Recursive description of the integers $c_j(a,b)$

Let $p$ be the characteristic of the field, and let $0\leq j<a\leq b$. For $e\geq 0$, set

$$q'=p^{e-1}<a\leq q=p^e,$$

and let $r$ satisfy

$$rq\leq a+b-1-j<(r+1)q.$$

If $b-j>rq$, choose $m$ such that $mq'\leq a<(m+1)q'$, set $a'=a-mq'$, and choose $i$ such that $iq'\leq j\leq (i+1)q'-1$.

Recursive description of the integers $c_j$. If $b-j\leq rq$, then $c_j(a,b)=rq$. Otherwise,

$$c_j(a,b)=c_{j-iq'}(a',b+(m-2i)q')$$

when $j\leq iq'+a'-1$, while

$$c_j(a,b)=c_{j-iq'-a'}(q'-a',b+(m-1-2i)q')$$

when $j\geq iq'+a'$.

This recursive description is used to compute the two-fold products $\delta_a\delta_b$ and hence the products $\delta_{a_1}\cdots\delta_{a_n}$ that determine the Jordan type of the Artinian complete intersection $A={\mathbf k}[T_1,\ldots,T_n]/\langle T_1^{a_1},\ldots,T_n^{a_n}\rangle$ with respect to $T_1+\cdots+T_n$. Its status is not resolved in the supplied material.