What is known about explicit (not necessarily closed-form) solutions to the recurrence
$$R^n\_k= (\alpha n) R^{n-1}\_k + (\alpha' n + \beta' k) R^{n-1} \_{k-1},$$
with initial condition $R\_0^0 = 1$ and with $R^n\_k = 0$ for $n < 0$ or $k < 0$? Special cases of this are closely related to recurrences satisfied by some interesting combinatorial numbers, such as the binomial coefficients and the Stirling numbers.


The more general recurrence
$$R^n\_k= (\alpha n + \beta k + \gamma) R^{n-1}\_k + (\alpha' n + \beta' k + \gamma') R^{n-1} \_{k-1},$$


is open Problem 6.94 in *Concrete Mathematics* (2nd edition, p. 319).


The closest published result I have found thus far is the following formula due to Neuwirth ("[Recursively defined combinatorial functions: Extending Galton's board](http://homepage.univie.ac.at/erich.neuwirth/papers/TechRep99-05.pdf)," *Discrete Mathematics*, 2001) for the case $\alpha' = 0$ of the *Concrete Mathematics* problem,


$$R^n\_k = \prod\_{i=1}^k (\beta' i + \gamma') \sum\_{i=0}^n \sum\_{j=0}^n s^n\_i \binom{i}{j} S^j\_k \alpha^{n-i} (\gamma - \alpha)^{i-j} \beta^{j-k},$$


which, of course, gives me an answer to my question when $\alpha'=0$. (Here, $s^n\_i$ and $S^j\_k$ are unsigned Stirling numbers of the first and second kinds, respectively.)


I have tried generating functions without any success thus far. An answer like Neuwirth's that involves sums and binomial coefficients or Stirling numbers would be fine, as would a partial answer or just another idea to try.



Solving a general two-term combinatorial recurrence relation
A general solution of the Graham-Knuth-Patashnik 6.94 problem 


$$
R^{n}\_k=(\alpha \, n + \beta\, k + \gamma)\, R^{n−1}\_k+ (\alpha′\, n+ \beta′\, k +\gamma′)\, R^{n−1}\_{k−1} + \delta\_{n,0}\delta\_{k,0}\,, 
$$


with $R\_{n}^k=0$ if $n<0$ or $k<0$, can be found in the paper ``Bivariate generating functions for a class of linear recurrences: General structure'', by J.F. Barbero G., J. Salas, and E.J.S. Villaseñor, published in J. Combin. Theory A 125 (2014) 146-165 (see also 
[arXiv:1307.2010](http://arxiv.org/abs/1307.2010)).
The solution was obtained by using generating functions.


