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February 21st, 2019, 10:46 PM   #1
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Recurrence relation

For second order linear homogeneous recurrence relation with constant coefficients, is it that the general sequence is only in the form of 1, t, t^2, t^3, ..., t^n or just one of the general sequences which satisfies the characteristic equation? Thank you.
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February 21st, 2019, 11:35 PM   #2
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See this article.
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February 22nd, 2019, 12:13 AM   #3
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Thank you for the reference. It seems that the characteristic equation for a second order linear homogeneous recurrence relation with constant coefficients should be a Quadratic equation.
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February 22nd, 2019, 06:40 AM   #4
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Can I say that it is first assumed that the indexed term a(n)=t^n, and it happens that the assumption satisfies the recurrence relation by solving the characteristic equation?
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February 22nd, 2019, 06:49 AM   #5
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Yes, it turns into a polynomial equation.
But I don’t know the reason behind it; let someone else prove it or give an article of proof.

Last edited by skipjack; February 22nd, 2019 at 09:31 AM.
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February 22nd, 2019, 09:29 AM   #6
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If the characteristic equation has distinct roots p and q,
the nth term of the sequence is a(p^n) + b(q^n), where a and b are constants.
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