The Factorial-Basis Method for Finding Definite-Sum Solutions of Linear Recurrences With Polynomial Coefficients
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The problem of finding a nonzero solution of a linear recurrence Ly = 0 with polynomial coefficients where y has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of [14][Sec. 8], has now been open for three decades. Here we present an algorithm (implemented in a SageMath package) which, given such a recurrence and a quasi-triangular, shift-compatible factorial basis ℬ = ⟨ P_k(n)⟩_k=0^∞ of the polynomial space 𝕂[n] over a field 𝕂 of characteristic zero, computes a recurrence satisfied by the coefficient sequence c = ⟨ c_k⟩_k=0^∞ of the solution y_n = ∑_k=0^∞ c_kP_k(n) (where, thanks to the quasi-triangularity of ℬ, the sum on the right terminates for each n ∈ℕ). More generally, if ℬ is m-sieved for some m ∈ℕ, our algorithm computes a system of m recurrences satisfied by the m-sections of the coefficient sequence c. If an explicit nonzero solution of this system can be found, we obtain an explicit nonzero solution of Ly = 0.
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