}$ (where $m$ is any given non-negative integer), and $\genfrac{\{}{\}}{0pt}{}{k}{j}_{k-j}$, respectively. Moreover, we provide a formula for the Bernoulli polynomials $B_k(x-1)$ in terms of $\genfrac{\{}{\}}{0pt}{}{k}{j}_{x}$ and the harmonic numbers.

math.NT↗

arXiv2022

A note on another approach on power sums

In this note, we first review the novel approach to power sums put forward recently by Muschielok in arXiv:2207.01935v1, which can be summarized by the formula $S_m^{(a)}(n) = \sum_{k} c_{mk} ψ_k^{(a)}(n)$, where the $c_{mk}$'s are the expansion coefficients and where the basis functions $ψ_m^{(a)}(n)$ fulfil the recursive property $ψ_m^{(a+1)}(n)= \sum_{i=1}^n ψ_m^{(a)}(i)$. Then, we point out a number of supplementary facts concerning the said approach not contemplated explicitly in Muschielok's paper. In particular, we show that, for any given $m$, the values of the $c_{mk}$'s can be obtained by inverting a matrix involving only binomial coefficients. This may be compared with the original approach of Muschielok, where the values of the $c_{mk}$'s can be obtained by inverting a lower triangular matrix involving the Stirling numbers of the first kind. Also, we make a conjecture about the functional form of the coefficients $c_{m\, m-k}$.

math.NT↗

arXiv2022

A determinantal formula for the hyper-sums of powers of integers

For non-negative integers $r$ and $m$, let $S_m^{(r)}(n)$ denote the $r$-fold summation (or hyper-sum) over the first $n$ positive integers to the $m$th powers, with the initial condition $S_m^{(0)}(n) =n^m$. In this paper, we derive a new determinantal formula for $S_m^{(r)}(n)$. Specifically, we show that, for all integers $r\geq 0$ and $m \geq 1$, $S_m^{(r)}(n)$ is proportional to $S_1^{(r)}(n)$ times the determinant of a lower Hessenberg matrix of order $m-1$ involving the Bernoulli numbers and the variable $N_r = n + \frac{r}{2}$. Furthermore, whenever $r\geq 1$, evaluating this determinant gives us $S_m^{(r)}(n)$ as $S_1^{(r)}(n)$ times an even or odd polynomial in $N_r$ of degree $m-1$, depending on whether $m$ is odd or even.

math.NT↗

arXiv2022

A note on the coefficients of power sums of arithmetic progressions

In this note we show a simple formula for the coefficients of the polynomial associated with the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial coefficients and binomial powers. Arguably, this is the simplest formula that can probably be found for the said coefficients. Furthermore, we give an explicit formula for the Bernoulli polynomials involving the Stirling numbers of the first and second kind.

math.NT↗

arXiv2022

Bernoulli and Faulhaber

In a recent work, Zielinski used Faulhaber's formula to explain why the odd Bernoulli numbers are equal to zero. Here, we assume that the odd Bernoulli numbers are equal to zero to explain Faulhaber's formula.

math.NT↗

arXiv2022

A note on the hyper-sums of powers of integers, hyperharmonic polynomials and r-Stirling numbers of the first kind

Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1} S_{k+i}(n), \end{equation*} where $S_k^{(0)}(n) \equiv S_k(n)$ is the ordinary power sum $1^k + 2^k + \cdots + n^k$. In this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly as a polynomial in $n$ of degree $m-i$. We find out the connection between this polynomial and the so-called $r$-Stirling polynomials of the first kind. Furthermore, we determine the hyperharmonic polynomials and their successive derivatives in terms of the $r$-Stirling polynomials of the first kind, and show the relationship between the (exponential) complete Bell polynomials and the $r$-Stirling numbers of the first kind. Finally, we derive some identities involving the Bernoulli numbers and polynomials, the $r$-Stirling numbers of the first kind, the Stirling numbers of both kinds, and the harmonic numbers.

