generalized Bernoulli polynomials

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21: Errata

… ►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. … ►

Several biographies had their publications updated.

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Originally all six integrands in these equations were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$. The new equations are:

25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$

Reported 2016-05-08 by Clemens Heuberger.

25.11.19 $${\zeta }^{\prime }(s,a)=-\frac{\ln a}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{{(s-1)}^2}+\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2)\ln \left(x+a\right)}{{(x+a)}^{s+2}}dx-\frac{(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$

Reported 2016-06-27 by Gergő Nemes.

25.11.20 $${(-1)}^k{\zeta }^{(k)}(s,a)=\frac{{(\ln a)}^k}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum _{r=0}^{k-1}\frac{{(\ln a)}^r}{r!{(s-1)}^{k-r+1}}-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^k}{{(x+a)}^{s+2}}dx+\frac{k(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-1}}{{(x+a)}^{s+2}}dx-\frac{k(k-1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-2}}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$

Reported 2016-06-27 by Gergő Nemes.

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Because of the use of the $O$ order symbol on the right-hand side, the asymptotic expansion for the generalized Laguerre polynomial $L_n^{(\alpha )}\left(\nu x\right)$ was rewritten as an equality.

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Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

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22: Bibliography G

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On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.

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External Links:

ISSN 0771-050X, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

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ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii), §25.18(i), 5th item.

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The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.

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ISSN 0002-9939, Link, MathReview (Raj Wilson), ZentralBlatt

Referenced by:

§18.28(xi).

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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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23: 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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Caustics: Mathematical curves generated by light shined through rippled plastic. Scientific American 249, pp. 146–153.

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Referenced by:

§36.14(ii).

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§4.12.

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Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

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Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math. 85 (1), pp. 87–112.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.34(iii).

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24: Bibliography S

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General relativity and quantum mechanics: towards a generalization of the Lambert $W$ function: a generalization of the Lambert $W$ function. Appl. Algebra Engrg. Comm. Comput. 17 (1), pp. 41–47.

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External Links:

ISSN 0938-1279, Document, FullText, MathReview (E. Capelas de Oliveira), ZentralBlatt

Referenced by:

§4.13, §4.13.

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Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

Referenced by:

§24.10(ii).

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About von Staudt congruences for Bernoulli numbers. Comment. Math. Univ. St. Paul. 48 (2), pp. 137–144.

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ISSN 0010-258X, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.10(ii).

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On the generalized Bernoulli numbers that belong to unequal characters. Rev. Mat. Iberoamericana 16 (3), pp. 459–475.

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External Links:

ISSN 0213-2230, MathReview (Aichi Kudo), ZentralBlatt

Referenced by:

§24.4(iii).

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25: Bibliography P

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Zonal Polynomials of Order $1$ Through $12$ . In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.

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External Links:

MathReview

Referenced by:

§35.11.

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Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.

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ISSN 0022-247X, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(iv), §18.33(v).

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

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Document, Code, ZentralBlatt

Referenced by:

1st item.

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:

Document

Referenced by:

§25.18(i).

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Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

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26: 3.5 Quadrature

… ►For the Bernoulli numbers $B_m$ see §24.2(i). … ►

Gauss–Legendre Formula

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Gauss–Laguerre Formula

… ►The $p_n(x)$ are the monic Hermite polynomials $H_n\left(x\right)$ (§18.3). … ► …