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relation to generalized Bernoulli polynomials


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1: 24.16 Generalizations
§24.16(ii) Character Analogs
2: 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. …For example, the tangent numbers T n can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. … For algorithms for computing B n , E n , B n ( x ) , and E n ( x ) see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). … 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 ) . …
3: 25.16 Mathematical Applications
which is related to the Riemann zeta function by … H ( s ) is the special case H ( s , 1 ) of the function …when both H ( s , z ) and H ( z , s ) are finite. … Related results are: … For further generalizations, see Flajolet and Salvy (1998).
4: Bibliography L
  • D. J. Leeming (1989) The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.
  • D. H. Lehmer (1940) On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.
  • 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 (1999c) Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.
  • 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: 18.2 General Orthogonal Polynomials
    §18.2 General Orthogonal Polynomials
    Orthogonality on General Sets
    §18.2(iv) Recurrence Relations
    6: Bibliography B
  • 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. J. Bushell (1987) On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.
  • 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.
  • 7: Bibliography C
  • F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial L n α ( x )  as the index α  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.
  • P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.
  • W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.
  • H. S. Cohl (2013b) On a generalization of the generating function for Gegenbauer polynomials. Integral Transforms Spec. Funct. 24 (10), pp. 807–816.
  • 8: Bibliography Z
  • F. A. Zafiropoulos, T. N. Grapsa, O. Ragos, and M. N. Vrahatis (1996) On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.
  • D. Zagier (1998) A modified Bernoulli number. Nieuw Arch. Wisk. (4) 16 (1-2), pp. 63–72.
  • A. H. Zemanian (1987) Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.
  • A. Zhedanov (1998) On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.
  • Q. Zheng (1997) Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.
  • 9: Bibliography
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • T. Agoh and K. Dilcher (2011) Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl. 381 (1), pp. 10–16.
  • W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.
  • T. M. Apostol (2008) A primer on Bernoulli numbers and polynomials. Math. Mag. 81 (3), pp. 178–190.
  • R. Askey and M. E. H. Ismail (1984) Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
  • 10: Software Index
  • Research Software.

    This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

  • Open Source Collections and Systems.

    These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

  • Software Associated with Books.

    An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • Guide to Available Mathematical Software

    A cross index of mathematical software in use at NIST.