Stern polynomials (Q2383531)

Stern polynomials \(B_k(t)\) (\(k\geq 0\), \(t\) are real numbers), are introduced in the following way: \[ B_0(t) = 0,\;B_1(t) = 1,\;B_{2n}(t) = tB_n(t),\;B_{2n+1}(t) = B_{n+1}(t) + B_n(t). \] Some properties and applications of these polynomials are formulated and discussed, e.g., for any \(m\geq 0\) and any \(r\geq 1\) \[ B_{2^{t-1}(2m+1)}(t) = \tfrac 1t\left(B_{2^tm}(t) + B_{2^t(m+1)}(t)\right). \]