Generalisation of a waiting-time relation (Q1378666)

The main result of the paper is the equality \[ \sum_{n=0}^\infty(-1^n) {(abe^{ab})^n(t+n)^{n+R-1}\over{n!}} =e^{-abt}\sum_{k=0}^{R-1} Q_{-(R-k)}(-1)^k{R-1\choose k}{t^{R-k-1}\over a^k}. \] It is a generalisation of the waiting-time relation corresponding to the case \(R=1\). The left side of the equality is the solution of a certain differential-difference equation obtained via Laplace transform techniques and is important in a number of applications. Its representation in a closed form is first conjectured using the residue theorem and then proved strictly using Burmann's theorem on expansion of a function in positive powers of another function. A recurrence relation for \(Q_r\) and some examples are given.