# generalized Bernoulli polynomials

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## 1—10 of 26 matching pages

##### 1: 24.16 Generalizations
When $x=0$ they reduce to the Bernoulli and Euler numbers of order $\ell$ :
$B^{(\ell)}_{n}=B^{(\ell)}_{n}\left(0\right),$
For extensions of $B^{(\ell)}_{n}\left(x\right)$ to complex values of $x$, $n$, and $\ell$, and also for uniform asymptotic expansions for large $x$ and large $n$, see Temme (1995b) and López and Temme (1999b, 2010b). … $B^{(x)}_{n}$ is a polynomial in $x$ of degree $n$. …
##### 2: 5.11 Asymptotic Expansions
5.11.8 $\operatorname{Ln}\Gamma\left(z+h\right)\sim\left(z+h-\tfrac{1}{2}\right)\ln z-% z+\tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=2}^{\infty}\frac{(-1)^{k}B_{k}\left% (h\right)}{k(k-1)z^{k-1}},$
In terms of generalized Bernoulli polynomials $B^{(\ell)}_{n}\left(x\right)$24.16(i)), we have for $k=0,1,\ldots$,
##### 3: Bibliography D
• K. Dilcher (1987a) Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. J. Approx. Theory 49 (4), pp. 321–330.
• K. Dilcher (1987b) Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.
• K. Dilcher (1988) Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. Mem. Amer. Math. Soc. 73 (386), pp. iv+94.
• ##### 4: Bibliography L
• J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.
• J. L. López and N. M. Temme (2010b) Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.
• ##### 5: Bibliography T
• N. M. Temme (1995b) Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.
• ##### 6: 24.19 Methods of Computation
###### §24.19(i) Bernoulli and Euler Numbers and Polynomials
For algorithms for computing $B_{n}$, $E_{n}$, $B_{n}\left(x\right)$, and $E_{n}\left(x\right)$ see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).
###### §24.19(ii) Values of $B_{n}$ Modulo $p$
We list here three methods, arranged in increasing order of efficiency.
• Tanner and Wagstaff (1987) derives a congruence $\pmod{p}$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

• ##### 7: 24.17 Mathematical Applications
###### §24.17(iii) Number Theory
Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)).
##### 9: Bibliography C
• F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial ${L}_{n}^{\alpha}(x)$ as the index $\alpha\rightarrow\infty$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
• CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.
• L. Carlitz (1953) Some congruences for the Bernoulli numbers. Amer. J. Math. 75 (1), pp. 163–172.
• P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.
• H. S. Cohl (2013b) On a generalization of the generating function for Gegenbauer polynomials. Integral Transforms Spec. Funct. 24 (10), pp. 807–816.
• ##### 10: Bibliography B
• B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
• C. Brezinski (1980) Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.
• J. Brillhart (1969) On the Euler and Bernoulli polynomials. J. Reine Angew. Math. 234, pp. 45–64.
• T. Burić and N. Elezović (2011) Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions. J. Comput. Appl. Math. 235 (11), pp. 3315–3331.
• P. L. Butzer, M. Hauss, and M. Leclerc (1992) Bernoulli numbers and polynomials of arbitrary complex indices. Appl. Math. Lett. 5 (6), pp. 83–88.