Stieltjes polynomials

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11: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

… ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain

w(x)

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Stieltjes Inversion via (approximate) Analytic Continuation

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Histogram Approach

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Derivative Rule Approach

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12: Bibliography K

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T. H. Koornwinder and F. Bouzeffour (2011) Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials. Appl. Anal. 90 (3-4), pp. 731–746.

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External Links:: ISSN 0003-6811, Link, MathReview (Masatoshi Iida), ZentralBlatt
Referenced by:: §18.38(iii), §18.38(iii).
Encodings:: BibTeX

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T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:: Document, MathReview (M. Mikolas), ZentralBlatt
Referenced by:: (18.14.8), §18.14(i), §18.17(ii), §18.18(ii).
Encodings:: BibTeX

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T. H. Koornwinder (1989) Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §18.21(ii).
Encodings:: BibTeX

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T. H. Koornwinder (2012) Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.

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Notes:: Electronic only
External Links:: Document, FullText
Referenced by:: §18.28(i), §18.28(i).
Encodings:: BibTeX

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T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §15.14, §15.14.
Encodings:: BibTeX

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13: Errata

… ►We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. … ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

Subsection 25.2(ii) Other Infinite Series

It is now mentioned that (25.2.5), defines the Stieltjes constants $\gamma_{n}$ . Consequently, $\gamma_{n}$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

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Section 1.14

There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$\mathscr{F}\left(f\right)\left(x\right)$	$\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$
Fourier Cosine	$\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Fourier Sine	$\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Laplace	$\mathscr{L}\left(f\right)\left(s\right)$	$\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{L}(f(t);s)$
Mellin	$\mathscr{M}\left(f\right)\left(s\right)$	$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{M}(f;s)$
Hilbert	$\mathcal{H}\left(f\right)\left(s\right)$	$\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{H}(f;s)$
Stieltjes	$\mathcal{S}\left(f\right)\left(s\right)$	$\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{S}(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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14: 25.2 Definition and Expansions

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25.2.4

\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\gamma_{\NVar{n}}$ : Stieltjes constants, $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer and $s$ : complex variable
Keywords:: infinite series, Laurent series
Sources:: Hardy (1912, p. 216); mistakenly missing factor of $(-1)^{n}/n!$ ; Ivić (1985, (1.11), p. 4)
A&S Ref:: 23.2.5 (with unnecessary constraint removed)
Referenced by:: Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
Permalink:: http://dlmf.nist.gov/25.2.E4
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: Originally this equation had the constraint $\Re s>0$ . This constraint is unnecessary because, as stated after (25.2.1), $\zeta\left(s\right)$ is meromorphic with a simple pole at $s=1$ .
Suggested 2017-12-05 by John Harper
See also:: Annotations for §25.2(ii), §25.2 and Ch.25

►where the Stieltjes constants

\gamma_{n}

are defined via ►

25.2.5

\gamma_{n}=\lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln k)^{n}}{k}-\frac{(% \ln m)^{n+1}}{n+1}\right).

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Defines:: $\gamma_{\NVar{n}}$ : Stieltjes constants
Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Keywords:: definition, Stieltjes constants
Sources:: Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)
Referenced by:: §25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
Permalink:: http://dlmf.nist.gov/25.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §25.2(ii), §25.2 and Ch.25

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25.2.10

\zeta\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0% pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{% 1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,

\Re s>-2n

n=1,2,3,\dots

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: Euler–Maclaurin formula
Proof sketch:: Derivable from (25.2.9) with $N=1$ .
A&S Ref:: 23.2.3
Permalink:: http://dlmf.nist.gov/25.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §25.2(iii), §25.2 and Ch.25

►For

B_{2k}

see §24.2(i), and for

\widetilde{B}_{n}\left(x\right)

see §24.2(iii). …

15: 25.6 Integer Arguments

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§25.6(i) Function Values

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25.6.6

\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}B_{% 2k+1}\left(t\right)\cot\left(\pi t\right)\,\mathrm{d}t,

k=1,2,3,\dots

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral and $k$ : nonnegative integer
Source:: Nörlund (1924, (81*), p. 66); with $\nu=k$
A&S Ref:: 23.2.17
Permalink:: http://dlmf.nist.gov/25.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(i), §25.6 and Ch.25

