Direct recurrence conjecture for the coefficients of Hermite subdivision schemes

We prove the stronger closed formula

$$\boxed{ \alpha^1_{k,\ell} = 2(\ell-1)!\binom{k}{\ell} } \qquad(k\ge1,\ 1\le\ell\le k). \tag{1}$$

Let $c(k,n)$ denote the unsigned Stirling number of the first kind. The auxiliary polynomials in the defining recurrence satisfy

$$q_k(-x) = \prod_{r=0}^{k-1}(-2x-r) = (-1)^k(2x)^{\overline{k}},$$

so their coefficients are

$$\gamma_n^k=(-1)^k2^n c(k,n). \tag{2}$$

Substituting (2) into the original recursion gives

$$\begin{aligned} \alpha^1_{k,k-n+1} &= (-1)^k2^{1-n} \left( n\gamma_n^k- \sum_{j=1}^{k-n} (-1)^j\alpha^1_{k,j}\gamma_{n-1}^{k-j} \right)\\ &= 2n c(k,n) - \sum_{j=1}^{k-n} \alpha^1_{k,j}c(k-j,n-1). \tag{3} \end{aligned}$$

Count permutations of $[k]$ with $n$ cycles and one distinguished cycle. There are $n c(k,n)$ such permutations. If the distinguished cycle has length $j$, its elements and cyclic order can be chosen in

$$\binom{k}{j}(j-1)!$$

ways, and the remaining elements can be arranged in $c(k-j,n-1)$ ways. Therefore

$$n c(k,n) = \sum_{j=1}^{k-n+1} \binom{k}{j}(j-1)!c(k-j,n-1). \tag{4}$$

The initial condition

$$\alpha^1_{k,1}=2k$$

agrees with (1). Assume (1) holds for all earlier indices

$$j<\ell=k-n+1.$$

Substituting these values into (3) and applying (4), all but the last marked-cycle term cancel:

$$\alpha^1_{k,\ell} = 2\binom{k}{\ell}(\ell-1)! c(k-\ell,n-1) = 2\binom{k}{\ell}(\ell-1)!,$$

because $k-\ell=n-1$. This proves (1) for all $k,\ell$.

Finally, for $2\le\ell\le k-1$, the proposed direct recurrence follows from the hockey-stick identity:

$$\begin{aligned} 2(\ell+1)(\ell-1)! + (\ell-1) \sum_{i=\ell}^{k-2} \alpha^1_{i+1,\ell-1} &= 2(\ell-1)! \left( \ell+1+ \sum_{t=\ell+1}^{k-1} \binom{t}{\ell-1} \right)\\ &= 2(\ell-1)!\binom{k}{\ell}\\ &= \alpha^1_{k,\ell}. \end{aligned}$$

The boundary values are likewise

$$\alpha^1_{k,1}=2k,\qquad \alpha^1_{k,k}=2(k-1)!,$$

and in particular

$$\alpha^1_{k,k-1}=2k(k-2)!.$$

Thus the complete direct-recurrence conjecture holds, together with the stronger explicit closed formula.