24 Bernoulli and Euler Polynomials Properties 24.15 Related Sequences of Numbers 24.17 Mathematical Applications

§24.16 Generalizations

Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.16

§24.16(i) Higher-Order Analogs
§24.16(ii) Character Analogs
§24.16(iii) Other Generalizations

§24.16(i) Higher-Order Analogs

Notes:: For (24.16.6), (24.16.7), and (24.16.8) see Howard (1996a).
Referenced by:: §5.11(iii), ¶ ‣ §8.18(ii)
Permalink:: http://dlmf.nist.gov/24.16.i

¶ Polynomials and Numbers of Integer Order

Defines:: $\mathop{E^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Euler polynomials and $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Bernoulli polynomials

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

24.16.1 $\left(\frac{t}{e^{t}-1}\right)^{\ell}e^{{xt}}=\sum_{{n=0}}^{{\infty}}\mathop{B% ^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)\frac{t^{n}}{n!},$ $|t|<2\pi$ ,

Symbols:: : base of exponential function, : : factorial, $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Bernoulli polynomials, $\ell$ : integer, : integer, : real or complex and : real or complex
Permalink:: http://dlmf.nist.gov/24.16.E1
Encodings:: TeX, pMML, png

24.16.2 $\left(\frac{2}{e^{t}+1}\right)^{\ell}e^{{xt}}=\sum_{{n=0}}^{{\infty}}\mathop{E% ^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)\frac{t^{n}}{n!},$ $|t|<\pi$ .

Symbols:: : base of exponential function, : : factorial, $\mathop{E^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Euler polynomials, $\ell$ : integer, : integer, : real or complex and : real or complex
Permalink:: http://dlmf.nist.gov/24.16.E2
Encodings:: TeX, pMML, png

When they reduce to the Bernoulli and Euler numbers of order $\ell$ :

24.16.3

$\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits=\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits% \!\left(0\right),$

$\mathop{E^{{(\ell)}}_{{n}}\/}\nolimits=\mathop{E^{{(\ell)}}_{{n}}\/}\nolimits% \!\left(0\right).$

Symbols:: $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Bernoulli polynomials, $\mathop{E^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Euler polynomials, $\ell$ : integer and : integer
Referenced by:: ¶ ‣ §24.16(i)
Permalink:: http://dlmf.nist.gov/24.16.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

Also for $\ell=1,2,3,\ldots$ ,

24.16.4 $\left(\frac{\mathop{\ln\/}\nolimits\!\left(1+t\right)}{t}\right)^{\ell}=\ell% \sum_{{n=0}}^{{\infty}}\frac{\mathop{B^{{(\ell+n)}}_{{n}}\/}\nolimits}{\ell+n}% \frac{t^{n}}{n!},$

Symbols:: : : factorial, $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Bernoulli polynomials, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\ell$ : integer, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.16.E4
Encodings:: TeX, pMML, png

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

For extensions of $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ to complex values of , , and $\ell$ , and also for uniform asymptotic expansions for large and large , see Temme (1995b).

¶ Bernoulli Numbers of the Second Kind

Keywords:: Bernoulli numbers

24.16.5 $\frac{t}{\mathop{\ln\/}\nolimits\!\left(1+t\right)}=\sum_{{n=0}}^{{\infty}}b_{% n}t^{n},$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : integer, : real or complex and $b_{n}$
Permalink:: http://dlmf.nist.gov/24.16.E5
Encodings:: TeX, pMML, png

24.16.6 $n!b_{n}=-\frac{1}{n-1}\mathop{B^{{(n-1)}}_{{n}}\/}\nolimits,$ $n=2,3,\dots$ .

Symbols:: : : factorial, $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Bernoulli polynomials, : integer and $b_{n}$
Referenced by:: §24.16(i)
Permalink:: http://dlmf.nist.gov/24.16.E6
Encodings:: TeX, pMML, png

¶ Degenerate Bernoulli Numbers

Keywords:: Bernoulli numbers

For sufficiently small ,

24.16.7 $\frac{t}{(1+\lambda t)^{{\ifrac{1}{\lambda}}}-1}=\sum_{{n=0}}^{{\infty}}\beta_% {n}(\lambda)\frac{t^{n}}{n!},$

