I am trying to find estimates for partial derivatives of a function. While I have found the parts containing the asymptotics, I am struggling to find the correct coefficients.
My calculations have led me to the recurrence relation
\begin{equation} y^{m+1}_{c+1} = \left( m + 2 c \right) y^m_{c+1} + y^m_c \end{equation}
for $m, c \in \mathbb{N}$ with the boundary conditions
\begin{aligned} y^1_2 &= 1 \\ y^1_{2+n} &= 0 \\ y^m_1 &= 0 \end{aligned}
for all $n \in \mathbb{N}$. It can be viewed as a generalized Pascal triangle where one of the items in the previous row is multiplied by a non-constant factor. (The troublesome factor comes from the application of a partial derivative to a monomial)
I seem to be unable to solve the recurrence relation. Any ideas?
$$ y_{c+1}^{m+1} = (m+2c)y_{c+1}^m +y_c^m = (m+2c)[(m-1+2c)y_{c+1}^{m-1} + y_c^{m-1}] + y_{c}^m = ...\quad = (m+2c)_m \cdot y_{c+1}^1 + \sum_{k=0}^{m-1} (m+2c)_{k} \cdot y_{c}^{m-k} $$ where $(x)_k = x(x-1)\cdots (x-k+1) $ is the falling Pochhammer symbol and $(x)_0=1$. If $c>1$, then $y_{c+1}^1 = 0$, hence the first term cancels. If $c=1$, then $y_1^m=0$, hence $$ y_2^{m+1} = (m+2)_m .$$ $$ y_3^{m+1} = \sum_{k=0}^{m-1}(m+4)_k \cdot y_2^{m-k} $$ From here things get ugly, I'm not sure if there is a nice solution, but I might be wrong.