For , Bernoulli and Euler polynomials of order are defined respectively by
When they reduce to the Bernoulli and Euler numbers of order :
Also for ,
For extensions of to complex values of , , and , and also for uniform asymptotic expansions for large and large , see Temme (1995b).
For sufficiently small ,
Here again denotes the Stirling number of the first kind.
is a polynomial in of degree . (This notation is consistent with (24.16.3) when .)
Let be a primitive Dirichlet character (see §27.8). Then is called the conductor of . Generalized Bernoulli numbers and polynomials belonging to are defined by
Let be the trivial character and the unique (nontrivial) character with ; that is, , , . Then
For further properties see Berndt (1975a).
In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); -adic integer order Bernoulli numbers (Adelberg (1996)); -adic -Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli-Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).