Dixon 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

… ►

Rogers–Dougall Very Well-Poised Sum

… ►

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^{\genfrac{(}{)}{0.0pt}{}{n}{2}}\right)^{s-r}

is not included in the sum. … ►

17.4.3

{{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q% ,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s% };q,z\right)=\sum_{n=-\infty}^{\infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q% \right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{% 1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{\infty}\frac{\left(a_{1},a_{2},% \dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{0.0pt}{}{n}{2}}z^{% n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}+\sum_{n=1}^{\infty}\frac{\left% (q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{\left(q/a_{1},q/a_{2},\dots,q/a_{% r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b_{s}}{a_{1}a_{2}\cdots a_{r}z}% \right)^{n}.

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Defines:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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

… ►

17.4.5

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\sum_{m,n\geq 0}\frac{\left(a;q% \right)_{m+n}\left(b;q\right)_{m}\left(b^{\prime};q\right)_{n}x^{m}y^{n}}{% \left(q;q\right)_{m}\left(q;q\right)_{n}\left(c;q\right)_{m+n}},

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Defines:: $\Phi^{(1)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x}% ,\NVar{y}\right)$ : first $q$ -Appell function
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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 $\Phi^{(1)}$ 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 ${{}_{r+1}\phi_{r}}$ Functions

… ►

Bailey’s Transformation of Very-Well-Poised ${{}_{8}\phi_{7}}$

… ►

§17.9(iv) Bibasic Series

… ►

17.9.19

\sum_{n=0}^{\infty}\frac{\left(a;q^{2}\right)_{n}\left(b;q\right)_{n}}{\left(q% ^{2};q^{2}\right)_{n}\left(c;q\right)_{n}}z^{n}=\frac{\left(b;q\right)_{\infty% }\left(az;q^{2}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{2}\right)% _{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{2n}\left(z;q^{2}\right)% _{n}b^{2n}}{\left(q;q\right)_{2n}\left(az;q^{2}\right)_{n}}+\frac{\left(b;q% \right)_{\infty}\left(azq;q^{2}\right)_{\infty}}{\left(c;q\right)_{\infty}% \left(zq;q^{2}\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{2n% +1}\left(zq;q^{2}\right)_{n}b^{2n+1}}{\left(q;q\right)_{2n+1}\left(azq;q^{2}% \right)_{n}}.

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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

►

17.9.20

\sum_{n=0}^{\infty}\frac{\left(a;q^{k}\right)_{n}\left(b;q\right)_{kn}z^{n}}{% \left(q^{k};q^{k}\right)_{n}\left(c;q\right)_{kn}}=\frac{\left(b;q\right)_{% \infty}\left(az;q^{k}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{k}% \right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{n}\left(z;q^{k}% \right)_{n}b^{n}}{\left(q;q\right)_{n}\left(az;q^{k}\right)_{n}},

k=1,2,3,\dots

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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

… ►

S. C. Milne (1985a) A

q

-analog of the

{}_{5}F_{4}(1)

summation theorem for hypergeometric series well-poised in

\mathit{SU}(n)

. Adv. in Math. 57 (1), pp. 14–33.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

S. C. Milne (1985d) A

q

-analog of hypergeometric series well-poised in

\mathit{SU}(n)

and invariant

G

-functions. Adv. in Math. 58 (1), pp. 1–60.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

… ►

S. C. Milne (1996) 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.

ⓘ

External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

… ►

S. Moch, P. Uwer, and S. Weinzierl (2002) Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43 (6), pp. 3363–3386.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Driss Essouabri), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

…

5: Peter A. Clarkson

… ► Dixon and J. …

6: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

… ►

Sum Related to (17.6.4)

… ►

$q$ -Pfaff–Saalschütz Sum

… ►

F. H. Jackson’s Terminating $q$ -Analog of Dixon’s Sum

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$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

… ►

Gosper’s Bibasic Sum

…

7: Bibliography D

… ►

K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

… ►

A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

ⓘ

External Links:: ISSN 0377-9017, Document, MathReview, ZentralBlatt
Referenced by:: §14.17(iv).
Encodings:: BibTeX

… ►

A. L. Dixon and W. L. Ferrar (1930) Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §10.32(iii).
Encodings:: BibTeX

►

J. M. Dixon, J. A. Tuszyński, and P. A. Clarkson (1997) From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics. Oxford University Press, Oxford.

ⓘ

Referenced by:: Profile Peter A. Clarkson.
Encodings:: BibTeX

… ►

T. M. Dunster (1990a) 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) $\phi(\zeta)$ 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

…

8: 4.27 Sums

§4.27 Sums

►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2015, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).

9: 4.11 Sums

§4.11 Sums

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10: 4.41 Sums

§4.41 Sums

►For sums of hyperbolic functions see Gradshteyn and Ryzhik (2015, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).