THE INTEGER RECURRENCE $P(n)=a+P(n-\phi(a)) $

The Integer Linear Recurrence Zero Set Under the Microscope

The zero set of a linear recurrence is {m | xm = 0}, where xm is the m-th term. For a linear recurrence over the complex numbers C it is classical that xm = P1(m) · α1 + · · ·+ Pd(m) ·αd where P1(m), . . . , Pd(m) are polynomials in m, and α1, . . . , αd are the distinct zeros of the recurrence’s characteristic polynomial. We show that if the zero set of a linear recurrence over C is infinite, ...

S-parts of terms of integer linear recurrence sequences

Let S = {q1, . . . , qs} be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q1 1 . . . q rs s M , where r1, . . . , rs are non-negative integers and M is an integer relatively prime to q1 . . . qs. We define the S-part [m]S of m by [m]S := q r1 1 . . . q rs s . Let (un)n≥0 be a linear recurrence sequence of integers. Under certain necessary conditions, we ...

Generated the Integer Null Space and Conditions for Determination of an Integer Basis Using the ABS Algorithms

On the WZ Factorization of the Real and Integer Matrices

The textit{QIF}  (Quadrant Interlocking Factorization) method of Evans and Hatzopoulos solves linear equation systems using textit{WZ}  factorization. The  WZ factorization can be faster than the textit{LU} factorization  because,  it performs the simultaneous evaluation of two columns or two rows. Here, we present a  method for computing the real and integer textit{WZ} and  textit{ZW} factoriz...

Explicit formulas for five-term recurrence coefficients of orthogonal trigonometric polynomials of semi-integer degree

Orthogonal systems of trigonometric polynomials of semi-integer degree with respect to a weight function w(x) on [0, 2π) have been considered firstly by Turetzkii in [Uchenye Zapiski, Vypusk 1(149) (1959), 31–55, (translation in English in East J. Approx. 11 (2005) 337–359)]. It is proved that such orthogonal trigonometric polynomials of semi-integer degree satisfy five-term recurrence relation...