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)