Reflection Symmetries of q-Bernoulli Polynomials

Abstract A large part of the theory of classical Bernoulli polynomials B n (x)’s follows from their reflection symmetry around x = 1/2: B n (1 − x) = (−1) n B n (x). This symmetry not only survives quantization but has two equivalent forms, classical and quantum, depending upon whether one reflects around 1/2 the classical x or quantum [x] q .