The Graham--Knuth--Patashnik recurrence: Symmetries and continued fractions

We study the triangular array defined by the Graham–Knuth–Patashnik recurrence $T(n,k) = (\alpha n + \beta k + \gamma), T(n-1,k)+(\alpha’ n + \beta’ k + \gamma’) , T(n-1,k-1)$withinitialcondition$T(0,k) = \delta_{k0}$andparameters$$\mathbf{\mu }$ = (\alpha,\beta,\gamma, \alpha’,\beta’,\gamma’)$\mathrm{.}Weshowthatthefamilyofarrays$T($\mathbf{\mu }$)$isinvariantundera48-elementdiscretegroupisomorphicto$S_3 \times D_4$. Our main result is to determine all parameter sets $$\mathbf{\mu }$ \in $\mathbb{C}$^6$forwhichtheordinarygeneratingfunction$f(x,t) = \sum_{n,k=0}^\infty T(n,k) , x^k t^n$ is given by a Stieltjes-type continued fraction in $t$ with coefficients that are polynomials in $x$. We also exhibit some special cases in which $f(x,t)$ is given by a Thron-type or Jacobi-type continued fraction in $t$ with coefficients that are polynomials in $x$.