Bailey 2ψ2 transformations

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11: 4.48 Software

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Bailey (1993). Fortran.

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

12: Bibliography W

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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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G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:: ZentralBlatt
Referenced by:: §15.8(v), §15.8(v).
Encodings:: BibTeX

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T. Weider (1999) Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.

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Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: ISSN 0098-3500, Document, GAMS (module 794; package TOMS), ZentralBlatt
Referenced by:: 7th item, 10th item.
Encodings:: BibTeX

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F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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

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J. Wimp (1964) A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

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External Links:: Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §13.23(iv).
Encodings:: BibTeX

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13: 16.12 Products

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16.12.1

{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\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:: (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

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16.12.2

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

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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, $z$ : 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, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

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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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14: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

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F. H. Jackson’s Transformations

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Transformations of ${{}_{3}\phi_{2}}$ -Series

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

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Sears–Carlitz Transformation

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Mixed-Base Heine-Type Transformations

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15: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

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§17.7(i) ${{}_{2}\phi_{2}}$ Functions

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

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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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16: 17.6 ${{}_{2}\phi_{1}}$ Function

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Bailey–Daum $q$ -Kummer Sum

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§17.6(ii) ${{}_{2}\phi_{1}}$ Transformations

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Heine’s Third Transformation

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Fine’s Second Transformation

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Three-Term ${{}_{2}\phi_{1}}$ Transformations

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17: Bibliography S

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J. L. Schiff (1999) The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:: ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt
Referenced by:: §1.14(iii), §1.14(vii).
Encodings:: BibTeX

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O. A. Sharafeddin, H. F. Bowen, D. J. Kouri, and D. K. Hoffman (1992) Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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External Links:: ISSN 0021-9991, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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I. Shavitt and M. Karplus (1965) Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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External Links:: Document, MathReview (R. K. Nesbet)
Referenced by:: §8.24(i).
Encodings:: BibTeX

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N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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External Links:: ISSN 0253-424X, MathReview
Referenced by:: §9.10(v).
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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18: Bibliography M

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J. P. McClure and R. Wong (1978) Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

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External Links:: ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt
Referenced by:: §2.6(ii).
Encodings:: BibTeX

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A. R. Miller and R. B. Paris (2011) Euler-type transformations for the generalized hypergeometric function

{}_{r+2}F_{r+1}(x)

. Z. Angew. Math. Phys. 62 (1), pp. 31–45.

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External Links:: ISSN 0044-2275, Document, FullText, MathReview (Mridula Garg), ZentralBlatt
Referenced by:: §16.6, §16.6.
Encodings:: BibTeX

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A. R. Miller (2003) On a Kummer-type transformation for the generalized hypergeometric function

{}_{2}F_{2}

. J. Comput. Appl. Math. 157 (2), pp. 507–509.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

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A. E. Milne, P. A. Clarkson, and A. P. Bassom (1997) Bäcklund transformations and solution hierarchies for the third Painlevé equation. Stud. Appl. Math. 98 (2), pp. 139–194.

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External Links:: ISSN 0022-2526, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.10(iii), §32.7(iii), §32.8(iii), §32.9(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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19: 5.24 Software

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

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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. … ►

17.4.6

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};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,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}},

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Defines:: $\Phi^{(2)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c},\NVar{c^{\prime}}% ;\NVar{q};\NVar{x},\NVar{y}\right)$ : second $q$ -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, Erratum (V1.1.3) for Equation (17.4.6)
Permalink:: http://dlmf.nist.gov/17.4.E6
Encodings:: pMML, png, TeX
Clarification (effective with 1.1.3):: The explicit usage of $\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}$ in the denominator of the right-hand side was used.
Correction (effective with 1.0.10):: The notation for $\Phi^{(2)}$ 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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17.4.11

a_{0}q=a_{1}b_{1}=a_{2}b_{2}=\dots=a_{s}b_{s}.

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Symbols:: $q$ : complex base and $s$ : nonnegative integer
Referenced by:: §17.4(iv)
Permalink:: http://dlmf.nist.gov/17.4.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(iv), §17.4 and Ch.17

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17.4.12

b_{1}=-b_{2}=\sqrt{a_{0}}.

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Permalink:: http://dlmf.nist.gov/17.4.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §17.4(iv), §17.4 and Ch.17

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Bailey 2ψ2 transformations

11—20 of 816 matching pages

11: 4.48 Software

12: Bibliography W

13: 16.12 Products

14: 17.9 Further Transformations of ϕ r r + 1 Functions

F. H. Jackson’s Transformations

Transformations of ϕ 2 3 -Series

Bailey’s Transformation of Very-Well-Poised ϕ 7 8

Sears–Carlitz Transformation

Mixed-Base Heine-Type Transformations

15: 17.7 Special Cases of Higher ϕ s r Functions

§17.7(i) ϕ 2 2 Functions

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

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

16: 17.6 ϕ 1 2 Function

Bailey–Daum q -Kummer Sum

§17.6(ii) ϕ 1 2 Transformations