Salas, J; Sokal, AD; (2021) The graham–knuth–patashnik recurrence: symmetries and continued fractions. Electronic Journal of Combinatorics , 28 (2) , Article P2.18. 10.37236/9766.

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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 µ = (α,β,γ,α',β',γ'). We show that the family of arrays T (µ) is invariant under a 48-element discrete group isomorphic to S_{3} × D_{4}. Our main result is to determine all parameter sets µ ∈ C^{6} for which the ordinary generating function f(x, t) = ∑^{∞}_{n,k=0}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.

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