math.NT↗

arXiv2021

Daehee, hyperharmonic, and power sums polynomials

In this paper we consider the Daehee numbers and polynomials of the first and second kind, and give several explicit representations for them. In particular, we express the Daehee polynomials as the derivative of a generalized binomial coefficient. This is done by performing the Stirling transform of the power sum polynomial $S_k(x)$ associated with the sum of $k$th powers of the first $n$ positive integers $S_k(n) = 1^k + 2^k + \cdots + n^k$. Furthermore, we show the relationship between the Daehee polynomials and the hyperharmonic polynomials. This allows us to also express the hyperharmonic polynomials as the derivative of a generalized binomial coefficient. In addition, we reformulate and generalize a number of identities involving the Stirling, Bernoulli, and harmonic numbers, in terms of the Daehee and hyperharmonic polynomials. Finally, in the light of a recent result of Kargin et al., we discuss a further extension of the obtained identities in terms of the $r$-Stirling numbers.

math.NT↗

arXiv2021

Binary quadratic forms and sums of powers of integers

In this methodological paper, we first review the classic cubic Diophantine equation $a^3 + b^3 + c^3 = d^3$, and consider the specific class of solutions $q_1^3 + q_2^3 + q_3^3 = q_4^3$ with each $q_i$ being a binary quadratic form. Next we turn our attention to the familiar sums of powers of the first $n$ positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and express the squares $S_k^2$, $S_m^2$, and the product $S_k S_m$ as a linear combination of power sums. These expressions, along with the above quadratic-form solution for the cubic equation, allows one to generate an infinite number of relations of the form $Q_1^3 + Q_2^3 + Q_3^3 = Q_4^3$, with each $Q_i$ being a linear combination of power sums. Also, we briefly consider the quadratic Diophantine equations $a^2 + b^2 + c^2 = d^2$ and $a^2 + b^2 = c^2$, and give a family of corresponding solutions $Q_1^2 + Q_2^2 + Q_3^2 = Q_4^2$ and $Q_1^2 + Q_2^2 = Q_3^2$ in terms of sums of powers of integers.

math.NT↗

arXiv2021

Sums of powers of integers and hyperharmonic numbers

In this paper, we derive a formula for the sums of powers of the first $n$ positive integers, $S_k(n)$, that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Moreover, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for $S_k(n)$ to negative values of $n$.

math.NT↗

arXiv2021

Figurate numbers and sums of powers of integers

Recently, Marko and Litvinov (ML) conjectured that, for all positive integers $n$ and $p$, the $p$-th power of $n$ admits the representation $n^p = \sum_{\ell =0}^{p-1} (-1)^{l} c_{p,\ell} F_{n}^{p-\ell}$, where $F_{n}^{p-\ell}$ is the $n$-th hyper-tetrahedron number of dimension $p-\ell$ and $c_{p,\ell}$ denotes the number of $(p -\ell)$-dimensional facets formed by cutting the $p$-dimensional cube $0 \leq x_1, x_2, \ldots, x_p \leq n-1$. In this paper we show that the ML conjecture is true for every natural number $p$. Our proof relies on the fact that the validity of the ML conjecture necessarily implies that $c_{p,\ell} = (p-\ell)! S(p, p-\ell)$, where $S(p,p-\ell)$ are the Stirling numbers of the second kind. Furthermore, we provide a number of equivalent formulas expressing the sum of powers $\sum_{i=1}^{n} i^p$ as a linear combination of figurate numbers.

math.NT↗

arXiv2021

An algorithm for the Faulhaber polynomials

Let $S_p(n)$ denote the sum of $p$th powers of the first $n$ positive integers $1^p + 2^p + \cdots + n^p$. In this paper, first we express $S_p(n)$ in the so-called Faulhaber form, namely, as an even or odd polynomial in $(n + 1/2)$, according as $p$ is odd or even. Then, using the relation $S_p(n) - S_p(n-1) = n^p$, we derive a recursive formula for the associated Faulhaber coefficients. Applying Cramer's rule to the corresponding system of equations, we obtain an explicit determinant formula for the said coefficients. Furthermore, we show how to convert the (even or odd) Faulhaber polynomials in $(n+ 1/2)$ into polynomials in $S_1(n)$ for any arbitrary $p$, and vice versa.

math.NT↗

arXiv2020

More on Faulhaber's theorem for sums of powers of integers

In this note we consider the theorem established in arXiv:1912.07171 concerning the sums of powers of the first $n$ positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and show that it can be used to demonstrate the classical theorem of Faulhaber for both cases of odd and even $k$.

math.NT↗