DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS IN GENUS(15)

Let #(u) and #(u) be the usual functions in the theory of elliptic functions. The following two formulae were found in the nineteenth-century. First one is

HYPERELLIPTICFUNCTIONSINGENUSTHREE15

Proof.Becauseof3.1wehave

x(u) x(v)

σ2(u)2σ2(v)2·σ3(u v)

u(j) v(j)

xj 1=dx(v)by(2.2).Theassertionfollowsfrom2.12.

SinceourproofofthefollwingTheoremobtainedbyquitesimilarargumentbyusing4.1asinthecaseofgenustwo(see[13]),weleavetheprooftothereader.

Theorem4.2.Letnbeanintegergreaterthan3.Letjbeanyoneof{1,2,3}.Assumethatubelongstoκ 1ι(C).Thenthefollowingformulaforthefunctionψn(u)of2.13holds:

(1!2!···(n 1)!)ψn(u)=x(j 1)n(n 1)/2(u)× (x2)′(x3)′y′(x4)′ x′

′′(x2)′′(x3)′′y′′(x4)′′ x ′′′(x2)′′′(x3)′′′y′′′(x4)′′′ x ..... ......... . (n 1)x(x2)(n 1)(x3)(n 1)y(n 1)(x4)(n 1) ··· ··· ··· (u)... . ···(yx)′(yx)′′(yx)′′′...

(yx)(n 1)(x5)′(x5)′′(x5)′′′...(x5)(n 1)

Herethesizeofthematrixisn 1byn 1.Thesymbols′,′′,···, 2 dd,···,du(j)(n 1)denote