José Luis Cereceda

Independent Researcher · None

For integer k ≥ 1, let S k (n) denote the sum of the kth powers of the first n positive integers 1 k + 2 k + · · · + n k. 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 kth row of the integer sequence A304330, which is closely related to the central factorial numbers of the sec...

In this paper, we obtain new relationships between the Cauchy polynomials, the Bernoulli polynomials, and the generalized (or weighted) Stirling numbers of both kinds. Furthermore, we derive some identities involving the Cauchy polynomials, the Daehee polynomials, and the hyperharmonic polynomials. In addition, we consider the poly-Cauchy polynomia...

Recently, Thomas and Namboothiri [1] derived a recurrence identity expressing an exponential power sum with negative powers in terms of another exponential power sum with positive powers. From this result, the authors obtained a corresponding recurrence relation for the ordinary power sums $S_k(n) = 1^k + 2^k + \cdots + n^k$. In this short note, we...

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 t...

We provide a new proof of a simple generalization of the famous identity 13+23+⋯+n3=(1+2+⋯+n)2 by making use of the hyper-sums 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 cas...

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 t...

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 $S_m^{(a)}(n) = \sum_{k} c_{mk} \psi_k^{(a)}(n)$, where the $c_{mk}$'s are the expansion coefficients and where the basis functions $\psi_m^{(a)}(n)$ fulfil the recursive property $\psi_m^{(...

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...

In this note, we consider the alternating sum of the mth powers of the first n positive integers Ωm(n)=nm−(n−1)m+⋯+(−1)n−11m. In 1989, Gessel and Viennot showed that, for even m = 2k, Ω2k(n) can be expressed as a polynomial of degree k in the triangular number Tn without constant term. Here, we offer an alternative demonstration of this result that...

Let Sk(n) denote the sum of the k-th powers of the first n positive integers. In this article, starting from two explicit formulas for Sk(n) involving the Stirling numbers of the second kind, we derive a pair of corresponding recursive formulas for Sk(n) involving the Stirling numbers of the first kind. Our derivation makes use of the orthogonality...

In this paper, we focus on the higher-order derivatives of the hyperharmonic polynomials, which are a generalization of the ordinary harmonic numbers. 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...

Recently, Karg{\i}n {\it 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 $S_k^{(0)}(n) \equiv S_k(n)$...

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 c...

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 wit...

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 + \cd...

In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(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...

For positive integers n and p, we give a short proof of a formula expressing np as a linear combination of figurate numbers with coefficients given by the numbers cp,l of (p−l)-dimensional facets of p-dimensional simplices obtained by cutting a p-dimensional cube, where l=0,1,…,p−1. This formula was formulated as Conjecture 16 in F. Marko and S. Li...

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 f...

In this note we exhibit a simple formula for the coefficients of the polynomial associated to the sums of powers of the terms of an arbitrary arithmetic progression. Our 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 c...

In this paper, we derive a formula for the sums of powers of the first $n$ positive integers 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, as...

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.

In this note we examine 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 $k$.

Recently, Marko and Litvinov conjectured that, for all positive integers $n$ and $p$, the $p$th power of $n$ admits the representation $n^p = \sum_{l=0}^{p-1} (-1)^{l} c_{p,l} F_{n}^{p-l}$, where $F_{n}^{p-l}$ is the $n$th hyper-tetrahedron number of dimension $p-l$ and $c_{p,l}$ denotes the number of $(p-l)$-dimensional facets of the $p$-dimension...

In this paper, we first focus on the sum of powers of the first n positive odd integers, Tk(n)=1k+3k+5k+⋯+(2n−1)k, and derive in an elementary way a polynomial formula for Tk(n) in terms of a specific type of generalized Stirling numbers. Then we consider the sum of powers of an arbitrary arithmetic progression and obtain the corresponding polynomi...

In this note, we study Cabello's nonlocality argument (CNA) for three-qubit systems configured in the generalized GHZ state. For this class of states, we show that CNA runs for almost all entangled ones, and that the maximum probability of success of CNA is $14\%$ (approx.), which is attained for the maximally entangled GHZ state. This maximum prob...

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, Pk(n) and Qk(n), such that Pk(n) = Qk(n) = fk(n) for n = 1, 2,… , k, where fk(1), fk(2),… , fk(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that fk(n) is given by the sum of powers of th...

Relativistic causality forbids superluminal signaling between distant observers. By exploiting the non-signaling principle, we derive the exact relationship between the chained Clauser-Horne-Shimony-Holt sum of correlations CHSH_K and the success probability P_K associated with Hardy's ladder test of nonlocality for two qubits and K+1 observables p...

