The Weierstrass-Whittaker integral transform (Q2927722)

The authors introduce the Weierstrass-Whittaker integral transform, defined as NEWLINE\[NEWLINE [\mathcal{W}_tf](x)=\int_0^\infty \mathcal{K}_t(x,y)f(y)e^{-\left(y+\frac{1}{y}\right)}y^\alpha dy, NEWLINE\]NEWLINE where \(\mathcal{K}_t(x,y)\) is the heat kernel associated with the Whittaker transform. It is defined as NEWLINE\[NEWLINE \mathcal{K}_t(x,y)=\int_0^\infty e^{-4\nu^2\tau t}e^{\frac{-y\tau}{2}}W_{\mu,\nu}(y\tau)e^{\frac{-x\tau}{2}}W_{\mu,\nu}(x\tau)e^{-\left(\tau+\frac{1}{\tau}\right)} \tau^\alpha d\tau, NEWLINE\]NEWLINE for \(t, x, y>0\). They study some properties of the transform and show that it is useful in solving a generalized non-stationary heat equation with an initial condition.