Dougall bilateral sum

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1: 16.4 Argument Unity

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Rogers–Dougall Very Well-Poised Sum

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Dougall’s Very Well-Poised Sum

… ►Denote, formally, the bilateral hypergeometric function …This is Dougall’s bilateral sum; see Andrews et al. (1999, §2.8).

2: 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:

pMML, png, TeX

See also:

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

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3: 17.1 Special Notation

§17.1 Special Notation

… ►The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${}_r{}^{}\varphi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, and the $q$-analogs of the Appell functions ${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$, ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$, ${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$, and ${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$. … ►

$$f({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)+\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)=\sum _{j=1}^{n}f({\chi }_j;{\chi }_1,{\chi }_2,\mathrm{\dots },{\chi }_{j-1},{\chi }_{j+1},\mathrm{\dots },{\chi }_n).$$

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4: 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 (2015, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). …

5: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

§17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Ramanujan’s ${}_1{}^{}\psi _1^{}$ Summation

… ►These identities are all given in terms of sums and products of basic bilateral hypergeometric series. ►

Bailey’s Bilateral Summations

… ►For similar formulas see Verma and Jain (1983).

6: 17.18 Methods of Computation

§17.18 Methods of Computation

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7: 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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8: 17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

§17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

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17.10.1 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,d/a,c/b,dq/(abz);q)}_{\mathrm{\infty }}}{{(z,d,q/b,cd/(abz);q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,abz/d}{az,c};q,\frac{d}{a}),$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.2 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,bz,cq/(abz),dq/(abz);q)}_{\mathrm{\infty }}}{{(q/a,q/b,c,d;q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{abz/c,abz/d}{az,bz};q,\frac{cd}{abz}).$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.3 $${}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},c,d,e,f,a{q}^{-n},q^{-n}}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},a{q}^{n+1}};q,\frac{a^2{q}^{2n+2}}{cdef})=\frac{{(aq,q/a,aq/(cd),aq/(ef);q)}_n}{{(q/c,q/d,aq/e,aq/f;q)}_n}{}_4{}^{}\psi _4^{}(\genfrac{}{}{0pt}{}{e,f,a{q}^{n+1}/(cd),q^{-n}}{aq/c,aq/d,q^{n+1},ef/(a{q}^n)};q,q),$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.4 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{e,f}{aq/c,aq/d};q,\frac{aq}{ef})=\frac{{(q/c,q/d,aq/e,aq/f;q)}_{\mathrm{\infty }}}{{(aq,q/a,aq/(cd),aq/(ef);q)}_{\mathrm{\infty }}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(1-a{q}^{2n}){(c,d,e,f;q)}_n}{(1-a){(aq/c,aq/d,aq/e,aq/f;q)}_n}{\left(\frac{q{a}^3}{cdef}\right)}^n{q}^{n^2}.$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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9: 17.4 Basic Hypergeometric Functions

§17.4 Basic Hypergeometric Functions

… ►In these references the factor ${\left({(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\right)}^{s-r}$ is not included in the sum. … ►

§17.4(ii) ${}_r{}^{}\psi _s^{}$ Functions

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17.4.3 $${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_s};q,z)={}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}=\sum _{n=0}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(q/b_1,q/b_2,\mathrm{\dots },q/b_s;q)}_n}{{(q/a_1,q/a_2,\mathrm{\dots },q/a_r;q)}_n}{\left(\frac{b_1{b}_2\mathrm{\cdots }{b}_s}{a_1{a}_2\mathrm{\cdots }{a}_{r}z}\right)}^n.$$

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Defines:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§17.8

Permalink:

http://dlmf.nist.gov/17.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.4(ii), §17.4 and Ch.17

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17.4.4 $$\underset{q\to 1-}{lim}{}_r{}^{}\psi _r^{}(\genfrac{}{}{0pt}{}{q^{a_1},q^{a_2},\mathrm{\dots },q^{a_r}}{q^{b_1},q^{b_2},\mathrm{\dots },q^{b_r}};q,z)={}_r{}^{}H_r^{}(\genfrac{}{}{0pt}{}{a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_r};z).$$

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Symbols:

${}_p{}^{}H_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: bilateral hypergeometric function, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, $q$: complex base, $r$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.4(ii), §17.4 and Ch.17

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10: Bibliography D

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Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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External Links:

ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

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External Links:

ISSN 0377-9017, Document, MathReview, ZentralBlatt

Referenced by:

§14.17(iv).

Encodings:

BibTeX

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

BibTeX

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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

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