scholarly journals On the Closed Forms of Z(pq), q = kp ± 10

2021 ◽  
Vol 41 (1) ◽  
pp. 96-105
Author(s):  
AAK Majumdar ◽  
LC Das
Keyword(s):  
Positive Integer ◽  
Closed Form

This paper derives the closed form expressions of Z(pq), where p and q ( > p) are primes, and q is of the forms q = kp ± 10 for some (positive) integer k, and Z(.) is the pseudo Smarandache function. The Chittagong Univ. J. Sci. 40(1) : 96-105, 2019

scholarly journals Rational Divide-and-Conquer Relations

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Charinthip Hengkrawit ◽  
Vichian Laohakosol ◽  
Watcharapon Pimsert
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Rational Function ◽  
Natural Generalization ◽  
Divide And Conquer ◽  
Recursive Equation ◽  
Closed Form Solutions ◽  
Tangent Function

A rational divide-and-conquer relation, which is a natural generalization of the classical divide-and-conquer relation, is a recursive equation of the form f(bn)=R(f(n),f(n),…,f(b−1)n)+g(n), where b is a positive integer ≥2; R a rational function in b−1 variables and g a given function. Closed-form solutions of certain rational divide-and-conquer relations which can be used to characterize the trigonometric cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived and their global behaviors are investigated.

scholarly journals The Hilbert-Kunz Function for Binomial Hypersurfaces

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Shyamashree Upadhyay
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Positive Characteristic ◽  
Closed Form Formula

I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any field of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated with any binomial hypersurface over any field of arbitrary positive characteristic is rational. As an example, I also prove the well known fact that for 1-dimensional binomial hypersurfaces the Hilbert-Kunz multiplicity is a positive integer and give a precise account of the integer.

scholarly journals Closed Form Continued Fraction Expansions of Special Quadratic Irrationals

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Fishman ◽  
Steven J. Miller
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Recurrence Relation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Fibonacci Numbers ◽  
Continued Fraction Expansions

We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation with , , a positive integer, and (note that gives the Fibonacci numbers). Let . We find simple closed form continued fraction expansions for for any integer by exploiting elementary properties of the recurrence relation and continued fractions.

DEFINING POWER SUMS OF n AND φ(n) INTEGERS

2009 ◽  
Vol 05 (01) ◽  
pp. 41-53 ◽  
Author(s):  
JITENDER SINGH
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Power Sums ◽  
Power Sum ◽  
Analytical Results ◽  
General Power ◽  
Integral Relations ◽  
Positive Integers

Let n be a positive integer and φ(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those φ(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and φ(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums.

scholarly journals Clausen’s Series 3F2(1) with Integral Parameter Differences

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1783
Author(s):  
Kwang-Wu Chen
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Open Problem ◽  
Group Symmetry ◽  
Complex Numbers ◽  
Integral Parameter ◽  
Arbitrary Complex

Ebisu and Iwassaki proved that there are three-term relations for 3F2(1) with a group symmetry of order 72. In this paper, we apply some specific three-term relations for 3F2(1) to partially answer the open problem raised by Miller and Paris in 2012. Given a known value 3F2((a,b,x),(c,x+1),1), if f−x is an integer, then we construct an algorithm to obtain 3F2((a,b,f),(c,f+n),1) in an explicit closed form, where n is a positive integer and a,b,c and f are arbitrary complex numbers. We also extend our results to evaluate some specific forms of p+1Fp(1), for any positive integer p≥2.

The Correlation Dimension of a Rectilinear Grid

2016 ◽  
Vol 16 (01) ◽  
pp. 1550010 ◽  
Author(s):  
ERIC ROSENBERG
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Correlation Dimension ◽  
Three Dimensions ◽  
Closed Form Expression ◽  
Formal Definition ◽  
Form Expression ◽  
Network Measures ◽  
Random Node ◽  
Definition Of

The correlation dimension dC of a finite network measures how the fraction of nodes at a given distance from a random node scales with the distance. However, there is no standardized formal definition of dC for a network. We consider various possible definitions of dC for a finite unweighted and undirected rectilinear grid in one, two, and three dimensions. We propose a simple “overall slope” definition for dC which yields an exact closed form expression for such grids. We prove that the overall slope definition satisfies two properties that should be satisfied by any definition of the correlation dimension of a network. Lastly, we present a conjecture giving a closed form expression for the overall slope correlation dimension of a finite rectilinear grid in E dimensions, for any positive integer E.

On the summation of certain trigonometric series

1945 ◽  
Vol 41 (2) ◽  
pp. 145-160 ◽  
Author(s):  
L. S. Goddard
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Trigonometric Series ◽  
Present Note ◽  
Rational Functions ◽  
Recurrent Relation ◽  
Polygamma Function ◽  
Special Values ◽  
Image Position

In the present note, which is introductory to the following paper, closed expressions, suitable for computational purposes, are found for the sums of the serieswhere α > 1, t = 1, 2, 3, …, and n is a positive integer. In each case a recurrent relation is found giving the values of and for t > 2 in terms of and the series Θκ(α) (κ = 1, 2, …, t), whereWhen κ is even the last series is expressed in closed form in terms of the Bernoullian polynomial φκ(l/α) and, when κ is odd and α is rational, a closed form is found involving the polygamma function Ψ(κ)(z), where The general expressions for and involve Ψ(z) and Ψ′(z) when α is rational, but for special values of α they reduce to a form independent of the Ψ-function. and are independent of n and are expressible as simple rational functions of α.

Identities for the shifted harmonic numbers and binomial coeffecients

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6071-6086
Author(s):  
Ce Xu
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Harmonic Numbers ◽  
Riemann Zeta

We develop new closed form representations of sums of (n+?)th shifted harmonic numbers and reciprocal binomial coeffecients through ?th shifted harmonic numbers and Riemann zeta function with positive integer arguments. Some interesting new consequences and illustrative examples are considered.

scholarly journals Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

2019 ◽  
Vol 33 (1) ◽  
pp. 55-65
Author(s):  
Göksal Bilgici ◽  
Tuncay Deniz Şentürk
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Arbitrary Positive Integer ◽  
Open Problems ◽  
Positive Integers ◽  
Addition Formulas

AbstractIn this paper, we obtain a closed form for ${F_{\sum\nolimits_{i = 1}^k {} }}$, ${P_{\sum\nolimits_{i = 1}^k {} }}$and ${J_{\sum\nolimits_{i = 1}^k {} }}$ for some positive integers k where Fr, Pr and Jr are the rth Fibonacci, Pell and Jacobsthal numbers, respectively. We also give three open problems for the general cases ${F_{\sum\nolimits_{i = 1}^n {} }}$, ${P_{\sum\nolimits_{i = 1}^n {} }}$ and ${J_{\sum\nolimits_{i = 1}^n {} }}$for any arbitrary positive integer n.

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