The Graham–Knuth–Patashnik Recurrence: Symmetries and Continued Fractions

We study the triangular array defined by Graham–Knuth–Patashnik recurrence $T(n,k) = (\alpha n + \beta k \gamma)\, T(n-1,k)+(\alpha' \beta' \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) \delta_{k0}$ and parameters $\mathbf{\mu} (\alpha,\beta,\gamma, \alpha',\beta',\gamma')$. show that family of arrays $T(\mathbf{\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result determine all parameter sets \in \mathbb{C}^6$ for which ordinary generating function $f(x,t) \sum_{n,k=0}^\infty T(n,k) x^k t^n$ given Stieltjes-type continued fraction in $t$ coefficients are polynomials $x$. also exhibit some special cases $f(x,t)$ Thron-type or Jacobi-type