  The chemical master equation (CME), which describes the discrete and
stochastic molecule number dynamics associated with biological processes like
transcription, is difficult to solve analytically. It is particularly hard to
solve for models involving bursting/gene switching, a biological feature that
tends to produce heavy-tailed single cell RNA counts distributions. In this
paper, we present a novel method for computing exact and analytic solutions to
the CME in such cases, and use these results to explore approximate solutions
valid in different parameter regimes, and to compute observables of interest.
Our method leverages tools inspired by quantum mechanics, including ladder
operators and Feynman-like diagrams, and establishes close formal parallels
between the dynamics of bursty transcription, and the dynamics of bosons
interacting with a single fermion. We focus on two problems: (i) the chemical
birth-death process coupled to a switching gene/the telegraph model, and (ii) a
model of transcription and multistep splicing involving a switching gene and an
arbitrary number of downstream splicing steps. We work out many special cases,
and exhaustively explore the special functionology associated with these
problems. This is Part I in a two-part series of papers; in Part II, we explore
an alternative solution approach that is more useful for numerically solving
these problems, and apply it to parameter inference on simulated RNA counts
data.
