Zeros and S-units in sums of terms of recurrence sequences in function fields

Let (Uₙ)n≥ 0 be a non-degenerate linear recurrence sequence with order at least two defined over a function field and OS^* be the set of S-units. In this paper, we use a result of Brownawell and Masser to prove effective results related to the Diophantine equations concerning linear recurrence sequences and S-units. In particular, we provide a finiteness result for the solutions of the Diophantine equation Un₁ + ⋯ + Unᵣ ∈ OS^* in nonnegative integers n₁, …, nᵣ. Furthermore, we study the finiteness result of the Diophantine equation Un₁⁽¹⁾+ Un₂⁽²⁾+⋯+Unₖ⁽ᵏ⁾ = 0 in (n₁, …, nₖ)∈ ᵏ, where Uₙ⁽¹⁾, …, Uₙ⁽ᵏ⁾ are simple linear recurrence sequences in the function field.