q-Bernoulli polynomials

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1: 17.3 $q$-Elementary and $q$-Special Functions

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$q$-Bernoulli Polynomials

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17.3.7 $${\beta }_n(x,q)={(1-q)}^{1-n}\sum _{r=0}^n{(-1)}^r\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)\frac{r+1}{(1-q^{r+1})}{q}^{rx}.$$

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Defines:

${\beta }_n(x,q)$: $q$-Bernoulli polynomial

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

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2: Bibliography C

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Some congruences for the Bernoulli numbers. Amer. J. Math. 75 (1), pp. 163–172.

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External Links:

FullText, MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§24.10(i).

Encodings:

BibTeX

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$q$-Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc. 76 (2), pp. 332–350.

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External Links:

FullText, MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§17.3(iii), §24.16(iii).

Encodings:

BibTeX

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A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.

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External Links:

MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§24.10(iv).

Encodings:

BibTeX

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Expansions of $q$-Bernoulli numbers. Duke Math. J. 25 (2), pp. 355–364.

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External Links:

Document, MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§17.3(iii).

Encodings:

BibTeX

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Accélération de calcul de nombres de Bernoulli. J. Number Theory 28 (3), pp. 347–362 (French).

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External Links:

ISSN 0022-314X, Document, MathReview (Louis Shapiro), ZentralBlatt

Referenced by:

§24.19(i).

Encodings:

BibTeX

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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

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Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.

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External Links:

ISSN 1246-7405, PostScript, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.15(iv).

Encodings:

BibTeX

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Remark on $p$-adic $q$-Bernoulli numbers. Adv. Stud. Contemp. Math. (Pusan) 1, pp. 127–136.

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Notes:

Algebraic number theory (Hapcheon/Saga, 1996)

External Links:

ISSN 1229-3067, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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On the degree of an irreducible factor of the Bernoulli polynomials. Acta Arith. 50 (3), pp. 243–249.

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External Links:

ISSN 0065-1036, Pdf, MathReview (Wells Johnson), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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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.

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External Links:

FullText

Referenced by:

R. Koekoek, P. A. Lesky, and R. F. Swarttouw (2010).

Encodings:

BibTeX

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