I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by y = 1/2$y=\frac{1}{2}$, 1$1$ and 2$2$) below. The common expression will be of fourth-degree in k $k$ and of arbitrary degree (probably between 3$3$ and 6$6$) in y$y$.
For y = 1/2$y=\frac{1}{2}$ (the real case), the polynomial is
-(15/4) (294 + 413 k + 213 k^2 + 48 k^3 + 4 k^4) = -(15/4) (2 + k) (3 + k)$-(y=\frac{15}{4})$ (7 + 2 k)^2.$(294 + 413 k + 213 k^2 + 48 k^3 + 4 k^4) = -(15/4) (2 + k) (3 + k) (7 + 2 k)^2.$
For y = 1 $y = 1$ (the complex case), the polynomial is
-24 (990 + 873 k + 280 k^2 + 39 k^3 + 2 k^4) = -24 (3 + k) (5 + k) (6 + k) (11 + 2 k).$-24 (990 + 873 k + 280 k^2 + 39 k^3 + 2 k^4) = -24 (3 + k) (5 + k) (6 + k) (11 + 2 k).$
For y = 2$y = 2$ (the quaternionic case), the polynomial (NOT now the product of linear factors) is
-60 (13134 + 6925 k + 1291 k^2 + 103 k^3 + 3 k^4) =
-60 (11 + k) (1194 + k (521 + k (70 + 3 k))).$-60 (13134 + 6925 k + 1291 k^2 + 103 k^3 + 3 k^4) = -60 (11 + k) (1194 + k (521 + k (70 + 3 k))).$
A common expression for y = 1/2$y=\frac{1}{2}$ and 1$1$ is
-4 y (1 + y) (2 + y) (1 + k + 2 y) (1 + k + 4 y) (1 + k + 5 y) (3 + 2 k + 8 y),$-4 y (1 + y) (2 + y) (1 + k + 2 y) (1 + k + 4 y) (1 + k + 5 y) (3 + 2 k + 8 y),$
but not for y = 2.$y = 2.$
This question has pertinence to random matrix theory and quantum information topics (cf. http://arxiv.org/abs/1109.2560Arxiv).
