Solutions of linear recurrence equations

Solutions are given to general homogeneous and non-homogeneous recurrence equations defined on the set of integers. Solutions to homogeneous recurrence equations are given as: (i) an infinite series of terms involving the partial ordinary Bell polynomial; (ii) an infinite series of terms involving the complete ordinary Bell polynomial; (iii) a weighted finite sum of terms involving powers. Solutions to non-homogeneous recurrence equations are given as: (i) a finite series of terms involving the complete ordinary Bell polynomial; (ii) a weighted infinite sum of terms involving powers. Computational issues of these solutions are also discussed.

[1]  Donato Trigiante,et al.  THEORY OF DIFFERENCE EQUATIONS Numerical Methods and Applications (Second Edition) , 2002 .

[2]  A. J. Jerri.  Linear Difference Equations with Discrete Transform Methods , 1996 .

[3]  Geertrui Klara Immink,et al.  Asymptotics of Analytic Difference Equations , 1984 .

[4]  Linear recurrence relations for binomial coefficients modulo a prime , 2005, math/0511417.

[5]  One and Two-Dimension Coupling Simulation for Dam-Break Flood , 2009 .

[6]  Robert McGinnis,et al.  Introduction to Difference Equations. With illustrative examples from Economics, Psychology, and Sociology. , 1959 .

[7]  David L. Jagerman.  Difference equations with applications to queues , 2000 .

[8]  H. Feshbach,et al.  Finite Difference Equations , 1959 .

[9]  Przemyslaw Stpiczynski,et al.  Solving Linear Recurrence Systems Using Level 2 and 3 BLAS Routines , 2003, PPAM.

[10]  K. S. Miller,et al.  An Introduction to the Calculus of Finite Differences and Difference Equations. , 1960 .

[11]  A. Nikoghossian.  Radiative Transfer in Inhomogeneous Atmospheres. III , 2004 .

[12]  Marko Petkovšek,et al.  Solving Linear Recurrence Equations with Polynomial Coefficients , 2013 .

[13]  F. H. Jackson.  q-Difference Equations , 1910 .

[14]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[15]  Kenneth S. Miller,et al.  Linear difference equations , 1968 .

[16]  Sanjoy K. Mitter,et al.  The Solution of Linear Probabilistic Recurrence Relations , 2002, Algorithmica.

[17]  S. Elaydi.  An introduction to difference equations , 1995 .

[18]  M. Rachidi,et al.  Linear recurrence relations in the algebra of matrices and applications , 2001 .

[19]  Wayne Luk,et al.  High Quality Uniform Random Number Generation Using LUT Optimised State-transition Matrices , 2007, J. VLSI Signal Process..

[20]  Jet Wimp,et al.  Computation with recurrence relations , 1986 .

[21]  A. Peterson,et al.  Difference Equations: An Introduction with Applications , 2000 .

[22]  Mark van Hoeij,et al.  Solving recurrence relations using local invariants , 2010, ISSAC.

[23]  Johann A. Makowsky,et al.  Linear Recurrence Relations for Graph Polynomials , 2008, Pillars of Computer Science.