DLMF: Untitled Document

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M. Kaneko (1997) 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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T. Kasuga and R. Sakai (2003) Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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External Links:: ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)
Referenced by:: §18.32.
Encodings:: BibTeX

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N. M. Katz (1975) The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers. Math. Ann. 216 (1), pp. 1–4.

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External Links:: ISSN 0025-5831, Document, MathReview (M. Basmakov), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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R. P. Kelisky (1957) 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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N. Kimura (1988) 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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M. Hauss (1997) 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:: BibTeX

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K. Horata (1989) 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.
Encodings:: BibTeX

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K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

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External Links:: ZentralBlatt
Referenced by:: §24.15(iii).
Encodings:: BibTeX

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F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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External Links:: ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(i).
Encodings:: BibTeX

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I. Huang and S. Huang (1999) 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:: BibTeX

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G. Nemes (2013c) Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

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P. G. Nevai (1979) Orthogonal polynomials. Mem. Amer. Math. Soc. 18 (213), pp. v+185 pp..

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External Links:: ISSN 0065-9266, Document, Link, MathReview (T. S. Chihara)
Referenced by:: §18.2(xi), §18.2(xi).
Encodings:: BibTeX

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N. Nielsen (1923) Traité Élémentaire des Nombres de Bernoulli. Gauthier-Villars, Paris.

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External Links:: ZentralBlatt
Referenced by:: §24.1.
Encodings:: BibTeX

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N. E. Nörlund (1922) Mémoire sur les polynomes de Bernoulli. Acta Math. 43, pp. 121–196 (French).

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External Links:: Document, ZentralBlatt
Referenced by:: §24.13(i), §24.13(ii), §24.14(i), Ch.24.
Encodings:: BibTeX

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M. Noumi and J. V. Stokman (2004) Askey-Wilson polynomials: an affine Hecke algebra approach. In Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, pp. 111–144.

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External Links:: MathReview (V. L. Deshpande), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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	Open Source								With Book			Commercial
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4.48(iii) General Precision	✓		✓	✓	✓		✓	✓					✓	✓	a	REDUCE
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16 Generalized Hypergeometric Functions & Meijer G-Function
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18 Orthogonal Polynomials
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24 Bernoulli and Euler Polynomials
24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , $E_{n}\left(x\right)$	✓	✓			✓	✓	a	✓	✓	✓	✓		✓	✓	✓	Derive, MuPAD
…

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H\left(s\right)

has a simple pole with residue

\zeta\left(1-2r\right)

(

=-B_{2r}/(2r)

) at each odd negative integer

s=1-2r

,

r=1,2,3,\dots

. ►

H\left(s\right)

is the special case

H\left(s,1\right)

of the function …when both

H\left(s,z\right)

and

H\left(z,s\right)

are finite. ►For further properties of

H\left(s,z\right)

see Apostol and Vu (1984). … ►For further generalizations, see Flajolet and Salvy (1998).

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Á. Elbert (2001) Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.

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External Links:: ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

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E. Elizalde (1986) An asymptotic expansion for the first derivative of the generalized Riemann zeta function. Math. Comp. 47 (175), pp. 347–350.

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External Links:: ISSN 0025-5718, FullText, MathReview (S. Audinarayana Moorthy), ZentralBlatt
Referenced by:: (25.11.44), (25.11.45), §25.11(xii), §25.11(xii).
Encodings:: BibTeX

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D. Elliott (1971) Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §18.15(i).
Encodings:: BibTeX

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R. Ernvall (1979) Generalized Bernoulli numbers, generalized irregular primes, and class number. Ann. Univ. Turku. Ser. A I 178, pp. 1–72.

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External Links:: ISSN 0082-7002, MathReview (Wells Johnson), ZentralBlatt
Referenced by:: §24.10(ii).
Encodings:: BibTeX

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W. N. Everitt (2008) Note on the

X_{1}

-Laguerre orthogonal polynomials.

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Notes:: arXiv:0811.3559v2
Referenced by:: §18.36(vi).
Encodings:: BibTeX

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D. Zagier (1998) A modified Bernoulli number. Nieuw Arch. Wisk. (4) 16 (1-2), pp. 63–72.

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External Links:: ISSN 0028-9825, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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A. H. Zemanian (1987) Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.

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Notes:: Reprint of 1965 McGraw-Hill Publication
External Links:: ISBN 0-486-65479-6, MathReview Entry
Referenced by:: §1.16(ix).
Encodings:: BibTeX

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A. S. Zhedanov (1991) “Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.

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External Links:: MathReview (Vadim B. Kuznetsov), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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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.

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External Links:: ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt
Referenced by:: §18.33(iii).
Encodings:: BibTeX

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Q. Zheng (1997) Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

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Referenced by:: §35.7(ii).
Encodings:: BibTeX

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A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

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T. Agoh and K. Dilcher (2011) Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl. 381 (1), pp. 10–16.

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External Links:: ISSN 0022-247X, Document, FullText, MathReview (Pablo Manuel Román), ZentralBlatt
Referenced by:: §24.13(i), §24.13(i).
Encodings:: BibTeX

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T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:: §24.4(iii).
Encodings:: BibTeX

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T. M. Apostol (2008) A primer on Bernoulli numbers and polynomials. Math. Mag. 81 (3), pp. 178–190.

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External Links:: ISSN 0025-570X, MathReview Entry, ZentralBlatt
Referenced by:: §24.5(ii), Ch.24.
Encodings:: BibTeX

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R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:: ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.
Encodings:: BibTeX

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K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

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A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:: ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.36(ii).
Encodings:: BibTeX

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M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

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M. E. H. Ismail (2000a) An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.

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External Links:: ISSN 0030-8730, FullText, MathReview (T. Erber), ZentralBlatt
Referenced by:: §18.39(v).
Encodings:: BibTeX

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M. E. H. Ismail (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt
Referenced by:: §1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

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S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

ⓘ

Referenced by:: §24.7(i).
Encodings:: BibTeX

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J. Raynal (1979) On the definition and properties of generalized

6

-

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

REDUCE (free interactive system)

ⓘ

Notes:: Symbolic and extended-precision general purpose computer algebra system.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

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W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

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E. Ya. Remez (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

ⓘ

Notes:: English translation, AEC-TR-4491, United States Atomic Energy Commission
External Links:: MathReview (A. S. Householder)
Referenced by:: §3.11(iii).
Encodings:: BibTeX

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generalized Bernoulli polynomials

11—20 of 26 matching pages

11: Bibliography K

12: Bibliography H

13: Bibliography N

14: Software Index

15: 25.16 Mathematical Applications

16: Bibliography E

17: Bibliography Z

18: Bibliography

19: Bibliography I

20: Bibliography R