Hersh recently showed that the Faulhaber polynomials, related to sums of consecutive powers, can be expressed as even or odd polynomials in the variable offset by one-half. In this article, we use generating functions to give explicit formulas for all terms in these polynomials.

In this paper, we consider Hardy’s ladder proof of non-locality for two qubits and K+1 observables per qubit, and show that the maximum success probability of Hardy’s ladder argument for non-locality allowed by generalized probabilistic theory reaches 50% irrespective of the value of K. This extends the known result for K = 1 to an arbitrary number...

The sums of powers of the first n positive integers S p ( n ) = 1 p + 2 p + …+ n p , ( p = 0, 1, 2, … )satisfy the fundamental identity (1) from which we can successively compute S 0 ( n ), S 1 ( n ), S 2 ( n ), etc. Identity (1) can easily be proved by using the binomial theorem; see e.g. [1, 2]. Several variations of (1) are also well known [3, 4...

In this note, we derive Binet's formula for the general term (Formula presented.) of the generalized tribonacci sequence. This formula gives (Formula presented.) explicitly as a function of the index n, the roots of the associated characteristic equation, and the initial terms (Formula presented.) , (Formula presented.) , and (Formula presented.)....

In this note, we deal with a generalization of the harmonic numbers proposed by Conway and Guy in their Book of Numbers, namely, the so-called hyperharmonic numbers. Our main aim is to prove, by mathematical induction, the formula defining the hyperharmonic numbers in terms of ordinary harmonic numbers. Moreover, we calculate the hyperharmonic numb...

In this paper, we consider the hypersum polynomial (Formula presented.), which is a generalization of the polynomial associated with the sums of powers of integers (Formula presented.). Using the Stolz-Cesàro lemma, we derive an explicit formula for the set of coefficients (Formula presented.) in terms of (Formula presented.) and the Bernoulli numb...

Relying on a recurrence relation for the hypersums of powers of integers put forward recently, we develop an iterative procedure which allows us to express a hypersum of arbitrary order in terms of ordinary (zeroth order) power sums. Then, we derive the coefficients of the hypersum polynomial as a function of the Bernoulli numbers and the Stirling...

In this paper we consider the hypersum polynomials P k (m) (n)=∑ r=0 k+m+1 c r,m r n r , and give an explicit formula for the coefficients c k,m r . We show that the c k,m r satisfy a generalized Akiyama-Tanigawa recurrence relation, thus extending some previous results due to Y. Inaba [J. Integer Seq. 8, No. 2, Art. 05.2.7, 6 p. (2005; Zbl 1073.11...

As an application of Faulhaber's theorem on sums of powers of integers and the associated Faulhaber polynomials, in this article we provide the solution to the following two questions: (1) when is the average of sums of powers of integers itself a sum of the first n integers raised to a power? and (2), when is the average of sums of powers of integ...

Deutsch's algorithm for two qubits (one control qubit plus one auxiliary qubit) is extended to two $d$-dimensional quantum systems or qudits for the case in which $d$ is equal to $2^n$, $n=1,2,...$ . This allows one to classify a certain oracle function by just one query, instead of the $2^{n-1}+1$ queries required by classical means. The given alg...

Recently Eibl et al. [PRL 92, 077901 (2004)] reported the experimental observation of the three-photon polarization-entangled W state. In this Comment we point out that, actually, the particular measurements involved in the experiment testing Mermin's inequality cannot be used for the verification of the existence of genuinely quantal tripartite co...

In this paper we extend Hardy's nonlocality proof for two spin-1/2 particles [PRL 71 (1993) 1665] to the case of n spin-1/2 particles configured in the generalized GHZ state. We show that, for all n \geq 3, any entangled GHZ state violates the Bell inequality associated with the Hardy experiment. This feature is important since it has been shown [P...

In a recent Brief Report, Zheng [S-B. Zheng, PRA 66, 014103 (2002)] has given a proof of nonlocality without using inequalities for three spin-1/2 particles in the nonmaximally entangled state |psi> = cos\theta |+++> + i sin\theta |-->. Here we show that Zheng's proof is not correct. Indeed it is the case that, for the experiment considered by Zhen...

In this paper, a new measure of entanglement for general pure bipartite states of two qutrits is formulated.

In this paper we show a Clauser-Horne (CH) inequality for two three-level quantum systems or qutrits, alternative to the CH inequality given by Kaszlikowski et al. [PRA 65, 032118 (2002)]. In contrast to this latter CH inequality, the new one is shown to be equivalent to the Clauser-Horne-Shimony-Holt (CHSH) inequality for two qutrits given by Coll...

