INFINITE SERIES WITH HARMONIC NUMBERS AND CENTRAL BINOMIAL COEFFICIENTS

Binomial series involving harmonic polynomials are expressed in terms of parametric integrals. By evaluating these parametric integrals, we establish several remarkable closed formulae for the infinite series containing both central binomial coefficients and harmonic numbers. Most of the values for binomial series found in this paper concern the dilogarithm and trilogarithm functions.

The aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functionsis considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequenceis proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functionsis also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

We show that an alternating binomial sum which is connected with the Catalan numbers is divisible by n. A natural generalization of this sum is connected with the generalized Catalan numbers and also divisible by n. A new class of binomial sum is used. In Appendix A, we consider a positive binomial sum connected with Fibonacci and Lucas numbers. In Appendix B, we consider an alternating binomial sum which is also connected with Catalan numbers and divisible by (a + 1)n + 1. Similar reasoning was already used by the author to reprove more simply Calkin's result for divisibility of the alternating sum of powers of binomials coefficients by the central binomial coefficient.

In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant $$\zeta (3)$$ . In addition, by an idea from De Amo et al. (Proc Am Math Soc 139:1441–1444, 2011) we derive some new rational series representations involving even zeta values and central binomial coefficients. These formulas are expressed in terms of odd and even values of the Riemann zeta function and odd values of the Dirichlet beta function. In particular cases, we recover some well-known series representations of $$\pi $$ .

Combinatorial Gray codes for classes of pattern avoiding permutations

We discuss the integer sequence transform a 1→ b, where bn is the number of real roots of the polynomial a 0 + a 1 x + a 2 x 2 + · · · + anxn . It is shown that several sequences a give the trivial sequence b = (0, 1, 0, 1, 0, 1, .), i.e., bn = n mod 2, among them the Catalan numbers, central binomial coefficients, n! and for a fixed k. We also look at some sequences a for which b is more interesting such as an = (n + 1) k for k ≥ 3. Further, general procedures are given for constructing real sequences an for which bn is either always maximal or minimal.

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics, and theoretical physics. Here we show how one can obtain further interesting and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for2F1(1).

By means of Abel’s lemma on summation by parts, we derive several infinite series identities, which involve the classical harmonic numbers and their variants.

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we show how one can obtain further interesting identities about certain finite series involving binomial coefficients, harmonic numbers and generalized harmonic numbers by applying the usual differential operator to a known identity. MSC:11M06, 33B15, 33E20, 11M35, 11M41, 40C15.

This chapter contains challenging problems on various topics discussed in the previous chapters of this book. A special attention is given to the calculation of series involving harmonic numbers and tails of Riemann zeta function values. The calculation of these series reduces to the evaluation of linear or nonlinear Euler sums. A series involving a single harmonic number is called a linear Euler sum, while a series involving a product of at least two harmonic numbers is called a nonlinear Euler sum. The study of these series has been initiated by the famous mathematicians Leonhard Euler and Christian Goldbach in the mid-eighteenth century. Most of the series in this chapter are new and we have included them here because they are simply unusual sums containing combinations of harmonic numbers and tails of zeta function values. These series, linear or quadratic, alternating or not, are evaluated in terms of zeta function, logarithms, and polylogarithm of special values. Proofs of special results of analysis, which are called gems, challenges, and several open problems are scattered throughout the chapter.

In this note, we employ two identities from symbolic calculations and combine several known series expansions for specific special functions. By utilizing these tools, we can generate numerous Euler-like sums that connect harmonic numbers, pi and zeta values etc. These results are aesthetically pleasing and widely used in Euler-like sums fields. The main step is to calculate some new Euler-like sums of the harmonic number by integration transformations. The definite integrals of some elementary functions are first calculated by symbol software Mathematica. On the other hand, we can write these elementary functions in the form of Maclaurin series and then integrate them term by term and finally get the series containing harmonic numbers. In the process, we need to deal with the series, integrals, differentiating and other operations skillfully. We can continue to use the integral transform in this paper to expand, or we can find another new integral transform to get more valuable results by using similar methods.

In this paper, we leverage the iterated integral representation of multiple polylogarithm functions to derive explicit expressions for series containing multiple harmonic sums. These expressions are formulated in terms of multiple harmonic star sums and (coloured) multiple zeta values. Building upon these results and utilizing the stuffle relations, we systematically investigate the evaluation of various Euler-type sums that incorporate harmonic numbers and binomial coefficients. Specifically, our analysis focuses on sums of the form: ∑ n = 1 ∞ ∏ j = 1 p H n ( i j ) ( m 1 n + s 1 ) ( m 2 n + s 2 ) , ∑ n = 1 ∞ ∏ j = 1 p H n ( i j ) ( mn + s ) ( n + k k ) and some other forms, where p , k , i 1 , … , i p ∈ N , m , m 1 , m 2 ∈ N ∖ { 1 } and 1 ≤ s ≤ m , 1 ≤ s 1 ≤ m 1 , 1 ≤ s 2 ≤ m 2 . We provide several explicit evaluations to illustrate our results. These examples demonstrate that our approach offers a unified framework for investigating such sums, while simultaneously extending and generalizing numerous previously established results in the literature. Additionally, in the final section, we provide a concise discussion on how our method can be extended to investigate analogous Euler-type sums of alternating versions.

This paper presents closed-form evaluations for a broad class of alternating Euler sums built from multiple harmonic numbers and binomial coefficients. The three variants S π , q ( k ) , S ¯ π , q ( k ) , and S ¯ π q ( k ) are treated within a single framework. Central to our method is the conversion of multiple polylogarithms into iterated integrals, followed by partial-fraction decomposition and summation reordering, facilitated by decompositions of multiple harmonic sums (including star values) and simple combinatorial identities. The applicability of our method is demonstrated through representative examples, providing a template for future extensions to higher weights and more general weight functions.

The thire paper examined the effect of turbulence by combustion causes to heat loss. But in real gasoline engine, the air fuel mixture, which inhaled it in cylinder includes initial turbulence. Therefore effect of initial turbulence of mixture was examined to estimate heat loss of gasoline engine in this paper. The initial turbulence was expressed using a simple model in which the momentum of turbulence damps when turbulence moves from unburned gas to burned gas. The expression to estimate heat loss is expressed as next by using the model. [numerical formula] Where the third term expressed the effect of initial turbulence and the coefficient: Cd≈0.24 has expressed damping of initial turbulence.

Among the square spline shafts which are in use generally, the profiles of the shafts for light and medium loads are complicated ones with double notches. And when these spline shaft are twisted, the stress distributions arround the notches or the torsional stress concentration factors at the bottom of notches have not been determined. Applying the method of the electrolytic tank analogy, which can be used on models of large scale, the author measured the stress distribution about the standardized spline shafts when twisted. And the following empirical formulas for the stress concentration factor αK0 and the torsional moment Mt are obtained. [numerical formula] From these equations, the torsional strength of the standardized square spline shaft can be determined.