Combinatorial and modular solutions of some sequences with links to a certain conformal map
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Pablo A. Panzone [1]
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[1] Universidad Nacional del Sur
Universidad Nacional del Sur
Argentina
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- Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 59, Nº. 2, 2018, págs. 389-414
- Idioma: inglés
- DOI: 10.33044/revuma.v59n2a09
- Enlaces
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Resumen
If fn is a free parameter, we give a combinatorial closed form solution of the recursion (n + 1)2un+1 − fnun − n 2un−1 = 0, n ≥ 1, and a related generating function. This is used to give a solution to the Apéry type sequence rnn3 + rn−1nαn3 −3α2n+α+ 2θon − θo+ rn−2(n − 1)3 = 0, n ≥ 2, for certain parameters α, θ. We show from another viewpoint two independent solutions of the last recursion related to certain modular forms associated with a problem of conformal mapping: Let f(τ) be a conformal map of a zero-angle hyperbolic quadrangle to an open half plane with values 0, ρ, 1, ∞ (0 < ρ < 1) at the cusps and define t = t(τ) := 1ρf(τ)f(τ)−ρ(τ)−1. Then the function E(τ) = 12πif0(τ)f(τ)11 −f(τ)ρ is a solution, as a generating function in the variable t, of the above recurrence. In other words, E(τ) = r0 +r1t+r2t 2 +. . . , where r0 = 1, r1 = −θ, α = 2− 4ρ.
