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- W3047660089 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 $x$." @default.
- W3047660089 created "2020-08-13" @default.
- W3047660089 creator A5057050233 @default.
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- W3047660089 date "2021-05-07" @default.
- W3047660089 modified "2023-09-26" @default.
- W3047660089 title "The Graham–Knuth–Patashnik Recurrence: Symmetries and Continued Fractions" @default.
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- W3047660089 doi "https://doi.org/10.37236/9766" @default.
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