contiguous%20relations

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31—40 of 378 matching pages

31: Bibliography

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M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

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V. S. Adamchik and H. M. Srivastava (1998) Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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External Links:: ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt
Referenced by:: (25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.
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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D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

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F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:: Document, MathReview (I. Marx), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

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32: 18.5 Explicit Representations

… ►Related formula: …See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials. ►

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Laguerre

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Hermite

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33: Bibliography N

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D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

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National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview, ZentralBlatt
Referenced by:: §28.1, §28.26(i), 6th item, Notations S, Notations S.
Encodings:: BibTeX

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J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §13.28(i).
Encodings:: BibTeX

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W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

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E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:: See for additions and corrections same journal, Sect. B 75, 149–163 (1975)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii), §7.21, §7.7(iii).
Encodings:: BibTeX

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34: 17 q-Hypergeometric and Related Functions

Chapter 17 $q$ -Hypergeometric and Related Functions

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35: 26.13 Permutations: Cycle Notation

… ►They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. …

36: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

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37: 28 Mathieu Functions and Hill’s Equation

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38: 26.10 Integer Partitions: Other Restrictions

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Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from

A_{j,k}

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	$p\left(\mathcal{D},n\right)$	$p\left(\mathcal{D}2,n\right)$	$p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)$	$p\left(\mathcal{D}^{\prime}3,n\right)$
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$20$	$64$	$31$	$20$	$18$

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§26.10(iii) Recurrence Relations

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39: Bibliography K

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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S. Koizumi (1976) Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

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External Links:: FullText, MathReview (T. Oda), ZentralBlatt
Referenced by:: §21.6(ii).
Encodings:: BibTeX

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T. H. Koornwinder (2007b) The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.

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External Links:: ISSN 0377-0427, Document, Link, MathReview Entry
Referenced by:: (18.9.18_5).
Encodings:: BibTeX

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40: 9.9 Zeros

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§9.9(ii) Relation to Modulus and Phase

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9.9.6

a_{k}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $T$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.7

\operatorname{Ai}'\left(a_{k}\right)=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)% \right),

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $V$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.8

a^{\prime}_{k}=-U\left(\tfrac{3}{8}\pi(4k-3)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ , $k$ : nonnegative integer and $U$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.9

\operatorname{Ai}\left(a^{\prime}_{k}\right)=(-1)^{k-1}W\left(\tfrac{3}{8}\pi(% 4k-3)\right).

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ , $k$ : nonnegative integer and $W$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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