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Max Alekseyev
Max Alekseyev
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Generalization of Lucas sequences to order 3 and(and above)

For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The two sequences can be defined via roots $x_{1,2}$ of $f(x)$ as $U_n:=\frac{x_1^n - x_2^n}{x_1-x_2}$ and $V_n:=x_1^n + x_2^n$.

Lucas sequences form a linear basis for the other sequences satisfying the same recurrence, and are tightly interconnected with numerous identities connecting arithmetic operations on terms to those on their indices.

Q: What would be a generalization of Lucas sequences to integer recurrences of higher order, in particular of order 3 with a characteristic polynomial $g(x):=x^3 - Px^2 + Qx - R$ with integer coefficients?

Generalized Lucas sequences should be integer, form a basis for the other sequences with the same characteristic polynomial, and have nice interconnecting propertiesrelations (generalizing those for order 2). I assume one

A natural extension of the sequences should stillorder 2 case seems to be the$U_n := h_{n-1}$ and $V_n := p_n$ (ie. complete homogeneous and power sum symmetric polynomial ofsymmetric polynomials) evaluated at the roots of $g(x)$, but. What other sequence(s) should we include into consideration and what would be the otherscorresponding relations?

Generalization of Lucas sequences to order 3 and above

For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The two sequences can be defined via roots $x_{1,2}$ of $f(x)$ as $U_n:=\frac{x_1^n - x_2^n}{x_1-x_2}$ and $V_n:=x_1^n + x_2^n$.

Lucas sequences form a linear basis for the other sequences satisfying the same recurrence, and are tightly interconnected with numerous identities connecting arithmetic operations on terms to those on their indices.

Q: What would be a generalization of Lucas sequences to integer recurrences of higher order, in particular of order 3 with a characteristic polynomial $g(x):=x^3 - Px^2 + Qx - R$ with integer coefficients?

Generalized Lucas sequences should be integer, form a basis for the other sequences with the same characteristic polynomial, and have nice interconnecting properties. I assume one of the sequences should still be the power sum symmetric polynomial of the roots of $g(x)$, but what would be the others?

Generalization of Lucas sequences to order 3 (and above)

For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The two sequences can be defined via roots $x_{1,2}$ of $f(x)$ as $U_n:=\frac{x_1^n - x_2^n}{x_1-x_2}$ and $V_n:=x_1^n + x_2^n$.

Lucas sequences form a linear basis for the other sequences satisfying the same recurrence, and are tightly interconnected with numerous identities connecting arithmetic operations on terms to those on their indices.

Q: What would be a generalization of Lucas sequences to integer recurrences of higher order, in particular of order 3 with a characteristic polynomial $g(x):=x^3 - Px^2 + Qx - R$ with integer coefficients?

Generalized Lucas sequences should be integer, form a basis for the other sequences with the same characteristic polynomial, and have nice interconnecting relations (generalizing those for order 2).

A natural extension of the order 2 case seems to be $U_n := h_{n-1}$ and $V_n := p_n$ (ie. complete homogeneous and power sum symmetric polynomials) evaluated at the roots of $g(x)$. What other sequence(s) should we include into consideration and what would the corresponding relations?

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Max Alekseyev
Max Alekseyev
  • 34.3k
  • 5
  • 74
  • 152

Generalization of Lucas sequences to order 3 and above

For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The two sequences can be defined via roots $x_{1,2}$ of $f(x)$ as $U_n:=\frac{x_1^n - x_2^n}{x_1-x_2}$ and $V_n:=x_1^n + x_2^n$.

Lucas sequences form a linear basis for the other sequences satisfying the same recurrence, and are tightly interconnected with numerous identities connecting arithmetic operations on terms to those on their indices.

Q: What would be a generalization of Lucas sequences to integer recurrences of higher order, in particular of order 3 with a characteristic polynomial $g(x):=x^3 - Px^2 + Qx - R$ with integer coefficients?

Generalized Lucas sequences should be integer, form a basis for the other sequences with the same characteristic polynomial, and have nice interconnecting properties. I assume one of the sequences should still be the power sum symmetric polynomial of the roots of $g(x)$, but what would be the others?