Nth Harmonic Number Research Articles

Lucky Cars and the Quicksort Algorithm

Quicksort is a classical divide-and-conquer sorting algorithm. It is a comparison sort that makes an average of 2 ( n + 1 ) H n − 4 n comparisons on an array of size n ordered uniformly at random, where H n : = ∑ i = 1 n 1 i is the nth harmonic number. Therefore it makes n ! [ 2 ( n + 1 ) H n − 4 n ] comparisons to sort all possible orderings of the array. In this article, we prove that this count also enumerates the parking preference lists of n cars parking on a one-way street with n parking spots resulting in exactly n − 1 lucky cars (i.e., cars that park in their preferred spot). For n ≥ 2 , both counts satisfy the second order recurrence relation f n = 2 n f n − 1 − n ( n − 1 ) f n − 2 + 2 ( n − 1 ) ! with f 0 = f 1 = 0 .

Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion

Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers. In this paper, we rewrite Ramanujan’s harmonic number expansion into a similar form of Euler’s asymptotic expansion as n approaches infinity: Hn∼γ+c0(h)log(q+h)−∑k=1∞ck(h)k·(q+h)k, where q=n(n+1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and ck(x)=∑j=0kkjcjxk−j, with ck is the negative of the median Bernoulli numbers. Then, 2cn=∑k=0nnkBn+k, where Bn is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the nth harmonic number. For example, Hn∼γ+12log(q+13)−1180(q+13)2∑j=0∞bj(r)(q+13)j1/r as n approaches infinity, where bj(r) can be determined.

On the Ramanujan Harmonic Number Expansion

In this paper, we give a recursive relation for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number, without the Bernoulli numbers and polynomials.

Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.

Ramanujan’s formula for the harmonic number

In this paper, we investigate certain asymptotic series used by Hirschhorn to prove an asymptotic expansion of Ramanujan for the nth harmonic number. We give a general form of these series with a recursive formula for its coefficients. By using the result obtained, we present a formula for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number. We also give a recurrence relation for determining the coefficients aj(r) such thatHn:=∑k=1n1k∼12ln(2m)+γ+112m(∑j=0∞aj(r)mj)1/ras n → ∞, where m=n(n+1)/2 is the nth triangular number and γ is the Euler–Mascheroni constant. In particular, for r=1, we obtain Ramanujan’s expansion for the harmonic number.

Ramanujan’s Harmonic Number Expansion and Two Identities for Bernoulli Numbers

By the Lagrange–Burmann formula, we provide a new explicit formula for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number. Based on the new explicit formula, we obtain two interesting identities for the Bernoulli numbers.

Accurate double inequalities for generalized harmonic numbers

For n∈N and p∈R the nth harmonic number of order pH(n,p):=∑k=1n1kpis expressed in the formH(n,p)=H˜q(m,n,p)+Rq(m,n,p)where m,q∈N are parameters controlling the magnitude of the error term. The function H˜q(m,n,p) consists of m+2q+1 simple summands and the remainder Rq(m, n, p) is estimated, for p ≥ 0, as0≤(−1)q+1Rq(m,n,p)<1π(1−2·4−q)(p2+q−1)πm)2q−1·1mp.Similar result is obtained also for p < 0 and for real zeta function (n=∞,p > 1) as well.

PROOF OF A CONGRUENCE FOR HARMONIC NUMBERS CONJECTURED BY Z.-W. SUN

For a positive integer n let [Formula: see text] be the nth harmonic number. In this paper, we prove that for any prime p ≥ 7, [Formula: see text] which confirms the conjecture recently proposed by Z.-W. Sun. Furthermore, we also prove two similar congruences modulo p2.

Sharp inequalities for the harmonic numbers

Let Hn be the nth harmonic number, and let γ be the Euler-Mascheroni constant. We prove that for all integers n ≥ 1, the double-inequality ... holds with the best possible constants ... We also establish inequality for the Euler-Mascheroni constant.

Series Involving Products of Two Harmonic Numbers

SummaryThe paper is about the calculation of the nonlinear harmonic series k), where k is a nonnegative integer and Hn denotes the nth harmonic number. We show that, when k ≥ 1, the sum Sk can be written as a rational linear combination of ζ(2), ζ(3), and the first k harmonic numbers.

On Turán's inequality for Legendre polynomials

Let Δ n ( x ) = P n ( x ) 2 - P n - 1 ( x ) P n + 1 ( x ) , where P n is the Legendre polynomial of degree n. A classical result of Turán states that Δ n ( x ) ⩾ 0 for x ∈ [ - 1 , 1 ] and n = 1 , 2 , 3 , … . Recently, Constantinescu improved this result. He established h n n ( n + 1 ) ( 1 - x 2 ) ⩽ Δ n ( x ) ( - 1 ⩽ x ⩽ 1 ; n = 1 , 2 , 3 , … ) , where h n denotes the nth harmonic number. We present the following refinement. Let n ⩾ 1 be an integer. Then we have for all x ∈ [ - 1 , 1 ] α n ( 1 - x 2 ) ⩽ Δ n ( x ) with the best possible factor α n = μ [ n / 2 ] μ [ ( n + 1 ) / 2 ] . Here, μ n = 2 - 2 n 2 n n is the normalized binomial mid-coefficient.

