Andrews%E2%80%93Askey%20sum

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21: 26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). …

22: 27 Functions of Number Theory

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23: 26.16 Multiset Permutations

… ►Additional information can be found in Andrews (1976, pp. 39–45). … ►

26.16.2 $$\sum _{\sigma \in {𝔖}_S}{q}^{inv(\sigma )}={\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q,$$

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Symbols:

$\in$: element of, ${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q$: $q$-multinomial coefficient, ${𝔖}_n$: set of permutations of $\{1,2,\mathrm{\dots },n\}$ , $n$: nonnegative integer, $a_j$: numbers, $S$: multiset and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.16.E2

Encodings:

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See also:

Annotations for §26.16 and Ch.26

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26.16.3 $$\sum _{\sigma \in {𝔖}_S}{q}^{maj(\sigma )}={\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q.$$

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Symbols:

$\in$: element of, ${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q$: $q$-multinomial coefficient, ${𝔖}_n$: set of permutations of $\{1,2,\mathrm{\dots },n\}$ , $n$: nonnegative integer, $a_j$: numbers, $S$: multiset and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.16.E3

Encodings:

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See also:

Annotations for §26.16 and Ch.26

24: 26.10 Integer Partitions: Other Restrictions

… ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the inner sum is the sum of all positive odd divisors of $t$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the inner sum is the sum of all positive divisors of $t$ that are in $S$. …

25: 4.10 Integrals

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26: 26.11 Integer Partitions: Compositions

… ►

26.11.4 $$\sum _{n=0}^{\mathrm{\infty }}{c}_m\left(n\right)q^n=\frac{q^m}{{(1-q)}^m}.$$

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Symbols:

$c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.11.E4

Encodings:

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Annotations for §26.11 and Ch.26

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27: David M. Bressoud

… ► Andrews and A. …

28: Richard B. Paris

… ► Andrews, U. …

29: 18.12 Generating Functions

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18.12.7 $$\frac{1-z^2}{1-2xz+z^2}=1+2\sum _{n=1}^{\mathrm{\infty }}{T}_n\left(x\right)z^n,$$ .

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E7

Encodings:

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See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.8 $$\frac{1-xz}{1-2xz+z^2}=\sum _{n=0}^{\mathrm{\infty }}{T}_n\left(x\right)z^n,$$ .

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.6

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E8

Encodings:

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Annotations for §18.12, §18.12 and Ch.18

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18.12.9 $$-\ln \left(1-2xz+z^2\right)=2\sum _{n=1}^{\mathrm{\infty }}\frac{T_n\left(x\right)}{n}{z}^n,$$ .

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\ln z$: principal branch of logarithm function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.8

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E9

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Annotations for §18.12, §18.12 and Ch.18

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18.12.10 $$\frac{1}{1-2xz+z^2}=\sum _{n=0}^{\mathrm{\infty }}{U}_n\left(x\right)z^n,$$ .

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.10

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E10

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Annotations for §18.12, §18.12 and Ch.18

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18.12.11 $$\frac{1}{\sqrt{1-2xz+z^2}}=\sum _{n=0}^{\mathrm{\infty }}{P}_n\left(x\right)z^n,$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.12

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E11

Encodings:

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See also:

Annotations for §18.12, §18.12 and Ch.18

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30: Bibliography Z

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The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:

MathReview (J. Browkin), ZentralBlatt

Referenced by:

§25.12(i).

Encodings:

BibTeX

… ►

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

… ►

Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

… ►

“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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