In this paper we extend the ladder proof of nonlocality without inequalities for two spin-half particles given by Boschi et al (Boschi D et al 1997 Phys. Rev. Lett.79 2755) to the case in which the measurement settings of the apparatus measuring one of the particles are different from the measurement settings of the apparatus measuring the other pa...

The notions of three-particle entanglement and three-particle nonlocality are discussed in the light of Svetlichny's inequality [Phys. Rev. D 35, 3066 (1987)]. It is shown that there exist sets of measurements which can be used to prove three-particle entanglement, but which are nevertheless useless at proving three-particle nonlocality. In particu...

Elaborating on a previous work by Simon et al. [PRL 85, 1783 (2000)] we propose a realizable quantum optical single-photon experiment using standard present day technology, capable of discriminating maximally between the predictions of quantum mechanics (QM) and noncontextual hidden variable theories (NCHV). Quantum mechanics predicts a gross viola...

We consider a situation in which two parties, Alice and Bob, share a 3-qubit system coupled in an initial maximally entangled, GHZ state. By manipulating locally two of the qubits, Alice can prepare any one of the eight 3-qubit GHZ states. Thus the sending of Alice's two qubits to Bob, entails 3 bits of classical information which can be recovered...

By using an alternative, equivalent form of the CHSH inequality and making extensive use of the experimentally testable property of physical locality we determine the 64 different Bell-type inequalities (each one involving four joint probabilities) into which Hardy's nonlocality theorem can be cast. This allows one to identify all the two-qubit cor...

Elaborating on a previous work by Han et al., we give a general, basis-independent proof of the necessity of negative probability measures in order for a class of local hidden-variable (LHV) models to violate the Bell-CHSH inequality. Moreover, we obtain general solutions for LHV-induced probability measures that reproduce any consistent set of pro...

The relationship between the noncommutativity of operators and the violation of the Bell inequality is exhibited in the light of the n-particle Bell-type inequality discovered by Mermin [PRL 65, 1838 (1990)]. It is shown, in particular, that the maximal amount of violation of Mermin's inequality predicted by quantum mechanics decreases exponentiall...

Relativistic causality, namely, the impossibility of signaling at superluminal speeds, restricts the kinds of correlations which can occur between different parts of a composite physical system. Here we establish the basic restrictions which relativistic causality imposes on the joint probabilities involved in an experiment of the Einstein-Podolsky...

By selecting a certain subensemble of joint detection events in a two-particle interferometer arrangement, a formal nonlocality contradiction of the Hardy type is derived for an ensemble of particle pairs configured in the maximally entangled state. It is argued, however, that the class of experiments exhibiting this kind of contradiction does not...

In this paper the failure of Hardy’s nonlocality proof for the class of maximally entangled states is considered. A detailed analysis shows that the incompatibility of the Hardy equations for this class of states physically originates from the fact that the existence of quantum perfect correlations for the three pairs of two-valued observables (D 1...

In this paper we provide a simple proof of the fact that for a system of two spin-1/2 particles, and for a choice of observables, there is a unique state which shows Hardy-type nonlocality. Moreover, an explicit expression for the probability that an ensemble of particle pairs prepared in such a state exhibits a Hardy-type nonlocality contradiction...

In a recent article [Phys. Rev. A 54, 1793 (1996)] Krenn and Zeilinger investigated the conditional two-particle correlations for the subensemble of data obtained by selecting the results of the spin measurements by two observers 1 and 2 with respect to the result found in the corresponding measurement by a third observer. In this paper we write ou...

This paper has been withdrawn. Comment: REVTeX, 4 pages, 1 figure

Kar's recent proof showing that a maximally entangled state of two spin-1/2 particles gives the largest violation of a Bell inequality is extended to N spin-1/2 particles (with N greater than or equal to 3). In particular, it is shown that all the states yielding a direct contradiction with the assumption of local realism do generally consist of a...

Here the probability for a nonlocality contradiction of the Hardy type is made to depend explicitly on both the quantum state describing the particles and the choice of one of the observables involved. One can then set the above probability to any prescribed value within a certain interval by varying a suitable experimental parameter, with the quan...

In a recent article [Phys. Rev. A 53, R1927 (1996)] Wu et al. give a proof of nonlocality for the maximally entangled state of two particles without using inequalities. Here we point out that in order for their argument to go through it is necessary to consider a minimum total of six dimensions in Hilbert space. Indeed, the above argument is found...

In this paper we present a systematic formulation of some recent results concerning the algebraic demonstration of the two major no-hidden-variables theorems for N spin-1/2 particles. We derive explicitly the GHZ states involved and their associated eigenvalues. These eigenvalues turn out to be undefined for N=, this fact providing a new proof show...