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

2021 ◽  
pp. 031
Author(s):  
Sibel Koparal ◽  
Neşe Ömür ◽  
Cemile Duygu Çolak
Keyword(s):  
Real Number ◽  
Nonnegative Integer ◽  
Nonzero Real Number ◽  
Alternating Sums ◽  
Sums Of Products ◽  
Binomial Sums

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.

Using Logarithms to Explore Power and Exponential Functions

1994 ◽  
Vol 87 (3) ◽  
pp. 161-170
Author(s):  
James R. Rahn ◽  
Barry A. Berndes
Keyword(s):  
Experimental Data ◽  
Real Number ◽  
Positive Real Number ◽  
Power Functions ◽  
Exponential Functions ◽  
Positive Real ◽  
Nonzero Real Number ◽  
Physical Phenomena ◽  
The Relationship

Power functions and exponential functions often describe the relationship between variables in physical phenomena. Power functions are equations of the form y = kxn (see fig. 1), where k is a nonzero real number and n is a nonzero real number not equal to 1. Exponential functions are equations of the form y = kbx (see fig. 2), where k is a nonzero real number and b is a positive real number. Students should be able visually to recognize these functions so that they can easily identify their appearance when experimental data are graphed. When physical phenomena appear to describe exponential and power functions, logarithms can be used to locate approximate functions that represent the phenomena.

scholarly journals Pair Correlations and Random Walks on the Integers

2016 ◽  
Vol 11 (1) ◽  
pp. 159-164
Author(s):  
Radhakrishnan Nair ◽  
Entesar Nasr
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Number ◽  
Pair Correlation ◽  
Real Numbers ◽  
Pair Correlations ◽  
Nonzero Real Number ◽  
Correlation Estimate

AbstractThe paper gives conditions for a sequence of fractional parts of real numbers $\left( {\{ a_n x\} } \right)_{n = 1}^\infty $ to satisfy a pair correlation estimate. Here x is a fixed nonzero real number and $\left( {a_n } \right)_{n = 1}^\infty $ is a random walk on the integers.

96.50 A note on a family of alternating sums of products of binomial numbers

2012 ◽  
Vol 96 (536) ◽  
pp. 337-342
Author(s):  
N. Gauthier ◽  
Paul S. Bruckman
Keyword(s):  
Alternating Sums ◽  
Sums Of Products

scholarly journals The nonexistence of a factorization formula for Cayley numbers

1983 ◽  
Vol 24 (2) ◽  
pp. 131-132 ◽  
Author(s):  
P. J. C. Lamont
Keyword(s):  
Real Number ◽  
Nonzero Element ◽  
Real Field ◽  
The Real ◽  
Cayley Algebra ◽  
Factorization Formula ◽  
Nonzero Real Number ◽  
One To One

Let C be the Cayley algebra denned over the real field. If, for given elements α, β, and γ of a quaternion subalgebra of C, α = βγ, it follows, by associativity, that for any nonzero element δ of the same quaternion subalgebra, α = (βδ)(δ-1γ). For Cayley numbers ζ ξ, and η with ζ = ξη, the relation ζ = (ξδ)(δ-1η) in general only holds when δ is a nonzero real number. Because of the existence of factorization results [1, 2] in the orders of C, the question naturally arises of whether it is possible to choose one-to-one mappings, θ and φ, of C onto itself such that ζ = θξ. φη whenever ζ = ξη. To discuss this question, we make the following definition.

scholarly journals FACTORS OF SUMS AND ALTERNATING SUMS INVOLVING BINOMIAL COEFFICIENTS AND POWERS OF INTEGERS

2011 ◽  
Vol 07 (07) ◽  
pp. 1959-1976 ◽  
Author(s):  
VICTOR J. W. GUO ◽  
JIANG ZENG
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Binomial Coefficients ◽  
Alternating Sums ◽  
Divisibility Properties ◽  
Positive Integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers n1,…,nm, nm+1 = n1, and any nonnegative integer r, there holds [Formula: see text] and conjecture that for any nonnegative integer r and positive integer s such that r + s is odd, [Formula: see text] where ε = ±1.

scholarly journals Sums of products of generalized Fibonacci and Lucas numbers

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
Zvonko Čerin
Keyword(s):  
Recent Work ◽  
Recurrence Relation ◽  
Fixed Number ◽  
Binomial Coefficients ◽  
Natural Numbers ◽  
Lucas Numbers ◽  
Alternating Sums ◽  
Explicit Formulae ◽  
Fibonacci And Lucas Numbers ◽  
Sums Of Products

AbstractIn this paper we obtain explicit formulae for sums of products of a fixed number of consecutive generalized Fibonacci and Lucas numbers. These formulae are related to the recent work of Belbachir and Bencherif. We eliminate all restrictions about the initial values and the form of the recurrence relation. In fact, we consider six different groups of three sums that include alternating sums and sums in which terms are multiplied by binomial coefficients and by natural numbers. The proofs are direct and use the formula for the sum of the geometric series.

scholarly journals Factors of alternating sums of products of binomial and q-binomial coefficients

2007 ◽  
Vol 127 (1) ◽  
pp. 17-31 ◽  
Author(s):  
Victor J. W. Guo ◽  
Frédéric Jouhet ◽  
Jiang Zeng
Keyword(s):  
Binomial Coefficients ◽  
Alternating Sums ◽  
Sums Of Products

scholarly journals On the Operator ⊕k,m Related to the Wave Equation and Laplacian

2021 ◽  
Vol 14 (3) ◽  
pp. 881-894
Author(s):  
Sudprathai Bupasiri
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Real Number ◽  
Nonnegative Integer ◽  
Unknown Function ◽  
Nonnegative Real Number ◽  
Generalized Function

In this article, we study the fundamental solution of the operator $\oplus _{m}^{k}$, iterated $k$-times and is defined by$$\oplus _{m}^{k} = \left[\left(\sum_{r=1}^{p} \frac{\partial^2} {\partial x_r^2}+m^{2}\right)^4 - \left( \sum_{j=p+1}^{p+q} \frac{\partial^2}{\partial x_{j}^2} \right)^4 \right ]^k,$$ where $m$ is a nonnegative real number, $p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n$,$x=(x_1,x_2,\ldots,x_n)\in\mathbb{R}^n$, $k$ is a nonnegative integer. At first we study the fundamental solution of the operator $\oplus _{m}^{k}$ and after that, we apply such the fundamental solution to solve for the solution of the equation $\oplus _{m}^{k}u(x)= f(x)$, where $f(x)$ is generalized function and $u(x)$ is unknown function for $ x\in \mathbb{R}^{n}$.

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