The

theta function of a characteristic m of degree g is the series

Ramanujan's general

theta function f(a, b) is defined by

Assuming that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are two Riemann

theta functions with [xi] = [alpha]x + [beta]y + wt + [sigma], then Hirota bilinear operators [D.sub.x], [D.sub.y], and [D.sub.t] exhibit the following perfect properties when they act on a pair of

theta functions:

The following trigonometric series expressions for the logarithmic derivative with respect to z of Jacobi

Theta functions will be very useful in this paper,

In the letter, Ramanujan defined four third order mock

theta functions, ten fifth order mock

theta functions and three seventh order mock

theta functions.

Key words: Jacobian

theta functions, especially products; normalizing factors.

The two subjects, elliptic functions and

theta functions, developed in the nineteenth century, are intimately connected with each other.

Mock

theta functions are mysterious functions and not much is known about them.

In the years before his death in 1920, Ramanujan studied

theta functions, which are numerical relationships that show special symmetries.

This information is obtained with the help of a relation that generalises the reciprocity law for Jacobi's

theta functions (see Lemma 2).

SAFE, Convergence of Pade approximants of partial

theta functions and the Rogers-Szego polynomials, Constr.

His topics include the q-binomial theorem, Heine's transformation, the Jacobi triple product identity, the Rogers-Fine identity, Bailey chains, WP-Bailey pairs and chains, further results on Bailey/WP-Bailey pairs and chains, bijective proofs of basic hypergeometric identities, q-continued fractions, Lambert series, and mock

theta functions. ([umlaut] Ringgold, Inc., Portland, OR)