Zhang-Zhang Polynomials of Multiple Zigzag Chains

Abstract

Generating functions of the Zhang-Zhang polynomials of multiple zigzag chains Z(m, n) and generalized multiple zigzag chains Z(k)(m, n) are derived for arbitrary values of the indices. These generating functions can be expressed in the form of highly regular finite continued fractions, Sigma(infinity)(m=0) ZZ(Z(m, n),z)t(m) = [0; -t,(-1)(2)zt,(-1)(3)zt,&,(-1)(n)zt, 1 + (-1)(n+1) zt], or, in the case of Z(k)(m, n), products of such continued fractions. For the particularly important case of the multiple zigzag chains Z(m, n), the generating functions are expanded to yield a closed form for the Zhang-Zhang polynomials of multiple zigzag chains Z(m, n) that is valid for arbitrary values of m and n.