  A new generalization of the modified Bessel function of the second kind
$K_{z}(x)$ is studied. Elegant series and integral representations, a
differential-difference equation and asymptotic expansions are obtained for it
thereby anticipating a rich theory that it may possess. The motivation behind
introducing this generalization is to have a function which gives a new pair of
functions reciprocal in the Koshliakov kernel $\cos \left( {{\pi z}}
\right){M_{2z}}(4\sqrt {x} ) - \sin \left( {{\pi z}} \right){J_{2z}}(4\sqrt {x}
)$ and which subsumes the self-reciprocal pair involving $K_{z}(x)$. Its
application towards finding modular-type transformations of the form $F(z, w,
\alpha)=F(z,iw,\beta)$, where $\alpha\beta=1$, is given. As an example, we
obtain a beautiful generalization of a famous formula of Ramanujan and Guinand
equivalent to the functional equation of a non-holomorphic Eisenstein series on
$SL_{2}(\mathbb{Z})$. This generalization can be considered as a higher level
analogue of the general theta transformation formula. We then use it to
evaluate an integral involving the Riemann $\Xi$-function and consisting of a
sum of products of two confluent hypergeometric functions.
