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relation to Bernoulli numbers

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1: 24.15 Related Sequences of Numbers
§24.15(i) Genocchi Numbers
§24.15(ii) Tangent Numbers
§24.15(iii) Stirling Numbers
2: 24.4 Basic Properties
§24.4(ix) Relations to Other Functions
For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).
3: 26.8 Set Partitions: Stirling Numbers
§26.8(vi) Relations to Bernoulli Numbers
4: 24.17 Mathematical Applications
§24.17(iii) Number Theory
5: 24.19 Methods of Computation
§24.19(i) Bernoulli and Euler Numbers and Polynomials
Equations (24.5.3) and (24.5.4) enable B n and E n to be computed by recurrence. …
§24.19(ii) Values of B n Modulo p
For number-theoretic applications it is important to compute B 2 n ( mod p ) for 2 n p - 3 ; in particular to find the irregular pairs ( 2 n , p ) for which B 2 n 0 ( mod p ) . We list here three methods, arranged in increasing order of efficiency. …
6: 25.6 Integer Arguments
§25.6(i) Function Values
7: 24.16 Generalizations
§24.16 Generalizations
Degenerate Bernoulli Numbers
§24.16(ii) Character Analogs
Generalized Bernoulli numbers and polynomials belonging to χ are defined by … 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)).
8: 25.1 Special Notation
k , m , n

nonnegative integers.

p

prime number.

B n , B n ( x )

Bernoulli number and polynomial (§24.2(i)).

B ~ n ( x )

periodic Bernoulli function B n ( x - x ) .

primes

on function symbols: derivatives with respect to argument.

The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
9: Bibliography T
  • J. W. Tanner and S. S. Wagstaff (1987) New congruences for the Bernoulli numbers. Math. Comp. 48 (177), pp. 341–350.
  • N. M. Temme (1995b) Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.
  • P. G. Todorov (1991) Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
  • P. G. Todorov (1984) On the theory of the Bernoulli polynomials and numbers. J. Math. Anal. Appl. 104 (2), pp. 309–350.
  • O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.
  • 10: Bibliography S
  • D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.
  • I. Sh. Slavutskiĭ (1995) Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.
  • I. Sh. Slavutskiĭ (1999) About von Staudt congruences for Bernoulli numbers. Comment. Math. Univ. St. Paul. 48 (2), pp. 137–144.
  • I. Sh. Slavutskiĭ (2000) On the generalized Bernoulli numbers that belong to unequal characters. Rev. Mat. Iberoamericana 16 (3), pp. 459–475.
  • S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.