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

Abstract

We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (αn + βk + γ)\, T(n-1,k)+(α' n + β' k + γ') \, T(n-1,k-1)$ with initial condition $T(0,k) = δ_{k0}$ and parameters $\mathbfμ = (α,β,γ, α',β',γ')$. We show that the family of arrays $T(\mathbfμ)$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result is to determine all parameter sets $\mathbfμ \in \mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \sum_{n,k=0}^\infty T(n,k) \, x^k t^n$ 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 $x$.