VANISHING COEFFICIENTS IN FOUR QUOTIENTS OF INFINITE PRODUCT EXPANSIONS

Motivated by Ramanujan’s continued fraction and the work of Richmond and Szekeres [‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged)40(3–4) (1978), 347–369], we investigate vanishing coefficients along arithmetic progressions in four quotients of infinite product expansions and obtain similar results. For example, $a_{1}(5n+4)=0$, where $a_{1}(n)$ is defined by $$\begin{eqnarray}\displaystyle {\displaystyle \frac{(q,q^{4};q^{5})_{\infty }^{3}}{(q^{2},q^{3};q^{5})_{\infty }^{2}}}=\mathop{\sum }_{n=0}^{\infty }a_{1}(n)q^{n}. & & \displaystyle \nonumber\end{eqnarray}$$

Continued Fractions of Order Six and New Eisenstein Series Identities

We prove two identities for Ramanujan’s cubic continued fraction and a continued fraction of Ramanujan, which are analogues of Ramanujan’s identities for the Rogers-Ramanujan continued fraction. We further derive Eisenstein series identities associated with Ramanujan’s cubic continued fraction and Ramanujan’s continued fraction of order six.

A Product of Theta-Functions Analogous to Ramanujan's Remarkable Product of Theta-Functions and Applications

We define a productlk,nfor any positive real numberskandninvolving Ramanujan's theta-functionsϕ(q)andψ(q)which is analogous to Ramanujan's remarkable product of theta-functions recorded by Ramanujan (1957) and study its several properties. We prove general theorems for the explicit evaluations oflk,nand find some explicit values. As application of the productlk,n, we also offer explicit formulas for explicit values of Ramanujan's continued fractionV(q)in terms oflk,nand give examples.

A q-CONTINUED FRACTION

We use the method of generating functions to find the limit of a q-continued fraction, with 4 parameters, as a ratio of certain q-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for (q2; q3)∞/(q; q3)∞and [Formula: see text]. In addition, we give a new proof of the famous Rogers–Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.