Matrix games in the multicast networks: maximum information flows with network switching

Network coding for achieving the maximum information flow in the multicast networks has been proposed by Ahlswede, Cai, Li, and Yeung. They have demonstrated that the conventional network switching, without resort to network coding, is in general not able to achieve the optimum information flow that has been promised by network coding. A basic problem arising here is that, for a given multicast network, what is the switching gap of the network defined as the ratio of the maximum information flow in the multicast network with network coding to that only with network switching. In the paper, by considering network switching as a special form of network coding, we make a complete theoretical and computational determination of the achievable information rate region for multisource multicast network switching. The multicast networks are allowed to be cyclic or acyclic with links having arbitrary positive integer or real-valued capacity. Network switching is essentially a problem of multicast-route packing in the multicast networks. Based on this, we use the theory of games to formulate the network switching as a matrix game between the first player of links and the second player of multicast routes in the multicast networks. We prove that the maximum achievable information rate at each probabilistic direction in the information rate region is the reciprocal of the value of the corresponding game. Consequently, the maximum achievable information rate can be computed in a simple way by applying the existing theory and algorithms for the computation of the value of a matrix game, especially for such multicast networks with links all having unit capacity as the Ahlswede-Cai-Li-Yeung multicast networks. For multicast networks with links having arbitrary positive real-valued capacity, by using convex optimization, we develop a simple and efficient iterative algorithm to find the maximum achievable information rates for multisource multicast network switching. For single-source multicast networks whose links have arbitrary positive real-valued capacity, we present two max-flow min-cut theorems. The maximum information flow for single-source network switching is the minimum capacity among all soft link-cuts of the multicast network, while the maximum information flow for single-source network coding is the minimum capacity among all hard link-cuts. Consequently, by applying the theory of approximation algorithms for the set-covering problem, we demonstrate that the switching gap of a single-source multicast network is upper-bounded by the nth harmonic number Hn=1+1/2+...+1/n where n is the largest number of multicast routes containing a given link in the network. This harmonic-number bound is asymptotically tight as Oscr(ln n) for the combination multicast network. For the special class of multisource multicast networks with the same set of sink nodes, we make a comparison between the achievable information rate regions for network switching and network coding

Sharp inequalities for the harmonic numbers

Let H n be the nth harmonic number. For all integers n ⩾ 1 we have a - log ( e 1 / ( n + 1 ) - 1 ) ⩽ H n < b - log ( e 1 / ( n + 1 ) - 1 ) , with the best possible constants a = 1 + log ( e - 1 ) = 0.5672 … and b = γ = 0.5772 … . This refines a recently published result of Batir, who proved that the double-inequality holds with a = log ( π 2 / 6 ) ( = 0.4977 … ) and b = γ .

Some completely monotonic functions involving polygamma functions and an application

By using the first Binet's formula the strictly completely monotonic properties of functions involving the psi and polygamma functions are obtained. As direct consequences, two inequalities are proved. As an application, the best lower and upper bounds of the nth harmonic number are established.

Graph Operations and Zipfian Degree Distributions

Abstract The probability distribution on a set S = {1, 2,…,n} defined by Pr (k) = 1 (H n K) , where Hn is the nth harmonic number, is commonly called a Zipfian distribution. In this note we look at the degree distribution of graphs created from graphs with Zipfian degree distributions using some standard graph theoretical operations.

Reliability theory for receptor–ligand bond dissociation

Cell adhesion in the presence of hydrodynamic forces is a critical factor in inflammation, cancer metastasis, and blood clotting. A number of assays have recently been developed to apply forces to small numbers of the receptor–ligand bonds responsible for adhesion. Examples include assays using hydrodynamic shear in flow chambers or elastic probe deflection assays such as the atomic force microscope or the biomembrane force probe. One wishes to use the data on the time distribution of dissociation from these assays to derive information on the force dependence of reaction rates, an important determinant of cell adhesive behavior. The dissociation process can be described using the theory developed for reliability engineering of electronic components and networks. We use this framework along with the Bell model for the reverse reaction rate (kr=kr0 exp[r0 f/kT], where f is the applied force and kr0 and r0 are Bell model parameters) to write closed form expressions for the probability distribution of break-up with multiple independent or interacting bonds. These expressions show that the average lifetime of n bonds scales with the nth harmonic number multiplied by the lifetime of a single bond. Results from calculation and simulations are used to describe the effect of experimental procedures in forced unbinding assays on the estimation of parameters for the force dependence of reverse reaction rates.

A regular interconnection network

A regular interconnection network for large distributed systems which is indicated as R n is presented, we call it R n. This interconnection network is based on a mixed radix number system. The system topology is formed by complete graph product and consists of N = n! points, n(n-1)n!/4 lines and is regular of degree n(n-1)/2. The performance of such network is measured through an analysis of its degree, diameter, connectivity, fault tolerance, routing algorithm and reliability. A number of interesting properties of this network are revealed. Those include its average message distance which is approximately equal to (n-H n), (H n is the nth Harmonic number), the number of point disjoint paths (n(n-1)/2), and an attractive routing algorithm that is similar to Boolean hypercube. Finally, we show that it posseses super-λ property. The reliability of R n can also be readily obtained.

Some identities involving harmonic numbers

Let H n {H_n} denote the nth harmonic number. Explicit formulas for sums of the form ∑ a k H k \sum {a_k}{H_k} or ∑ a k H k H n − k \sum {a_k}{H_k}{H_{n - k}} are derived, where the a k {a_k} are simple functions of k. These identities are generalized in a natural way by means of generating functions.