Binomial Coefficients and Triangular Numbers

We produce formulas of sums the product of the binomial coefficients and triangular numbers. And we apply our formula to prove an identity of Wang and Zhang. Further, we provide an analogue of our identity for the alternating sums.

On the alternating sums of reciprocal generalized Fibonacci numbers

In this paper, we consider finite alternating sums derived from the generalized Fibonacci numbers [Formula: see text] [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers with [Formula: see text], [Formula: see text]. Applying the greatest integer function to these sums, we obtain some equalities involving the generalized Fibonacci numbers.

ON BINOMIAL SUMS WITH THE TERMS OF SEQUENCES {g_kn} AND {h_kn}

In this paper, we derive sums and alternating sums of products of terms ofthe sequences $\left\{ g_{kn}\right\} $ and $\left\{ h_{kn}\right\} $ withbinomial coefficients. For example,\begin{eqnarray*} &\sum\limits_{i=0}^{n}\binom{n}{i}\left( -1\right) ^{i} \left(c^{2k}\left(-q\right) ^{k}+c^{k}v_{k}+1\right)^{-ai}h_{k\left( ai+b\right) }h_{k\left(ai+e\right) } \\ &=\left\{ \begin{array}{clc} -\Delta ^{\left( n+1\right) /2}g_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is odd,} & \\ \Delta ^{n/2}h_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is even,} & \end{array}% \right.\end{eqnarray*}%and\begin{eqnarray*} &&\sum\limits_{i=0}^{n}\binom{n}{i}i^{\underline{m}}g_{k\left( n-ti\right) }h_{kti} \\ &&=2^{n-m}n^{\underline{m}}g_{kn}-n^{\underline{m}}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{n\left( 1-t\right) }h_{kt}^{n-m}g_{k\left( tm+tn-n\right) },\end{eqnarray*}%where $a, b, e$ is any integer numbers, $c$ is nonzero real number and $m$is nonnegative integer.

On estimation of Hausdorff deviation of convex polygons in $\mathbb{R}^2$ from their differences with disks

We study a problem concerning the estimation of the Hausdorff deviation of convex polygons in $\mathbb R^2$ from their geometric difference with circles of sufficiently small radius. Problems with such a subject, in which not only convex polygons but also convex compacts in the Euclidean space $\mathbb R^n$ are considered, arise in various fields of mathematics and, in particular, in the theory of differential games, control theory, convex analysis. Estimates of Hausdorff deviations of convex compact sets in $\mathbb R^n$ in their geometric difference with closed balls in $\mathbb R^n$ are presented in the works of L.S. Pontryagin, his staff and colleagues. These estimates are very important in deriving an estimate for the mismatch of the alternating Pontryagin’s integral in linear differential games of pursuit and alternating sums. Similar estimates turn out to be useful in deriving an estimate for the mismatch of the attainability sets of nonlinear control systems in $\mathbb R^n$ and the sets approximating them. The paper considers a specific convex heptagon in $\mathbb R^2$. To study the geometry of this heptagon, we introduce the concept of a wedge in $\mathbb R^2$. On the basis of this notion, we obtain an upper bound for the Hausdorff deviation of a heptagon from its geometric difference with the disc in $\mathbb R^2$ of sufficiently small radius.

Moments of Catalan Triangle Numbers

In this chapter, we consider the Catalan numbers, C n = 1 n + 1 2 n n , and two of their generalizations, Catalan triangle numbers, B n , k and A n , k , for n , k ∈ N . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: ∑ k = 1 n k m B n , k j , ∑ k = 1 n + 1 2 k − 1 m A n , k j , for j , n ∈ N and m ∈ N ∪ 0 . We present their closed expressions for some values of m and j . Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section.

Congruences involving alternating sums related to harmonic numbers and binomial coefficients

In 2017, Bing He investigated arithmetic properties to obtain various basic congruences modulo a prime for several alternating sums involving harmonic numbers and binomial coefficients. In this paper we study how we can obtain more congruences modulo a power of a prime number p (super congruences) in the ring of p-integer \mathbb{Z}_{p} involving binomial coefficients and generalized harmonic numbers.

On the Means of Jordan’s Totient Function

In this paper, we study some important means of Jordan’s totient function, especially, we obtain asymptotic formula for geometric mean and harmonic mean. We also study alternating sums of Jordan’s totient function and Carleman’s inequality for this function.

Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.