scholarly journals The Polynomial Solutions of Quadratic Diophantine Equation X 2 − p t Y 2 + 2 K t X + 2 p t L t Y  = 0

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Hasan Sankari ◽  
Ahmad Abdo
Keyword(s):  
Recurrence Relation ◽  
Diophantine Equation ◽  
Recurrence Relations ◽  
Polynomial Solution ◽  
Pell Equation ◽  
Polynomial Solutions ◽  
Quadratic Diophantine Equation

In this study, we consider the number of polynomial solutions of the Pell equation x 2 − p t y 2 = 2 is formulated for a nonsquare polynomial p t using the polynomial solutions of the Pell equation x 2 − p t y 2 = 1 . Moreover, a recurrence relation on the polynomial solutions of the Pell equation x 2 − p t y 2 = 2 . Then, we consider the number of polynomial solutions of Diophantine equation E :   X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0 . We also obtain some formulas and recurrence relations on the polynomial solution X n , Y n of E .

scholarly journals Arithmetic properties of polynomial solutions of the Diophantine equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$

2021 ◽  
Author(s):  
Karl Dilcher ◽  
Maciej Ulas
Keyword(s):  
Differential Equations ◽  
Diophantine Equation ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Polynomial Solutions ◽  
Arithmetic Properties

AbstractFor each integer $$n\ge 1$$ n ≥ 1 we consider the unique polynomials $$P, Q\in {\mathbb {Q}}[x]$$ P , Q ∈ Q [ x ] of smallest degree n that are solutions of the equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$ P ( x ) x n + 1 + Q ( x ) ( x + 1 ) n + 1 = 1 . We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.

scholarly journals ELEMENTARY OSCILLATION CRITERIA FOR A THREE TERM RECURRENCE RELATION WITH OSCILLATORY COEFFICIENT SEQUENCE

1998 ◽  
Vol 29 (3) ◽  
pp. 227-232
Author(s):  
GUANG ZHANG ◽  
SUI-SUN CHENG
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Qualitative Properties ◽  
Oscillation Criteria ◽  
Three Term Recurrence Relation

Qualitative properties of recurrence relations with coefficients taking on both positive and negative values are difficult to obtain since mathematical tools are scarce. In this note we start from scratch and obtain a number of oscillation criteria for one such relation : $x_{n+1}-x_n+p_nx_{n-r}\le 0$.

scholarly journals On the quotient moment of lower generalized order statistics and characterization

2015 ◽  
Vol 11 (1) ◽  
pp. 73-89
Author(s):  
Devendra Kumar
Keyword(s):  
Order Statistics ◽  
Recurrence Relation ◽  
Conditional Expectation ◽  
General Class ◽  
Recurrence Relations ◽  
Generalized Order Statistics ◽  
Lower Records ◽  
Moments Of Order Statistics ◽  
Generalized Order

Abstract In this paper we consider general class of distribution. Recurrence relations satisfied by the quotient moments and conditional quotient moments of lower generalized order statistics for a general class of distribution are derived. Further the results are deduced for quotient moments of order statistics and lower records and characterization of this distribution by considering the recurrence relation of conditional expectation for general class of distribution satisfied by the quotient moment of the lower generalized order statistics.

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

scholarly journals Some Recurrence Relations and Hilbert Series of Right-Angled Affine Artin Monoid M(D~n∞)

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Young Chel Kuwn ◽  
Zaffar Iqbal ◽  
Abdul Rauf Nizami ◽  
Mobeen Munir ◽  
Sana Riaz ◽  
...  
Keyword(s):  
Growth Rate ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Hilbert Series ◽  
The Right

We find the Hilbert series of the right-angled affine Artin monoid M(D~n∞). We also discuss its recurrence relation and the growth rate.

The geometric unfolding of recurrence relations

2020 ◽  
Vol 104 (561) ◽  
pp. 403-411
Author(s):  
Stan Dolan
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Initial Values ◽  
Almost All

In 1942 R. C. Lyness challenged readers of the Gazette to find a recurrence relation of order 2 which would generate a cycle of period 7 for almost all initial values [1].

scholarly journals On a class of insoluble binary quadratic diophantine equations

1991 ◽  
Vol 123 ◽  
pp. 141-151 ◽  
Author(s):  
Franz Halter-Koch
Keyword(s):  
Diophantine Equation ◽  
Class Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Quadratic Number ◽  
Number Problem ◽  
Quadratic Number Fields ◽  
Quadratic Diophantine Equation ◽  
Real Quadratic

The binary quadratic diophantine equationis of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

The Positive Integer Solutions of a Diophantine Equation

2015 ◽  
Vol 713-715 ◽  
pp. 1483-1486
Author(s):  
Yi Wu ◽  
Zheng Ping Zhang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Pell Equation ◽  
Integer Solutions ◽  
Basic Equations

In this paper, we studied the positive integer solutions of a typical Diophantine equation starting from two basic equations including a Diophantine equation and a Pell equation, and we will prove all the positive integer solutions of the typical Diophantine equation.

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