Bailey–Daum q-Kummer sum

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1: 17.6 ${{}_{2}\phi_{1}}$ 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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Bailey–Daum $q$ -Kummer Sum

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2: 17.12 Bailey Pairs

§17.12 Bailey Pairs

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Bailey Transform

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Bailey Pairs

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Weak Bailey Lemma

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Strong Bailey Lemma

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3: 17 q-Hypergeometric and Related Functions

Chapter 17 $q$ -Hypergeometric and Related Functions

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4: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

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17.8.1

\sum_{n=-\infty}^{\infty}(-z)^{n}q^{n(n-1)/2}=\left(q,z,q/z;q\right)_{\infty};

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Symbols:: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

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17.8.3

\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}z^{3n}(1+zq^{n})=\left(q,-z,-q/z% ;q\right)_{\infty}\left(qz^{2},q/{z^{2}};q^{2}\right)_{\infty}.

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Symbols:: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

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Bailey’s Bilateral Summations

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Sum Related to (17.6.4)

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5: 4.48 Software

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Bailey (1993). Fortran.

… ►See also Bailey (1995), Hull and Abrham (1986), Xu and Li (1994). …

6: 17.1 Special Notation

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f(\chi_{1};\chi_{2},\dots,\chi_{n})+\operatorname{idem}\left(\chi_{1};\chi_{2}% ,\dots,\chi_{n}\right)=\sum_{j=1}^{n}f(\chi_{j};\chi_{1},\chi_{2},\dots,\chi_{% j-1},\chi_{j+1},\dots,\chi_{n}).

… ►A slightly different notation is that in Bailey (1964) and Slater (1966); see §17.4(i). …

7: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

§17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

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Bailey’s ${{}_{2}\psi_{2}}$ Transformations

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17.10.4

{{}_{2}\psi_{2}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.

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Symbols:: ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : 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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8: 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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9: 16.12 Products

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16.12.3

\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},

|z|<1

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : 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:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

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10: Bibliography B

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D. H. Bailey (1993) Algorithm 719: Multiprecision translation and execution of Fortran programs. ACM Trans. Math. Software 19 (3), pp. 288–319.

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External Links:: ISSN 0098-3500, Document, GAMS (module 719; package TOMS), C++ wrapper, ZentralBlatt
Referenced by:: 3rd item, 2nd item.
Encodings:: BibTeX

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D. H. Bailey (1995) A Fortran-90 based multiprecision system. ACM Trans. Math. Software 21 (4), pp. 379–387.

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External Links:: ISSN 0098-3500, Document, ZentralBlatt
Referenced by:: §4.48(iii).
Encodings:: BibTeX

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W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

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W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

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W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.

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External Links:: MathReview
Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.
Encodings:: BibTeX

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Bailey–Daum q-Kummer sum

1—10 of 373 matching pages

1: 17.6 ϕ 1 2 Function

q -Gauss Sum

First q -Chu–Vandermonde Sum

Second q -Chu–Vandermonde Sum

Andrews–Askey Sum

Bailey–Daum q -Kummer Sum

2: 17.12 Bailey Pairs

§17.12 Bailey Pairs

Bailey Transform

Bailey Pairs

Weak Bailey Lemma

Strong Bailey Lemma

3: 17 q-Hypergeometric and Related Functions

Chapter 17 q -Hypergeometric and Related Functions

4: 17.8 Special Cases of ψ r r Functions

Bailey’s Bilateral Summations

Sum Related to (17.6.4)

5: 4.48 Software

6: 17.1 Special Notation

7: 17.10 Transformations of ψ r r Functions

§17.10 Transformations of ψ r r Functions

Bailey’s ψ 2 2 Transformations

8: 17.7 Special Cases of Higher ϕ s r Functions

q -Analog of Bailey’s F 1 2 ⁡ ( − 1 ) Sum

Sum Related to (17.6.4)

First q -Analog of Bailey’s F 3 4 ⁡ ( 1 ) Sum

Second q -Analog of Bailey’s F 3 4 ⁡ ( 1 ) Sum

Bailey’s Nonterminating Extension of Jackson’s ϕ 7 8 Sum

9: 16.12 Products

10: Bibliography B

1: 17.6 ${{}_{2}\phi_{1}}$ Function

$q$ -Gauss Sum

First $q$ -Chu–Vandermonde Sum

Second $q$ -Chu–Vandermonde Sum

Bailey–Daum $q$ -Kummer Sum

Chapter 17 $q$ -Hypergeometric and Related Functions

4: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

7: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

§17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

Bailey’s ${{}_{2}\psi_{2}}$ Transformations

8: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

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

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

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

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