relation to Bernoulli numbers

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21: Bibliography C

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

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

Referenced by:

§24.10(i).

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

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

Referenced by:

§17.3(iii), §24.16(iii).

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

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

Referenced by:

§17.3(iii).

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

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

Referenced by:

§24.19(i).

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

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22: Bibliography I

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

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A Classical Introduction to Modern Number Theory. 2nd edition, Springer-Verlag, New York.

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ISBN 0-387-97329-X, MathReview (Glenn Stevens), ZentralBlatt

Referenced by:

§24.10(i), §24.10(ii), §24.10(iii), §24.17(iii).

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Special Functions, $q$-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

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ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Mourad E.H. Ismail.

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

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23: 26.14 Permutations: Order Notation

… ► … ►The Eulerian number is equal to the number of permutations in ${𝔖}_n$ with exactly $k$ excedances. It is also equal to the number of permutations in ${𝔖}_n$ with exactly $k+1$ weak excedances. … ►

§26.14(iii) Identities

►In this subsection $S(n,k)$ is again the Stirling number of the second kind (§26.8), and $B_m$ is the $m$th Bernoulli number (§24.2(i)). …

24: Bibliography W

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Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

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External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Generating functions of class-numbers. Compositio Math. 1, pp. 39–68.

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ISSN 0010-437X, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§20.7(v), §20.8.

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

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Computation with Recurrence Relations. Pitman, Boston, MA.

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ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

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25: Bibliography G

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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

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A theorem on the numerators of the Bernoulli numbers. Amer. Math. Monthly 97 (2), pp. 136–138.

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ISSN 0002-9890, FullText, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.10(iv), §24.16(iii).

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

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26: 25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: … ►For ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►where $H_n$ are the harmonic numbers: … ►When $a=1$, (25.11.35) reduces to (25.2.3). …

27: Bibliography K

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

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

Referenced by:

§24.16(iii).

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The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers. Math. Ann. 216 (1), pp. 1–4.

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ISSN 0025-5831, Document, MathReview (M. Basmakov), ZentralBlatt

Referenced by:

§24.16(iii).

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

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MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.15(iv).

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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)

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

Referenced by:

§24.16(iii).

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Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 (100), pp. 663–688.

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FullText, MathReview, ZentralBlatt

Referenced by:

§24.15(ii), §24.19(i).

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28: Bibliography D

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Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

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Notes:

A frequently updated version, with links to reviews, exists online

External Links:

Online version, MathReview, ZentralBlatt

Referenced by:

Ch.24.

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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. J. Approx. Theory 49 (4), pp. 321–330.

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ISSN 0021-9045, Document, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§24.11.

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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

Encodings:

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Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

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Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.

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ISBN 1-56881-126-8, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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29: 3.5 Quadrature

… ►For the Bernoulli numbers $B_m$ see §24.2(i). … ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … …

30: Bibliography H

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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External Links:

ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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External Links:

ZentralBlatt

Referenced by:

§24.6.

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On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

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ZentralBlatt

Referenced by:

§24.15(iii).

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Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(i).

Encodings:

BibTeX

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Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.

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External Links:

ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

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