Goldbach conjecture

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11: Bibliography Z

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A proof of Andrews’ $q$-Dyson conjecture. Discrete Math. 54 (2), pp. 201–224.

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External Links:

ISSN 0012-365X, Document, MathReview (Ira Gessel), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

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12: Bibliography N

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Proofs of two conjectures on the real zeros of the cylinder and Airy functions. SIAM J. Math. Anal. 53 (4), pp. 4328–4349.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(10.18.17), (10.18.18), §10.18(iii), (9.8.22), (9.8.23), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv), Erratum (V1.1.8) for Section 10.18(iii).

Encodings:

BibTeX

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13: 18.14 Inequalities

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18.14.25 $$\sum _{m=0}^n\frac{{\left(\lambda +1\right)}_{n-m}}{(n-m)!}\frac{{\left(\lambda +1\right)}_m}{m!}\frac{P_m^{(\alpha ,\beta )}\left(x\right)}{P_m^{(\beta ,\alpha )}\left(1\right)}\ge 0,$$ $x\ge -1$, $\alpha +\beta \ge \lambda \ge 0$, $\beta \ge -\frac{1}{2}$, $n=0,1,\mathrm{\dots }$.

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 5)(proved)

Source:

Askey and Gasper (1976, Conjecture 1); conjectured

Referenced by:

Erratum (V1.2.0) Section 18.14

Permalink:

http://dlmf.nist.gov/18.14.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

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18.14.26 $$\sum _{m=0}^n\frac{P_m^{(\alpha ,\beta )}\left(x\right)}{P_m^{(\beta ,\alpha )}\left(1\right)}\ge 0,$$ $x\ge -1$, $n=0,1,\mathrm{\dots }$,

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 4); also proved there for $\alpha +\beta \ge -2$, $\beta \ge 0$(proved)

Source:

Askey and Gasper (1976, (1.7), (1.12)); conjectured

Referenced by:

§18.14(iv), §18.38(ii)

Permalink:

http://dlmf.nist.gov/18.14.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

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18.14.27 $$\sum _{m=0}^n\frac{{\left(\lambda +1\right)}_{n-m}}{(n-m)!}\frac{{\left(\lambda +1\right)}_m}{m!}\frac{{(-1)}^m{L}_m^{(\beta )}\left(x\right)}{L_m^{(\beta )}\left(0\right)}\ge 0,$$ $x\ge 0$,  $\beta ,\lambda \ge -\frac{1}{2}$,  $n=0,1,\mathrm{\dots }$.

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 1)(proved)

Source:

Askey and Gasper (1976, Conjecture 2); conjectured

Referenced by:

Erratum (V1.2.0) Section 18.14

Permalink:

http://dlmf.nist.gov/18.14.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

14: Bibliography H

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La $q$-conjecture de Macdonald-Morris pour $G_2$ . C. R. Acad. Sci. Paris Sér. I Math. 303 (6), pp. 211–213 (French).

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External Links:

ISSN 0249-6291, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

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15: 18.40 Methods of Computation

… ►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that …

16: 27.2 Functions

… ►Gauss and Legendre conjectured that $\pi \left(x\right)$ is asymptotic to $x/\ln x$ as $x\to \mathrm{\infty }$: …

17: Bibliography

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Plane partitions. III. The weak Macdonald conjecture. Invent. Math. 53 (3), pp. 193–225.

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External Links:

ISSN 0020-9910, Document, MathReview (Edward A. Bender), ZentralBlatt

Referenced by:

§26.12(i), §26.12(ii).

Encodings:

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18: Bibliography D

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A proof of the Bieberbach conjecture. Acta Math. 154 (1-2), pp. 137–152.

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External Links:

ISSN 0001-5962, Document, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

§16.23(iii), §18.38(ii).

Encodings:

BibTeX

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19: 27.14 Unrestricted Partitions

… ►Lehmer (1947) conjectures that $\tau \left(n\right)$ is never 0 and verifies this for all by studying various congruences satisfied by $\tau \left(n\right)$, for example: …

20: 28.6 Expansions for Small $q$

… ►It is conjectured that for large $n$, the radii increase in proportion to the square of the eigenvalue number $n$; see Meixner et al. (1980, §2.4). …