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

TL;DR

This paper investigates the symmetries of the Graham--Knuth--Patashnik recurrence array and characterizes when its generating function can be expressed as specific continued fractions, revealing deep algebraic structures and special cases.

Contribution

It identifies the symmetry group of the recurrence array family and classifies parameter sets for which the generating function has continued fraction representations.

Findings

The array family is invariant under a 48-element symmetry group.

Characterization of parameters yielding Stieltjes continued fractions.

Identification of special cases with Thron or Jacobi continued fractions.

Abstract

We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\alpha n + \beta k + \gamma)\, T(n-1,k)+(\alpha' n + \beta' k + \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) = \delta_{k0}$ and parameters $\mathbf{\mu} = (\alpha,\beta,\gamma, \alpha',\beta',\gamma')$. We show that the family of arrays $T(\mathbf{\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result is to determine all parameter sets $\mathbf{\mu} \in \mathbb{C}^6$ 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 $x$.