Summation formulas involving generalized harmonic numbers

By means of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.     

Certain summation formulas involving harmonic numbers and generalized harmonic numbers

Harmonic numbers and generalized harmonic numbers have been studied 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 aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function ₅F₄(1).     

Binomial sums involving harmonic numbers

This paper develops the approach to the evaluation of a class of infinite series that involve special products of binomial type, generalized harmonic numbers of order 1 and rational functions. We give new summation results for certain infinite series of non-hypergeometric type. New formulas for the number π are included.     

Four families of summation formulas involving generalized harmonic numbers

In terms of Abel’s transformation on difference operators, we establish four families of summation formulas involving generalized harmonic numbers. They include several known and numerous new harmonic number identities as special cases.     

Minton-Karlsson identities and summation formulae involving generalized harmonic numbers

By applying the derivative operator to Minton and Karlsson's hypergeometric series identities, several interesting summation formulae involving generalized harmonic numbers are established.     

The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form.     

A new class of identities involving Cauchy numbers,harmonic numbers and zeta values

Improving an old idea of Hermite, we associate to each natural number k a modified zeta function of order k. The evaluation of the values of these functions F _k at positive integers reveals a wide class of identities linking Cauchy numbers, harmonic numbers and zeta values.     

Summation formulae involving the Laguerre polynomial

A general result involving the generalized hypergeometric function is deduced by the elementary manipulation of series. Kummer's first theorem for the confluent hypergeometric function and two summation formulae for the Gauss hypergeometric function are then applied and new summation formulae involving the Laguerre polynomial are deduced.     

Ramanujan-like series for $$\frac{1}{\pi }$$ involving harmonic numbers

We introduce new classes of Ramanujan-like series for \(\frac{1}{\pi }\), by devising methods for evaluating harmonic sums involving squared central binomial coefficients, such as the Ramanujan-type series

$$\begin{aligned} \sum _{n=1}^{\infty } \frac{\left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 \left( H_n^2+H_n^{(2)}\right) }{16^n (2 n-1)} = \frac{4 \pi }{3}-\frac{32 \ln ^2(2) - 32 \ln (2) + 16 }{\pi } \end{aligned}$$

introduced in this article. While the main technique used in this article is based on the evaluation of a parameter derivative of a beta-type integral, we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for \(\frac{1}{\pi }\) containing harmonic numbers.

     

Sharp bounds for harmonic numbers

In the paper, we collect some inequalities and establish a sharp double inequality for bounding the n-th harmonic number.     

Inequalities for the harmonic numbers

We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}$ . One of our results states that we have for all integers n ???2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\alpha= \frac{6 \sqrt{6}-2 \sqrt[3]{396}}{3 \log(\log{2}+\gamma)}=0.0140\ldots \quad\mbox{and} \quad\beta=1.$$ Here, ?? denotes Euler??s constant.     

Summation of double series using the Euler-Maclaurin sum formula

A certain variation of the Euler-Maclaurin sum formula is used to deduce a corresponding formula, suitable for the summation of finite or infinite double series.     

Further congruences involving Bernoulli numbers

Congruences of Voronoi's type for Bernoulli numbers are proved via Bernoulli distributions and connections to some already known congruences of a similar type are briefly discussed.     

Size Ramsey numbers involving stars

We calculate some size Ramsey numbers involving stars. For example we prove that for t ? k ? 2 and n sufficiently large the size Ramsey number.

$\text{r}{}_{\mathrm{n}}\text{(K}{}_{\mathrm{1,k}}\text{K}{}_{\mathrm{t}}\text{+}\text{K}{}_{\mathrm{n}}\text{=}\text{k(t?1)+1}\text{2}\text{+(k(t?1)+1)(n+k?1).}$