Some Combinatorial Identities of Bernstein
L. Carlitz · SIAM Journal on Mathematical Analysis · 1978
For g a rational integer such that $\Delta = 4g^3 + 27$ is square-free, let w denote the real root of $u^3 + gu - 1 = 0$ and put $w^n = r_n + s_n w + t_n w^2 $, $w^{ - n} = x_n + y_n w + z_n w^2 $, $n \geqq 0$. Making use of the theory of units in an algebraic number field, Bernstein obtained quadratic relations involving the $r_n $ and $x_n $ as well as explicit formulas. These lead to certain combinatorial identities. In the present paper these and related identities are proved using only some elementary algebra.