  A new class of integrals involving the confluent hypergeometric function
${}_1F_{1}(a;c;z)$ and the Riemann $\Xi$-function is considered. It generalizes
a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar
and gives as by-products, transformation formulas of the form
$F(z,\alpha)=F(iz,\beta)$, where $\alpha\beta=1$. As particular examples, we
derive an extended version of the general theta transformation formula and
generalizations of certain formulas of Ferrar and Hardy. A one-variable
generalization of a well-known identity of Ramanujan is also given. We conclude
with a generalization of a conjecture due to Ramanujan, Hardy and J.E.
Littlewood involving infinite series of M\"obius functions.
