scholarly journals Some new identities involving Laguerre polynomials

2021 ◽  
Vol 6 (11) ◽  
pp. 12713-12717
Author(s):  
Xiaowei Pan ◽  
◽  
Xiaoyan Guo ◽  
Keyword(s):  
Positive Integer ◽  
Laguerre Polynomials ◽  
Elementary Method

<abstract><p>In this paper, we use elementary method and some sort of a counting argument to show the equality of two expressions. That is, let $ f(n) $ and $ g(n) $ be two functions, $ k $ be any positive integer. Then $ f(n) = \sum\limits_{r = 0}^n(-1)^r\cdot \frac{n!}{r!}\cdot \binom{n+k-1}{r+k-1}\cdot g(r) $ if and only if $ g(n) = \sum\limits_{r = 0}^n(-1)^r\cdot \frac{n!}{r!}\cdot \binom{n+k-1}{r+k-1}\cdot f(r) $ for all integers $ n\geq0 $. As an application of this formula, we obtain some new identities involving the famous Laguerre polynomials.</p></abstract>

scholarly journals On the Odd Prime Solutions of the Diophantine Equationxy+yx=zz

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Yuanyuan Deng ◽  
Wenpeng Zhang
Keyword(s):  
Positive Integer ◽  
Pell’S Equation ◽  
Elementary Method ◽  
Integer Solutions ◽  
Least Solution

Using the elementary method and some properties of the least solution of Pell’s equation, we prove that the equationxy+yx=zzhas no positive integer solutions (x,y,z) withxandybeing odd primes.

An Arithmetic Function and the k-th Power Part of a Positive Integer

2012 ◽  
Vol 490-495 ◽  
pp. 1941-1944
Author(s):  
Ming Jun Wang
Keyword(s):  
Positive Integer ◽  
Arithmetic Function ◽  
Mean Value ◽  
Asymptotic Formulas ◽  
Elementary Method ◽  
Large Exponent ◽  
Power Part ◽  
The Mean

For any positive integer denotes the largest - power less than or equal to ,and denotes the smallest - power greater than or equal to . Let be a prime, denotes the large exponent of power which divides .In this paper we use elementary method to study the mean value properties of and ,and give two interesting asymptotic formulas.

scholarly journals On the Galois groups of generalized Laguerre Polynomials

2014 ◽  
Vol Volume 37 ◽  
Author(s):  
Shanta Laishram
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Positive Integer ◽  
Laguerre Polynomials ◽  
Galois Groups ◽  
Newton Polygons ◽  
Short Article ◽  
Generalized Laguerre Polynomials ◽  
Algebraic Properties ◽  
International Audience

International audience For a positive integer n and a real number α, the generalized Laguerre polynomials are defined by L (α) n (x) = n j=0 (n + α)(n − 1 + α) · · · (j + 1 + α)(−x) j j!(n − j)!. These orthogonal polynomials are solutions to Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for their interesting algebraic properties. In this short article, it is shown that the Galois groups of Laguerre polynomials L(α)(x) is Sn with α ∈ {±1,±1,±2,±1,±3} except when (α,n) ∈ {(1,2),(−2,11),(2,7)}. The proof is based on ideas of p−adic Newton polygons.

On the Diophantine System $x^2 - Dy^2 = 1-D$ and $x=2z^2-1$

2004 ◽  
Vol 95 (2) ◽  
pp. 171 ◽  
Author(s):  
Maohua Le
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
System Of Equations ◽  
Elementary Method ◽  
Diophantine System ◽  
All Solutions ◽  
Integer Solutions

Let $D$ be a positive integer such that $D-1$ is an odd prime power. In this paper we give an elementary method to find all positive integer solutions $(x, y, z)$ of the system of equations $x^2-Dy^2=1-D$ and $x=2z^2-1$. As a consequence, we determine all solutions of the equations for $D=6$ and $8$.

Extended State Space of the Rational sl(2) Gaudin Model in Terms of Laguerre Polynomials

2013 ◽  
Vol 58 (11) ◽  
pp. 1084-1091
Author(s):  
Yu.V. Bezvershenko ◽  
◽  
P.I. Holod ◽  
Keyword(s):  
State Space ◽  
Laguerre Polynomials ◽  
Gaudin Model ◽  
Extended State

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

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