Let $B_t$ be real net government debt, let $G_t$ be public spending, let $T$ be net tax revenue (i.e, tax income minus transfers), and ignore everything else. One can show, $$B_{t+1} = (1+r)B_t + G - T$$
If I say that deficits are financed by debt issuance, what does this mean? How would I represent that in the formula?
Your formula can be simply reaaranged to
$$(G+rB_t)-T=B_{t+1}-B_t.$$
The left-hand side of this equation is the deficit (the difference between public spending—including the spending needed to pay interest on existing debt—and public revenue from taxation). The right-hand side is the amount of new debt issued. Thus, this equation tells you that you must issue enough new debt to pay the deficit—in other words, deficits are financed by debt issuance.
