ARITHMETIC PROPERTIES OF THE SEQUENCE OF DEGREES OF STERN POLYNOMIALS AND RELATED RESULTS

Let B-n(t) be an nth Stern polynomial and let e(n) = deg B-n(t) be its degree. In this note we continue our study started in [On certain arithmetic properties of Stern polynonials, Publ. Math. Debrecen 79(1-2) (2011) 55-81] of the arithmetic properties of the sequence of Stern polynomials and the sequence {e(n)}(n=1)(infinity). We also study the sequence d(n) = ord(t=0) B-n(t). Among other things we prove that d(n) = nu(n), where nu(n) is the maximal power of 2 which divides the number n. We also count the number of the solutions of the equations e(m) = i and e(m) - d(m) = i in the interval [1, 2(n)]. We also obtain an interesting closed expression for a certain sum involving Stern polynomials.