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25.6.12

\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{% \gamma}^{2}-\tfrac{1}{24}\pi^{2}+\gamma_{1},

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Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\gamma_{\NVar{n}}$ : Stieltjes constants, $\pi$ : the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$ : principal branch of logarithm function
Source:: Apostol (1985a, pp. 226, 227, 229)
Referenced by:: Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
Permalink:: http://dlmf.nist.gov/25.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(ii), §25.6 and Ch.25

►where

\gamma_{1}

is given by (25.2.5). …

16: Bibliography B

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W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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External Links:: ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt
Referenced by:: §18.18(vii).
Encodings:: BibTeX

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H. Bateman (1905) A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.

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Notes:: Reviewed in JFM 36.0512.01
External Links:: Document, ZentralBlatt
Referenced by:: §18.12.
Encodings:: BibTeX

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G. Baxter (1961) Polynomials defined by a difference system. J. Math. Anal. Appl. 2 (2), pp. 223–263.

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External Links:: Document, MathReview (F. L. Spitzer), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

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S. L. Belousov (1962) Tables of Normalized Associated Legendre Polynomials. Pergamon Press, The Macmillan Co., Oxford-New York.

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External Links:: MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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W. G. C. Boyd (1990b) Stieltjes transforms and the Stokes phenomenon. Proc. Roy. Soc. London Ser. A 429, pp. 227–246.

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External Links:: ISSN 0962-8444, FullText, MathReview, ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

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17: Bibliography N

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G. Nemes (2015b) On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:: ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt
Referenced by:: §11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).
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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P. Nevai (1986) Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.

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External Links:: ISSN 0021-9045, Document, MathReview (T. S. Chihara), ZentralBlatt
Referenced by:: §18.32.
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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M. Noumi and Y. Yamada (1999) Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J. 153, pp. 53–86.

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External Links:: ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

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18: Bibliography J

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L. Jager (1997) Fonctions de Mathieu et polynômes de Klein-Gordon. C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).

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Notes:: Translated title: Mathieu functions and Klein-Gordon polynomials.
External Links:: ISSN 0764-4442, Document, MathReview, ZentralBlatt
Referenced by:: 7th item.
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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External Links:: ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §18.24, §2.4(vi).
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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External Links:: ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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W. B. Jones and W. Van Assche (1998) Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:: MathReview (Olav Njåstad), ZentralBlatt
Referenced by:: §5.10, §5.10.
Encodings:: BibTeX

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19: Bibliography C

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L. Carlitz (1954b) 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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J. M. Carnicer, E. Mainar, and J. M. Peña (2020) Stability properties of disk polynomials. Numer. Algorithms.

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External Links:: Document
Referenced by:: (18.14.3_5).
Encodings:: BibTeX

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B. W. Char (1980) On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §5.10.
Encodings:: BibTeX

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P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.

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External Links:: ISSN 0021-9045, Document, Link, MathReview (Maria das Neves Rebocho)
Referenced by:: §18.32.
Encodings:: BibTeX

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P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

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20: Bibliography H

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B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

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External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.16.
Encodings:: BibTeX

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E. Hendriksen and H. van Rossum (1986) Orthogonal Laurent polynomials. Nederl. Akad. Wetensch. Indag. Math. 48 (1), pp. 17–36.

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External Links:: ISSN 0019-3577, Document, MathReview (W. J. Thron), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

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T. H. Hildebrandt (1938) Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

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External Links:: ISSN 0002-9890, Document, Link, MathReview Entry
Referenced by:: §1.16(ix).
Encodings:: BibTeX

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F. T. Howard (1976) Roots of the Euler polynomials. Pacific J. Math. 64 (1), pp. 181–191.

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External Links:: MathReview (M. Marden), ZentralBlatt
Referenced by:: §24.12(ii).
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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