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," 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.
Best Answer
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). The solution was obtained by using generating functions.