that the sum converges, so that g(x) exists, one then has that it obeys the multiplication theorem; that is, that That is, g(x) is an eigenfunction of Bernoulli transfer operator, with eigenvalue f(k). The multiplication theorem for the Bernoulli polynomials then follows as a special case of the multiplicative function . The Dirichlet characters are fully multiplicative, and thus can be readily used to obtain additional identities of this form. Characteristic zero The multiplication theorem over a field of characteristic zero does not close after a finite number of terms, but requires an infinite series to be expressed. Examples include that for the Bessel function : where and may be taken as arbitrary complex numbers. Such characteristic-zero identities follow generally from one of many possible identities on the hypergeometric series. Notes References Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York. (Multiplication theorems are individually listed chapter by chapter) C. Truesdell, "On the Addition and Multiplication Theorems for the Special Functions", Proceedings of the National Academy of Sciences, Mathematics, (1950) pp. 752–757. Special functions Zeta and L-functions Gamma and related functions Mathematical theorems