A note on the arithmetic properties of Stern Polynomials

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We investigate the Stern polynomials defined by B₀ (t) =0,B₁ (t) =1, and for n ≥ 2 by the recurrence relations B₂ₙ(t) =tBₙ(t), B₂ₙ₊₁(t) =Bₙ(t) +Bₙ₊₁(t). We prove that all possible rational roots of that polynomials are 0,-1,-1/2,-1/3. We give complete characterization of n such that deg(Bₙ) = deg(Bₙ₊₁) and deg(Bₙ) =deg(Bₙ₊₁) =deg(Bₙ₊₂). Moreover, we present some result concerning reciprocal Stern polynomials.