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1: Bibliography K

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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2: 24.20 Tables

§24.20 Tables

►Abramowitz and Stegun (1964, Chapter 23) includes exact values of ${\sum }_{k=1}^m{k}^n$, $m=1(1)100$, $n=1(1)10$; ${\sum }_{k=1}^{\mathrm{\infty }}{k}^{-n}$, ${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}{k}^{-n}$, ${\sum }_{k=0}^{\mathrm{\infty }}{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 20D; ${\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 18D. ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

3: Bibliography C

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Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.

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Notes:

Code written by the second author to compute symmetric elliptic integrals numerically in the Derive computer algebra system is available.

External Links:

ISSN 0377-0427, Document, Code, MathReview, ZentralBlatt

Referenced by:

§19.29(ii), §19.36(i), §19.36(i), §19.39(iv).

Encodings:

BibTeX

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

Encodings:

BibTeX

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122 (1), pp. 119–145.

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Notes:

Erratum: see same journal 125(1996), no. 2, p. 391.

External Links:

ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

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A dispersion analysis for difference schemes: Tables of generalized Airy functions. Math. Comp. 32 (144), pp. 1163–1170.

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External Links:

ISSN 0025-5718, FullText, MathReview (Philip Brenner), ZentralBlatt

Referenced by:

§9.13(ii).

Encodings:

BibTeX

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4: 26.5 Lattice Paths: Catalan Numbers

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Table 26.5.1: Catalan numbers.

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$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
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6	132	13	7 42900	20	65641 20420

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26.5.2 $$\sum _{n=0}^{\mathrm{\infty }}C\left(n\right)x^n=\frac{1-\sqrt{1-4x}}{2x},$$ .

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Symbols:

$C\left(n\right)$: Catalan number, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E2

Encodings:

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See also:

Annotations for §26.5(ii), §26.5 and Ch.26

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26.5.3 $$C\left(n+1\right)=\sum _{k=0}^{n}C\left(k\right)C\left(n-k\right),$$

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Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E3

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See also:

Annotations for §26.5(iii), §26.5 and Ch.26

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26.5.5 $$C\left(n+1\right)=\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0.0pt}{}{n}{2k}\right)2^{n-2k}C\left(k\right).$$

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Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E5

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See also:

Annotations for §26.5(iii), §26.5 and Ch.26

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5: 23.9 Laurent and Other Power Series

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23.9.1 $$c_n=(2n-1)\sum _{w\in 𝕃\setminus \{0\}}{w}^{-2n},$$ $n=2,3,4,\mathrm{\dots }$.

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Symbols:

$\in$: element of, $\setminus$: set subtraction, $𝕃$: lattice, $n$: integer and $c_n$

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/23.9.E1

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See also:

Annotations for §23.9 and Ch.23

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23.9.2 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+\sum _{n=2}^{\mathrm{\infty }}{c}_n{z}^{2n-2},$$ ,

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $𝕃$: lattice, $n$: integer, $z$: complex and $c_n$

A&S Ref:

18.5.1

Referenced by:

§23.9, §23.9

Permalink:

http://dlmf.nist.gov/23.9.E2

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See also:

Annotations for §23.9 and Ch.23

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23.9.3 $$\zeta \left(z\right)=\frac{1}{z}-\sum _{n=2}^{\mathrm{\infty }}\frac{c_n}{2n-1}{z}^{2n-1},$$ .

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice, $n$: integer, $z$: complex and $c_n$

A&S Ref:

18.5.5

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E3

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See also:

Annotations for §23.9 and Ch.23

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$c_2={\displaystyle \frac{1}{20}}{g}_2,$

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23.9.5 $$c_n=\frac{3}{(2n+1)(n-3)}\sum _{m=2}^{n-2}{c}_m{c}_{n-m},$$ $n\ge 4$.

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Symbols:

$n$: integer, $m$: integer and $c_n$

A&S Ref:

18.5.3

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E5

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See also:

Annotations for §23.9 and Ch.23

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6: 8.17 Incomplete Beta Functions

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8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

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Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

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7: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …It can be expressed as a sum over all primes $p\le x$: … ►the sum of the $k$th powers of the positive integers $m\le n$ that are relatively prime to $n$. … ►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

8: Bibliography M

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:

ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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On the representation of numbers as a sum of $2r$ squares. Quarterly Journal of Math. 48, pp. 93–104.

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External Links:

ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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On the zeros of a cross-product of Bessel functions of different orders. Z. Angew. Math. Mech. 59 (6), pp. 272–273.

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External Links:

ISSN 0044-2267, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

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9: 26.14 Permutations: Order Notation

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26.14.1

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Symbols:

$j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer and $\sigma$: permutation

Permalink:

http://dlmf.nist.gov/26.14.E1

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See also:

Annotations for §26.14(i), §26.14 and Ch.26

►Equivalently, this is the sum over of the number of integers less than $\sigma (j)$ that lie in positions to the right of the $j$th position: $inv(35247816)=2+3+1+1+2+2+0=11.$ … ►The major index is the sum of all positions that mark the first element of a descent: ►

26.14.2

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Symbols:

$j$: nonnegative integer, $n$: nonnegative integer, $\sigma$: permutation and $maj(\sigma )$: major index

Permalink:

http://dlmf.nist.gov/26.14.E2

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See also:

Annotations for §26.14(i), §26.14 and Ch.26

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26.14.3 $$\sum _{\sigma \in {𝔖}_n}{q}^{inv(\sigma )}=\sum _{\sigma \in {𝔖}_n}{q}^{maj(\sigma )}=\prod _{j=1}^n\frac{1-q^j}{1-q}.$$

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Symbols:

$\in$: element of, ${𝔖}_n$: set of permutations of $\{1,2,\mathrm{\dots },n\}$ , $j$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.14.E3

Encodings:

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See also:

Annotations for §26.14(ii), §26.14 and Ch.26

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10: 26.2 Basic Definitions

… ►A partition of a nonnegative integer $n$ is an unordered collection of positive integers whose sum is $n$. … ►The integers whose sum is $n$ are referred to as the parts in the partition. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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3	3	20	627	37	21637
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