Dougall very well-poised sum

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1: 16.4 Argument Unity

… ►It is very well-poised if it is well-poised and $a_1=b_1+1$. … ►

Dixon’s Well-Poised Sum

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Rogers–Dougall Very Well-Poised Sum

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Dougall’s Very Well-Poised Sum

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2: 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. … ►

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

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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:

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See also:

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

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17.4.5 $${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)=\sum _{m,n\ge 0}\frac{{(a;q)}_{m+n}{(b;q)}_m{(b^{\prime };q)}_n{x}^m{y}^n}{{(q;q)}_m{(q;q)}_n{(c;q)}_{m+n}},$$

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Defines:

${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$: first $q$-Appell function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.4.E5

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

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

… ►The series (17.4.1) is said to be well-poised when $r=s$ and … ►The series (17.4.1) is said to be very-well-poised when $r=s$, (17.4.11) is satisfied, and …

3: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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Bailey’s Transformation of Very-Well-Poised ${}_8{}^{}\varphi _7^{}$

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§17.9(iv) Bibasic Series

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17.9.19 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(a;q^2)}_n{(b;q)}_n}{{(q^2;q^2)}_n{(c;q)}_n}{z}^n=\frac{{(b;q)}_{\mathrm{\infty }}{(az;q^2)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(z;q^2)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_{2n}{(z;q^2)}_n{b}^{2n}}{{(q;q)}_{2n}{(az;q^2)}_n}+\frac{{(b;q)}_{\mathrm{\infty }}{(azq;q^2)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(zq;q^2)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_{2n+1}{(zq;q^2)}_n{b}^{2n+1}}{{(q;q)}_{2n+1}{(azq;q^2)}_n}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

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17.9.20 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(a;q^k)}_n{(b;q)}_{kn}{z}^n}{{(q^k;q^k)}_n{(c;q)}_{kn}}=\frac{{(b;q)}_{\mathrm{\infty }}{(az;q^k)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}{(z;q^k)}_{\mathrm{\infty }}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(c/b;q)}_n{(z;q^k)}_n{b}^n}{{(q;q)}_n{(az;q^k)}_n},$$ $k=1,2,3,\mathrm{\dots }$.

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.9.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

4: Bibliography M

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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External Links:

ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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External Links:

ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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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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Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43 (6), pp. 3363–3386.

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External Links:

ISSN 0022-2488, Document, MathReview (Driss Essouabri), ZentralBlatt

Referenced by:

§16.24(ii).

Encodings:

BibTeX

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5: 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). …

6: 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

…

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: 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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9: 27.19 Methods of Computation: Factorization

… ►The snfs can be applied only to numbers that are very close to a power of a very small base. …

10: 17.18 Methods of Computation

… ►Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. …