Symbols:: : : factorial, : integer, : real or complex and $\beta_{n}(\lambda)$
Referenced by:: §24.16(i)
Permalink:: http://dlmf.nist.gov/24.16.E7
Encodings:: TeX, pMML, png

24.16.8 $\beta_{n}(\lambda)=n!b_{n}\lambda^{n}+\sum_{{k=1}}^{{\left\lfloor\ifrac{n}{2}% \right\rfloor}}\frac{n}{2k}\mathop{B_{{2k}}\/}\nolimits\mathop{s\/}\nolimits\!% \left(n-1,2k-1\right)\lambda^{{n-2k}},$ $n=2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , : integer, : integer and $\beta_{n}(\lambda)$
Referenced by:: §24.16(i)
Permalink:: http://dlmf.nist.gov/24.16.E8
Encodings:: TeX, pMML, png

Here $\mathop{s\/}\nolimits\!\left(n,m\right)$ again denotes the Stirling number of the first kind.

¶ Nörlund Polynomials

Keywords:: Nörlund polynomials

24.16.9 $\left(\frac{t}{e^{t}-1}\right)^{x}=\sum_{{n=0}}^{{\infty}}\mathop{B^{{(x)}}_{{% n}}\/}\nolimits\frac{t^{n}}{n!},$ $|t|<2\pi$ .

Symbols:: : base of exponential function, : : factorial, $\mathop{B^{{(\ell)}}_{{n}}\/}\nolimits\!\left(x\right)$ : generalized Bernoulli polynomials, : integer, : real or complex and : real or complex
Permalink:: http://dlmf.nist.gov/24.16.E9
Encodings:: TeX, pMML, png

$\mathop{B^{{(x)}}_{{n}}\/}\nolimits$ is a polynomial in of degree . (This notation is consistent with (24.16.3) when $x=\ell$ .)

§24.16(ii) Character Analogs

Notes:: See Washington (1997, pp. 31–34). For (24.16.12) and (24.16.13) see Dilcher (1988, pp. 8 and 9).
Keywords:: conductor, primitive Dirichlet characters
Permalink:: http://dlmf.nist.gov/24.16.ii

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

24.16.10 $\sum_{{a=1}}^{f}\frac{\chi(a)te^{{at}}}{e^{{ft}}-1}=\sum_{{n=0}}^{{\infty}}B_{% {n,\chi}}\frac{t^{n}}{n!},$

Symbols:: : base of exponential function, : : factorial, : integer, : real or complex, $\chi(a)$ : Dirichlet character, : conductor and $B_{{n,\chi}}$ : Generalized Bernoulli numbers
Permalink:: http://dlmf.nist.gov/24.16.E10
Encodings:: TeX, pMML, png

24.16.11 $B_{{n,\chi}}(x)=\sum_{{k=0}}^{n}{n\choose k}B_{{k,\chi}}x^{{n-k}}.$

Symbols:: $\binom{m}{n}$ : binomial coefficient, : integer, : integer, : real or complex, $\chi(a)$ : Dirichlet character and $B_{{n,\chi}}$ : Generalized Bernoulli numbers
Permalink:: http://dlmf.nist.gov/24.16.E11
Encodings:: TeX, pMML, png

Let $\chi_{0}$ be the trivial character and $\chi_{4}$ the unique (nontrivial) character with ; that is, $\chi_{4}(1)=1$ , $\chi_{4}(3)=-1$ , $\chi_{4}(2)=\chi_{4}(4)=0$ . Then

24.16.12 $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)=B_{{n,\chi_{0}}}(x-1),$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : integer, : real or complex, $\chi(a)$ : Dirichlet character and $B_{{n,\chi}}$ : Generalized Bernoulli numbers
Referenced by:: §24.16(ii)
Permalink:: http://dlmf.nist.gov/24.16.E12
Encodings:: TeX, pMML, png

24.16.13 $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)=-\frac{2^{{1-n}}}{n+1}B_{{n+1,\chi% _{4}}}(2x-1).$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : integer, : real or complex, $\chi(a)$ : Dirichlet character and $B_{{n,\chi}}$ : Generalized Bernoulli numbers
Referenced by:: §24.16(ii)
Permalink:: http://dlmf.nist.gov/24.16.E13
Encodings:: TeX, pMML, png

For further properties see Berndt (1975a).

§24.16(iii) Other Generalizations

Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.16.iii

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)).

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