Bailey 2ψ2 transformations

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3: 16.6 Transformations of Variable

§16.6 Transformations of Variable

►

Quadratic

… ►

Cubic

►

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).

4: 17 q-Hypergeometric and Related Functions

…

5: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

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

►

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

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17.10.1

{{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),

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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 $z$ : 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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Other Transformations

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17.10.3

{{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),

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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.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.10, §17.10 and Ch.17

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6: Bibliography B

… ►

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

►

W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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External Links:: ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt
Referenced by:: §18.18(vii).
Encodings:: BibTeX

… ►

J. L. Burchnall and T. W. Chaundy (1948) The hypergeometric identities of Cayley, Orr, and Bailey. Proc. London Math. Soc. (2) 50, pp. 56–74.

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External Links:: Document, MathReview (N. A. Hall), ZentralBlatt
Referenced by:: §15.16.
Encodings:: BibTeX

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7: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

… ►

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

►

Bailey’s Bilateral Summations

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17.8.4

{{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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 and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

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17.8.6

{{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},

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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 and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §17.8, §17.8 and Ch.17

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17.8.8

{{}_{2}\psi_{2}}\left({b^{2},\ifrac{b^{2}}{c}\atop q,cq};q^{2},\ifrac{cq^{2}}{% b^{2}}\right)=\frac{1}{2}\frac{\left(q^{2},qb^{2},\ifrac{q}{b^{2}},\ifrac{cq}{% b^{2}};q^{2}\right)_{\infty}}{\left(cq,\ifrac{cq^{2}}{b^{2}},\ifrac{q^{2}}{b^{% 2}},\ifrac{c}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{c\sqrt{q}}% {b};q\right)_{\infty}}{\left(b\sqrt{q};q\right)_{\infty}}+\frac{\left(\ifrac{-% c\sqrt{q}}{b};q\right)_{\infty}}{\left(-b\sqrt{q};q\right)_{\infty}}\right),

\left|cq^{2}\right|<\left|b^{2}\right|

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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 and $q$ : complex base
Source:: Verma and Jain (1983, (3.9))
Referenced by:: §17.6(i), §17.8, Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/17.8.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.8, §17.8 and Ch.17

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8: Bibliography

… ►

F. V. Andreev and A. V. Kitaev (2002) Transformations

RS^{2}_{4}(3)

of the ranks

\leq 4

and algebraic solutions of the sixth Painlevé equation. Comm. Math. Phys. 228 (1), pp. 151–176.

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External Links:: ISSN 0010-3616, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

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G. E. Andrews and A. Berkovich (1998) A trinomial analogue of Bailey’s lemma and

N=2

superconformal invariance. Comm. Math. Phys. 192 (2), pp. 245–260.

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External Links:: ISSN 0010-3616, Document, MathReview (Anne Schilling), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

… ►

G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

… ►

G. E. Andrews (2000) Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.

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Notes:: In memory of Gian-Carlo Rota
External Links:: ISSN 0097-3165, Document, MathReview (Alessandro Di Bucchianico), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

►

G. E. Andrews (2001) Bailey’s Transform, Lemma, Chains and Tree. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.

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External Links:: ISBN 0-7923-7119-4, MathReview, ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

…

9: 17.1 Special Notation

… ►The main functions treated in this chapter are the basic hypergeometric (or

q

-hypergeometric) function

{{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, the bilateral basic hypergeometric (or bilateral

q

-hypergeometric) function

{{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, and the

q

-analogs of the Appell functions

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)

, and

\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)

. … ►Another function notation used is the “idem” function: ►

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). Fine (1988) uses

F(a,b;t:q)

for a particular specialization of a

{{}_{2}\phi_{1}}

function.

10: 16.4 Argument Unity

… ►See Bailey (1964, pp. 19–22). … ►See Raynal (1979), Wilson (1978), and Bailey (1964). … ►See Bailey (1964, §4.4(4)). ►Transformations for both balanced

{{}_{4}F_{3}}\left(1\right)

and very well-poised

{{}_{7}F_{6}}\left(1\right)

are included in Bailey (1964, pp. 56–63). …See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …

Bailey 2ψ2 transformations

1—10 of 816 matching pages

1: 1.14 Integral Transforms

§1.14 Integral Transforms

§1.14(i) Fourier Transform

§1.14(iii) Laplace Transform

Fourier Transform

Laplace Transform

2: 17.12 Bailey Pairs

§17.12 Bailey Pairs

Bailey Transform

Bailey Pairs

Weak Bailey Lemma

Strong Bailey Lemma

3: 16.6 Transformations of Variable

§16.6 Transformations of Variable

Quadratic

Cubic

4: 17 q-Hypergeometric and Related Functions

5: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

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

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

Other Transformations

6: Bibliography B

7: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

Bailey’s Bilateral Summations

8: Bibliography

9: 17.1 Special Notation

10: 16.4 Argument Unity

Bailey 2ψ2 transformations

1—10 of 816 matching pages

1: 1.14 Integral Transforms

§1.14 Integral Transforms

§1.14(i) Fourier Transform

§1.14(iii) Laplace Transform

Fourier Transform

Laplace Transform

2: 17.12 Bailey Pairs

§17.12 Bailey Pairs

Bailey Transform

Bailey Pairs

Weak Bailey Lemma

Strong Bailey Lemma

3: 16.6 Transformations of Variable

§16.6 Transformations of Variable

Quadratic

Cubic

4: 17 q-Hypergeometric and Related Functions

5: 17.10 Transformations of ψ r r Functions

§17.10 Transformations of ψ r r Functions

Bailey’s ψ 2 2 Transformations

Other Transformations

6: Bibliography B

7: 17.8 Special Cases of ψ r r Functions

Bailey’s Bilateral Summations

8: Bibliography

9: 17.1 Special Notation

10: 16.4 Argument Unity

5: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions

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

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

7: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions