DLMF: Untitled Document

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

… ►

Mehler–Sonine and Related Integrals

… ►

Mellin–Barnes Type Integrals

… ►For (10.9.22)–(10.9.25) and further integrals of this type see Paris and Kaminski (2001, pp. 114–116). … ►

Mellin–Barnes Type

…

… ►

K. W. J. Kadell (1994) A proof of the

q

-Macdonald-Morris conjecture for

BC_{n}

. Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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External Links:: ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

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E. H. Kaufman and T. D. Lenker (1986) Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.

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External Links:: ISSN 0002-9890, FullText, MathReview (J. G. Herriot), ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

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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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… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►

Table 27.2.1: Primes.

► ►►

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…

► ►

Table 27.2.2: Functions related to division.

► ►►►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

►

… ►

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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D. E. Amos, S. L. Daniel, and M. K. Weston (1977) Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions

I_{\nu}(x)

and

J_{\nu}(x)

,

x\geq 0

,

\nu\geq 0

. ACM Trans. Math. Software 3 (1), pp. 93–95.

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Notes:: Single-precision Fortran, maximum accuracy 16S.
External Links:: ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)
Referenced by:: 1st item, 1st item.
Encodings:: BibTeX

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D. E. Amos (1990) Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:: Double- and single-precision Fortran, maximum accuracy 18S or 14S.
External Links:: ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt
Referenced by:: 1st item, 1st item.
Encodings:: BibTeX

… ►

G. E. Andrews, R. Askey, and R. Roy (1999) Special Functions. Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-62321-9, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §1.10(vii), §1.15(iii), §1.15(vi), §1.15(vii), §10.22(vi), §13.6(v), §13.9(i), Ch.13, §14.3(iii), §14.30(iv), §15.10(i), §15.11(i), §15.11(ii), §15.14, §15.17(iv), §15.2(i), §15.4(ii), §15.5(i), §15.5(ii), §15.6, §15.7, §15.8(iv), §15.8(iii), Ch.15, §16.4(ii), §16.4(iii), §16.4(v), §16.4(v), Ch.16, §17.12, Ch.17, §18.10(ii), §18.10(iv), §18.11(i), §18.12, (18.17.41_5), §18.17(iv), §18.17(viii), §18.18(iv), §18.18(iv), §18.18(v), §18.18(vii), §18.2(i), §18.2(iv), §18.2(vi), §18.20(ii), §18.22(i), §18.25(ii), §18.26(i), Table 18.3.1, §18.38(ii), §18.5(iii), (18.7.25), Ch.18, §26.19, §4.10(i), §4.10(ii), §5.13, §5.14, §5.16, §5.16, §5.18(i), §5.18(ii), §5.18(iii), §5.20, §5.4(ii), §5.5(iv), Ch.5, Ch.6, Profile Richard A. Askey, Profile Ranjan Roy, Erratum (V1.0.28) for Table 18.3.1.
Encodings:: BibTeX

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M. J. Atia, A. Martínez-Finkelshtein, P. Martínez-González, and F. Thabet (2014) Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt
Referenced by:: §18.15(vi), §18.15(vi).
Encodings:: BibTeX

…

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P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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D. R. Hartree (1936) Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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External Links:: ZentralBlatt
Referenced by:: §7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).
Encodings:: BibTeX

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K. Dilcher (1987b) Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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External Links:: ISSN 0022-314X, Document, MathReview (T. Metsänkylä)
Referenced by:: §24.12(iii).
Encodings:: BibTeX

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B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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B. A. Dubrovin (1981) Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:: With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)
External Links:: ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.
Encodings:: BibTeX

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T. M. Dunster (1997) Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

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External Links:: ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §2.11(iii), §8.20(ii).
Encodings:: BibTeX

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T. M. Dunster (2001b) Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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

…

… ►An element of

\mathfrak{S}_{n}

with

a_{1}

fixed points,

a_{2}

cycles of length

2,\ldots,a_{n}

cycles of length

n

, where

n=a_{1}+2a_{2}+\cdots+na_{n}

, is said to have cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

. The number of elements of

\mathfrak{S}_{n}

with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

is given by (26.4.7). … ►A permutation with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

can be written as a product of

a_{2}+2a_{3}+\cdots+(n-1)a_{n}=n-(a_{1}+a_{2}+\cdots+a_{n})

transpositions, and no fewer. …

… ►

32.8.3

w(z;3)=\frac{3z^{2}}{z^{3}+4}-\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80},

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Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

►

32.8.4

w(z;4)=-\frac{1}{z}+\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80}-\frac{9z^{5}(z^{% 3}+40)}{z^{9}+60z^{6}+11200}.

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

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Q_{3}(z)=z^{6}+20z^{3}-80,

…

… ►

§14.28(ii) Heine’s Formula

…

Mehler%E2%80%93Heine%20type%20formulas

11—20 of 358 matching pages

11: 18.7 Interrelations and Limit Relations

12: 10.9 Integral Representations

Mehler–Sonine and Related Integrals

Mellin–Barnes Type Integrals

Mellin–Barnes Type

13: Bibliography K

14: 27.2 Functions

15: Bibliography

16: Bibliography H

17: Bibliography D

18: 26.13 Permutations: Cycle Notation

19: 32.8 Rational Solutions

20: 14.28 Sums

§14.28(ii) Heine’s Formula

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…