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24 Bernoulli and Euler PolynomialsProperties

§24.16 Generalizations


§24.16(i) Higher-Order Analogs

Polynomials and Numbers of Integer Order

For =0,1,2,, Bernoulli and Euler polynomials of order are defined respectively by

24.16.1 (tet-1)ext =n=0Bn()(x)tnn!,
24.16.2 (2et+1)ext =n=0En()(x)tnn!,

When x=0 they reduce to the Bernoulli and Euler numbers of order :

24.16.3 Bn() =Bn()(0),
En() =En()(0).

Also for =1,2,3,,

24.16.4 (ln(1+t)t)=n=0Bn(+n)+ntnn!,

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

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

Bernoulli Numbers of the Second Kind

24.16.5 tln(1+t)=n=0bntn,
24.16.6 n!bn=-1n-1Bn(n-1),

Degenerate Bernoulli Numbers

For sufficiently small |t|,

24.16.7 t(1+λt)1/λ-1=n=0βn(λ)tnn!,
24.16.8 βn(λ)=n!bnλn+k=1n/2n2kB2ks(n-1,2k-1)λn-2k,

Here s(n,m) again denotes the Stirling number of the first kind.

Nörlund Polynomials

24.16.9 (tet-1)x=n=0Bn(x)tnn!,

Bn(x) is a polynomial in x of degree n. (This notation is consistent with (24.16.3) when x=.)

§24.16(ii) Character Analogs

Let χ be a primitive Dirichlet character modf (see §27.8). Then f is called the conductor of χ. Generalized Bernoulli numbers and polynomials belonging to χ are defined by

24.16.10 a=1fχ(a)teateft-1=n=0Bn,χtnn!,
24.16.11 Bn,χ(x)=k=0n(nk)Bk,χxn-k.

Let χ0 be the trivial character and χ4 the unique (nontrivial) character with f=4; that is, χ4(1)=1, χ4(3)=-1, χ4(2)=χ4(4)=0. Then

24.16.12 Bn(x)=Bn,χ0(x-1),
24.16.13 En(x)=-21-nn+1Bn+1,χ4(2x-1).

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 (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)); p-adic integer order Bernoulli numbers (Adelberg (1996)); p-adic q-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)).