scholarly journals New Tribonacci recurrence relations and addition formulas

2020 ◽  
Vol 26 (4) ◽  
pp. 164-172
Author(s):  
Kunle Adegoke ◽  
◽  
Adenike Olatinwo ◽  
Winning Oyekanmi ◽  
◽  
...  
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Simple Relation ◽  
Addition Formula ◽  
Lucas Numbers ◽  
Tribonacci Numbers ◽  
Three Term Recurrence Relation ◽  
Special Case ◽  
Addition Formulas

Only one three-term recurrence relation, namely, W_{r}=2W_{r-1}-W_{r-4}, is known for the generalized Tribonacci numbers, W_r, r\in Z, defined by W_{r}=W_{r-1}+W_{r-2}+W_{r-3} and W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}, where W_0, W_1 and W_2 are given, arbitrary integers, not all zero. Also, only one four-term addition formula is known for these numbers, which is W_{r + s} = T_{s - 1} W_{r - 1} + (T_{s - 1} + T_{s-2} )W_r + T_s W_{r + 1}, where ({T_r})_{r\in Z} is the Tribonacci sequence, a special case of the generalized Tribonacci sequence, with W_0 = T_0 = 0 and W_1 = W_2 = T_1 = T_2 = 1. In this paper we discover three new three-term recurrence relations and two identities from which a plethora of new addition formulas for the generalized Tribonacci numbers may be discovered. We obtain a simple relation connecting the Tribonacci numbers and the Tribonacci–Lucas numbers. Finally, we derive quadratic and cubic recurrence relations for the generalized Tribonacci numbers.

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 Numerical construction of the generalized Hermite polynomials

2003 ◽  
pp. 49-63
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Stable Method ◽  
Numerical Construction ◽  
Three Term Recurrence Relation

In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x ? z|? exp(?x2) on R, where z?R and ? > ? 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.

scholarly journals Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity

10.37236/5913 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Lily Li Liu ◽  
Ya-Nan Li
Keyword(s):  
Recurrence Relation ◽  
Linear Transformation ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Sufficient Condition ◽  
Linear Transformations ◽  
Whitney Numbers ◽  
Three Term Recurrence Relation

Let $[T(n,k)]_{n,k\geqslant0}$ be a triangle of positive numbers satisfying the three-term recurrence relation\[T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1).\]In this paper, we give a new sufficient condition for linear transformations\[Z_n(q)=\sum_{k=0}^{n}T(n,k)X_k(q)\]that preserves the strong $q$-log-convexity of polynomials sequences. As applications, we show linear transformations, given by matrices of the binomial coefficients, the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind, preserving the strong $q$-log-convexity in a unified manner.

scholarly journals On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Lucas Numbers ◽  
Polynomial Sequences ◽  
Fibonacci And Lucas Numbers ◽  
Linear Recurrence Relation ◽  
Humbert Polynomials ◽  
General Method

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

scholarly journals Recurrence relations for polynomials obtained by arithmetic functions

2019 ◽  
Vol 15 (06) ◽  
pp. 1291-1303 ◽  
Author(s):  
Bernhard Heim ◽  
Florian Luca ◽  
Markus Neuhauser
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Fourier Coefficients ◽  
Arithmetic Functions ◽  
Explicit Formulas ◽  
Dedekind Eta Function ◽  
Special Case

Families of polynomials associated to arithmetic functions [Formula: see text] are studied. The case [Formula: see text], the divisor sum, dictates the non-vanishing of the Fourier coefficients of powers of the Dedekind eta function. The polynomials [Formula: see text] are defined by [Formula: see text]-term recurrence relations. For the case that [Formula: see text] is a polynomial of degree [Formula: see text], we prove that at most a [Formula: see text] term recurrence relation is needed. For the special case [Formula: see text], we obtain explicit formulas and results.

scholarly journals A class of big (p,q)-Appell polynomials and their associated difference equations

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3085-3121
Author(s):  
H.M. Srivastava ◽  
B.Y. Yaşar ◽  
M.A. Özarslan
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Recurrence Relation ◽  
Difference Equations ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Equivalence Theorem ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Special Case

In the present paper, we introduce and investigate the big (p,q)-Appell polynomials. We prove an equivalance theorem satisfied by the big (p, q)-Appell polynomials. As a special case of the big (p,q)- Appell polynomials, we present the corresponding equivalence theorem, recurrence relation and difference equation for the big q-Appell polynomials. We also present the equivalence theorem, recurrence relation and differential equation for the usual Appell polynomials. Moreover, for the big (p; q)-Bernoulli polynomials and the big (p; q)-Euler polynomials, we obtain recurrence relations and difference equations. In the special case when p = 1, we obtain recurrence relations and difference equations which are satisfied by the big q-Bernoulli polynomials and the big q-Euler polynomials. In the case when p = 1 and q ? 1-, the big (p,q)-Appell polynomials reduce to the usual Appell polynomials. Therefore, the recurrence relation and the difference equation obtained for the big (p; q)-Appell polynomials coincide with the recurrence relation and differential equation satisfied by the usual Appell polynomials. In the last section, we have chosen to also point out some obvious connections between the (p; q)-analysis and the classical q-analysis, which would show rather clearly that, in most cases, the transition from a known q-result to the corresponding (p,q)-result is fairly straightforward.

scholarly journals Balancing and Lucas-Balancing Numbers and their Application to Cryptography

2016 ◽  
Vol 5 (1) ◽  
pp. 29-36
Author(s):  
Sujata Swain ◽  
Chidananda Pratihary ◽  
Prasanta Kumar Ray
Keyword(s):  
Finite Number ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Recursive Relation ◽  
Lucas Numbers ◽  
Counting Problems ◽  
Balancing Numbers

It is well known that, a recursive relation for the sequence  is an equation that relates  to certain of its preceding terms . Initial conditions for the sequence  are explicitly given values for a finite number of the terms of the sequence. The recurrence relation is useful in certain counting problems like Fibonacci numbers, Lucas numbers, balancing numbers, Lucas-balancing numbers etc. In this study, we use the recurrence relations for both balancing and Lucas-balancing numbers and examine their application to cryptography.

scholarly journals A recurrence relation generalizing those of Apéry

1984 ◽  
Vol 36 (2) ◽  
pp. 267-278 ◽  
Author(s):  
Richard Askey ◽  
J. A. Wilson
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Image Position ◽  
Three Term Recurrence Relation

AbstractA three term recurrence relation is found forwhen a + d = b + c. This includes the recurrence relations of Apéry associated with ζ(3), ζ(2) and log 2 as special or limiting cases.

scholarly journals An ubiquitous three-term recurrence relation

2021 ◽  
Vol 62 (3) ◽  
pp. 032106
Author(s):  
Paolo Amore ◽  
Francisco M. Fernández
Keyword(s):  
Recurrence Relation ◽  
Three Term Recurrence Relation

Orthogonal polynomials: applications and computation

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 45-119 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Numerical Methods ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Rational Function ◽  
Recurrence Relations ◽  
Recurrence Coefficients ◽  
Basic Task ◽  
Sobolev Orthogonal Polynomials ◽  
Gauss Type ◽  
Three Term Recurrence Relation

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

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