The first completed-theta shell can be written
$$ \phi_1(u) =2\pi^{-1/4}y^{5/4}(2y-3)e^{-y}, \qquad y=\pi e^{2u}. $$
Factor the common envelope from its derivatives:
$$ \phi_1^{(n)}(u) =2\pi^{-1/4}y^{5/4}e^{-y}Q_n(y). $$
Then the polynomials \(Q_n\) are reflected generalized-Bell polynomials, and they form an all-order Sturm chain. Every \(Q_n\) has only simple positive zeros, and consecutive orders strictly interlace.
The proof is almost offensively short once the correct recurrence is visible.
The exact recurrence#
Since \(d/du=2y\,d/dy\),
$$ Q_{n+1} =\left(\frac52-2y\right)Q_n+2yQ_n'. $$
Define monic polynomials
$$ H_1(y)=y-\frac32, $$
$$ H_{m+1}(y) =\left(y-\frac54\right)H_m(y)-yH_m'(y). $$
Then
$$ Q_n(y)=2^{n+1}(-1)^nH_{n+1}(y). $$
This \(H_m\) is a generalized-Bell sequence with parameter list
$$ \left(\frac32,\frac54,\frac54,\ldots\right). $$
So the shell derivatives are not an isolated special-function accident. They sit inside a stable polynomial field with a fixed differential recurrence.
The one-line root sign identity#
Suppose \(H_m\) has simple positive zeros
$$ 0<\rho_1<\cdots<\rho_m. $$
At a zero \(\rho_i\), the recurrence collapses to
$$ H_{m+1}(\rho_i) =-\rho_iH_m'(\rho_i). $$
The derivative signs alternate across the simple zeros, and \(\rho_i>0\), so the values \(H_{m+1}(\rho_i)\) alternate too.
At the origin,
$$ H_{m+1}(0)=-\frac54H_m(0), $$
which creates one additional sign change before \(\rho_1\). Monicity creates one more after \(\rho_m\). There is therefore one zero of \(H_{m+1}\) in every interval
$$ (0,\rho_1),\ (\rho_1,\rho_2),\ldots,\ (\rho_{m-1},\rho_m),\ (\rho_m,\infty). $$
Degree counting says these are all the zeros. Induction from \(H_1=y-3/2\) proves simple positive roots and strict interlacing for every order.
I love proofs where evaluating a recurrence at the zeros deletes almost the whole recurrence.
A separate Chebyshev tail through rank 23#
The every-other derivative flag uses Wronskians
$$
W_r(y)=\det[Q_{2i+j}(y)]_{0\le i,j Exact elimination gives $$
W_r(y)=y^{r(r-1)/2}P_r(y).
$$ For \(2\le r\le23\), every coefficient of $$
P_r(x+2r-1)
$$ is a strictly positive integer. Hence $$
\phi_1,\phi_1'',\ldots,\phi_1^{(2r-2)}
$$ form a strict extended complete Chebyshev system on the tail \(y\ge2r-1\) through rank \(23\). The all-order Sturm theorem and the rank-23 Chebyshev theorem are distinct. Consecutive derivative interlacing does not automatically prove every-other Wronskian positivity. There is an even sharper boundary: the integrated first-shell transform itself has an off-axis zero. The first shell has gorgeous local derivative geometry, but it cannot replace the full completed-theta kernel in an RH criterion. The shells have to be assembled before the global zero geometry becomes honest. Notebook references: C-0202, C-0201, K-0062, O-0228