Rogers–Dougall very well-poised sum

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11: 17.2 Calculus

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17.2.36 $$\sum _{j=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right){(-z)}^j={(1-z)}^n.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: nonnegative integer, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.2.E36

Encodings:

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See also:

Annotations for §17.2(iii), §17.2 and Ch.17

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17.2.45 $${\int }_0^{1}f(x)d_{q}x=(1-q)\sum _{j=0}^{\mathrm{\infty }}f(q^j)q^j,$$

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Symbols:

$\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base, $j$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.2.E45

Encodings:

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See also:

Annotations for §17.2(v), §17.2 and Ch.17

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17.2.46 $${\int }_0^{a}f(x)d_{q}x=a(1-q)\sum _{j=0}^{\mathrm{\infty }}f(a{q}^j)q^j.$$

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Symbols:

$\int$: integral, $d_{q}x$: $q$-differential, $q$: complex base, $j$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.2.E46

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(v), §17.2 and Ch.17

… ►provided that ${\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}f(q^j)q^j$ converges. ►

§17.2(vi) Rogers–Ramanujan Identities

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12: Bibliography

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$q$-identities of Auluck, Carlitz, and Rogers. Duke Math. J. 33 (3), pp. 575–581.

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External Links:

Document, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§17.9(iv).

Encodings:

BibTeX

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

Encodings:

BibTeX

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Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:

§24.4(iii).

Encodings:

BibTeX

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Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.

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External Links:

FullText, MathReview (F. Schipp), ZentralBlatt

Referenced by:

§16.23(iii), (18.14.25), (18.14.26), (18.14.27), (18.38.3), Profile Richard A. Askey.

Encodings:

BibTeX

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13: Bibliography R

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

BibTeX

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

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Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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External Links:

ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

Referenced by:

§34.6.

Encodings:

BibTeX

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Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii).

Encodings:

BibTeX

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14: 14.18 Sums

§14.18 Sums

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§14.18(iii) Other Sums

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Dougall’s Expansion

… ►For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2000, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). …

15: Bibliography B

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A new method for investigating Euler sums. Ramanujan J. 4 (4), pp. 397–419.

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External Links:

ISSN 1382-4090, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

(25.6.16), (25.6.17), (25.6.18), (25.6.19), (25.6.20), §25.6(iii), §25.6(iii), §25.6.

Encodings:

BibTeX

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Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

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External Links:

ISSN 0022-4715, Document, MathReview, ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

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Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.

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External Links:

ISSN 1431-0643, FullText, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

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16: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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$q$-Gauss Sum

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First $q$-Chu–Vandermonde Sum

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Second $q$-Chu–Vandermonde Sum

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Andrews–Askey Sum

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Rogers–Fine Identity

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17: Bibliography L

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A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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Note sur la fonction $𝔎(w,x,s)={\sum }_{k=0}^{\mathrm{\infty }}\frac{e^{2k\pi ix}}{{(w+k)}^s}$ . Acta Math. 11 (1-4), pp. 19–24 (French).

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§25.14(i), Notations K.

Encodings:

BibTeX

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18: 15.4 Special Cases

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Dougall’s Bilateral Sum

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15.4.25 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+n\right)\mathrm{\Gamma }\left(b+n\right)}{\mathrm{\Gamma }\left(c+n\right)\mathrm{\Gamma }\left(d+n\right)}=\frac{{\pi }^2}{\sin \left(\pi a\right)\sin \left(\pi b\right)}\frac{\mathrm{\Gamma }\left(c+d-a-b-1\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(d-a\right)\mathrm{\Gamma }\left(c-b\right)\mathrm{\Gamma }\left(d-b\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.4(ii)

Permalink:

http://dlmf.nist.gov/15.4.E25

Encodings:

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See also:

Annotations for §15.4(ii), §15.4(ii), §15.4 and Ch.15

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19: 10.23 Sums

§10.23 Sums

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§10.23(i) Multiplication Theorem

… ►For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). … ► …

20: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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Sum Related to (17.6.4)

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$q$-Pfaff–Saalschütz Sum

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F. H. Jackson’s $q$-Analog of Dougall’s ${}_7{}^{}F_6^{}\left(1\right)$ Sum

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Gasper–Rahman $q$-Analogs of the Karlsson–Minton Sums

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Gosper’s Bibasic Sum

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