The Binet-Pexider functional equation for rectangular matrices

Summary

IfK is a field of characteristic 0 then the following is shown. Iff, g, h: M _n (K) →K are non-constant solutions of the Binet—Pexider functional equation

$$f(AB) = \frac{1}{{n!}}\sum\limits_{\left| s \right| = n} {\left( {\begin{array}{*{20}c} n \\ {s_1 \cdots s_{n{\mathbf{ }} + {\mathbf{ }}r} } \\ \end{array} } \right)g(A^s )h(B_s )}$$

for rectangular matricesA ∈ M _{n ×(n + r)} (K) andB ∈ M _{(n + r) ×n} (K), and a fixed non-negative integerr thenf(X) = ab d(X), g(X) = a d(X), andh(X) = b d(X) wherea andb are arbitrary constants fromK andd: M _n (K) →K is a non-constant solution of the Binet—Cauchy functional equation

$$d(AB) = \frac{1}{{n!}}\sum\limits_{\left| s \right| = n} {\left( {\begin{array}{*{20}c} n \\ {s_1 \cdots s_m } \\ \end{array} } \right)d(A^s )d(B_s )}$$

forA ∈ M _{n ×m} (K) andB ∈ M _{M ×n} (K) wheren ⩽ m ⩽ n + r. The general non-constant solution to (2) has been shown by Heuvers, Cummings, and K. P. S. bhaskara Rao, to bed(X) = φ(perX), whereφ is an isomorphism ofK, provided thatd(E) ≠ 0 for then × n matrixE with all entries 1/n. Heuvers and Moak have shown that the general non-constant solution to (2) is given byd(x) = μ(detX), whereμ is a non-constant multiplicative function onK ifm = n andd(E) = 0. Ifn ⩽ m ⩽n + 1 andd(E) = 0 they have shown that it is given byd(X) = φ(detX), where φ is an isomorphism ofK.