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Decomposable Forms Generated by Linear Recurrences
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Kálmán Győry

Institute of Mathematics

University of Debrecen

Egyetem tér 1

Debrecen H4032

Hungary

Attila Pethő

Institute of Computer Sciences

University of Debrecen

Kassai út 26

Debrecen H4028

Hungary

László Szalay

Department of Mathematics

J. Selye University

Hradná 21

Komárno 945 01

Slovakia

and

Institute of Informatics and Mathematics

University of Sopron

Bajcsy Zs. utca 4

Sopron H9400

Hungary

**Abstract:**

Consider *k* ≥ 2 distinct, linearly independent, homogeneous
linear recurrences of order *k* satisfying the same recurrence
relation. We prove that the recurrences are related to a decomposable form
of degree *k*, and there is a general identity with a suitable
exponential expression depending on the recurrences. This identity is a
common and very broad generalization of several known identities. Further,
if the recurrences are integer sequences, then the diophantine equation
associated with the decomposable form and the exponential term
has infinitely many integer solutions generated by the terms of the
recurrences. We describe a method for the complete factorization of the
decomposable form. Both the form and its decomposition are explicitly
given if *k* = 2, and we present a typical example for *k* =
3. The basic tool we use is the matrix method.

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(Concerned with sequences
A000032
A000045
A000930
A001609.)

Received November 10 2023; revised versions received February 14 2024; February 22 2024.
Published in *Journal of Integer Sequences*,
March 1 2024.

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