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Divided Differences and Linearly Recursive Sequences
Author(s) -
VerdeStar Luis
Publication year - 1995
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1995954433
Subject(s) - mathematics , monic polynomial , degree (music) , isomorphism (crystallography) , polynomial , divided differences , space (punctuation) , algebra over a field , operator (biology) , pure mathematics , vector space , simple (philosophy) , combinatorics , discrete mathematics , mathematical analysis , linguistics , chemistry , physics , philosophy , biochemistry , epistemology , acoustics , crystal structure , transcription factor , gene , crystallography , repressor
We show that the theory of divided differences is a natural tool for the study of linearly recurrent sequences. The divided differences functional associated with a monic polynomial w on degree n + 1 yields a vector space isomorphism between the space of polynomials of degree at most equal to n and the space of linearly recurrent sequences f that satisfy the difference equation w ( E ) f =0 where E is the usual shift operator. Using such isomorphisms, we can translate problems about recurrent sequences into simple problems about polynomials. We present here a new approach to the theory of divided differences, using only generating functions and elementary linear algebra, which clarifies the connections of divided differences with rational functions, polynomial interpolation, residues, and partial fractions decompositions.