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q-Bernoulli polynomials

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1: 17.3 q -Elementary and q -Special Functions
q -Bernoulli Polynomials
17.3.7 β n ( x , q ) = ( 1 - q ) 1 - n r = 0 n ( - 1 ) r ( n r ) r + 1 ( 1 - q r + 1 ) q r x .
2: Bibliography C
  • L. Carlitz (1953) Some congruences for the Bernoulli numbers. Amer. J. Math. 75 (1), pp. 163–172.
  • L. Carlitz (1954a) q -Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc. 76 (2), pp. 332–350.
  • L. Carlitz (1954b) A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.
  • L. Carlitz (1958) Expansions of q -Bernoulli numbers. Duke Math. J. 25 (2), pp. 355–364.
  • M. Chellali (1988) Accélération de calcul de nombres de Bernoulli. J. Number Theory 28 (3), pp. 347–362 (French).
  • 3: 24.16 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)).
    4: Bibliography K
  • M. Kaneko (1997) Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.
  • T. Kim and H. S. Kim (1999) Remark on p -adic q -Bernoulli numbers. Adv. Stud. Contemp. Math. (Pusan) 1, pp. 127–136.
  • N. Kimura (1988) On the degree of an irreducible factor of the Bernoulli polynomials. Acta Arith. 50 (3), pp. 243–249.
  • R. Koekoek and R. F. Swarttouw (1998) The Askey-scheme of hypergeometric orthogonal polynomials and its q -analogue. Technical report Technical Report 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.
  • R. Koekoek, P. A. Lesky, and R. F. Swarttouw (2010) Hypergeometric Orthogonal Polynomials and Their q -Analogues. Springer Monographs in Mathematics, Springer-Verlag, Berlin.