Related papers: Some Harmonic Number Identities involving certain …
By means of partial fraction method, we investigate the decomposition of rational functions. Several striking identities on harmonic numbers and generalized Apery numbers will be established, including the binomial-harmonic number identity…
In this paper we present some interesting results involving summation of series in particular trigonometric ones. We failed to locate these results in existing literature or in the web like MathWorld (http://mathworld.wolfram.com/) nor…
A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with both…
We introduce notions of bi-unitary, bi*-unitary and bi**-unitary harmonic numbers, along with their preliminary study.
The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the M\"{o}bius function is used to formulate this divisor sum and…
We investigate the zeros of two one-parameter families of harmonic functions and describe how the number of zeros depends on the parameter. Our functions have the property that all zeros lie on certain rays in the complex plane and thus we…
Let A, B, C, D be given finite sets of pairs of n-by-n complex matrices. We describe an algorithm to determine, with finitely many computations, whether there is a single unitary matrix U such that each pair of matrices in A is unitarily…
By the theory of elliptic curves, we study the integers representable as the product of the sum of four integers with the sum of their reciprocals and give a sufficient condition for the integers with a positive representation.
Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of…
The regularized product of the Fibonacci numbers is evaluated.
Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…
A generalization of the Chu-Vandermonde convolution is presented and proved with the integral representation method. This identity can be transformed into another identity, which has as special cases two known identities. Another identity…
We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial…
All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm…
We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
An element $g$ in a group $G$ is called \emph{reciprocal} if there exists $h \in G$ such that $g^{-1}=hgh^{-1}$. The reciprocal elements are also known as `real elements' or `reversible elements' in the literature. We classify the…
We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the…
The alternating and non-alternating harmonic sums and other algebraic objects of the same equivalence class are connected by algebraic relations which are induced by the product of these quantities and which depend on their index calss…