Bailey%E2%80%93Daum%20q-Kummer%20sum

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11: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

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

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Sum Related to (17.6.4)

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First $q$ -Analog of Bailey’s ${{}_{4}F_{3}}\left(1\right)$ Sum

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Second $q$ -Analog of Bailey’s ${{}_{4}F_{3}}\left(1\right)$ Sum

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Bailey’s Nonterminating Extension of Jackson’s ${{}_{8}\phi_{7}}$ Sum

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12: 16.4 Argument Unity

… ►Methods of deriving such identities are given by Bailey (1964), Rainville (1960), Raynal (1979), and Wilson (1978). … ►See Bailey (1964, pp. 19–22). … ►See Raynal (1979), Wilson (1978), and Bailey (1964). … ►See Bailey (1964, §4.4(4)). … ►See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …

13: 5.24 Software

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•

Bailey (1993). Fortran and C++ wrapper.

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14: Bibliography W

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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S. O. Warnaar (1998) A note on the trinomial analogue of Bailey’s lemma. J. Combin. Theory Ser. A 81 (1), pp. 114–118.

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External Links:: ISSN 0097-3165, Document, MathReview (Anne Schilling), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

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E. J. Weniger (1996) Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:: Document
Referenced by:: §2.11(vi), §2.11(vi).
Encodings:: BibTeX

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15: Bibliography M

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L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §14.17(i).
Encodings:: BibTeX

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S. C. Milne and G. M. Lilly (1992) The

A_{l}

and

C_{l}

Bailey transform and lemma. Bull. Amer. Math. Soc. (N.S.) 26 (2), pp. 258–263.

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External Links:: ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

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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.

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

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S. C. Milne (1997) Balanced

\sideset{{}_{3}}{{}_{2}}{\mathop{\Theta}}

summation theorems for

U(n)

basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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

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L. J. Mordell (1917) 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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16: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

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

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§17.9(iv) Bibasic Series

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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

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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

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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

17: 16.6 Transformations of Variable

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16.6.1

{{}_{3}F_{2}}\left({a,b,c\atop a-b+1,a-c+1};z\right)=(1-z)^{-a}{{}_{3}F_{2}}% \left({a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)\atop a-b+1,a-c+1};\frac{-4z}{(1-z% )^{2}}\right).

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

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16.6.2

{{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

►For Kummer-type transformations of

{{}_{2}F_{2}}

functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).

18: Bibliography S

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K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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R. Sips (1949) Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales. Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).

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External Links:: FullText, MathReview (M. Strutt), ZentralBlatt
Referenced by:: §28.8(ii).
Encodings:: BibTeX

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V. P. Spiridonov (2002) An elliptic incarnation of the Bailey chain. Int. Math. Res. Not. 2002 (37), pp. 1945–1977.

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External Links:: ISSN 1073-7928, Document, MathReview (S. Ole Warnaar), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

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J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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19: 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\leq x

: … ►the sum of the

k

th powers of the positive integers

m\leq 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

\phi\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

. …

20: 17.4 Basic Hypergeometric Functions

… ►It is slightly at variance with the notation in Bailey (1964) and Slater (1966). 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
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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17.4.5

\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

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17.4.7

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\sum_{m,n\geq 0}\frac% {\left(a,b;q\right)_{m}\left(a^{\prime},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:: $\Phi^{(3)}\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},\NVar{b^{\prime}};\NVar{c}% ;\NVar{q};\NVar{x},\NVar{y}\right)$ : third $q$ -Appell function
Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : 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.E7
Encodings:: pMML, png, TeX
Correction (effective with 1.0.10):: The notation for $\Phi^{(3)}$ 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

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