# §24.16 Generalizations

## §24.16(i) Higher-Order Analogs

### ¶ Polynomials and Numbers of Integer Order

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 this and other properties see Milne-Thomson (1933, pp. 126–153) or Nörlund (1924, pp. 144–162).

For extensions of to complex values of , , and , and also for uniform asymptotic expansions for large and large , see Temme (1995b).

### ¶ Degenerate Bernoulli Numbers

For sufficiently small ,

24.16.7

Here again denotes the Stirling number of the first kind.

### ¶ Nörlund Polynomials

is a polynomial in of degree . (This notation is consistent with (24.16.3) when .)

## §24.16(ii) Character Analogs

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).

## §24.16(iii) Other Generalizations

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 (1954b), 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)).