Bailey transformation of very-well-poised 8ϕ7

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11—20 of 342 matching pages

11: Bibliography

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N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:: Translated from the Russian by H. H. McFaden
External Links:: ISBN 0-8218-4524-1, MathReview, ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

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W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

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Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0098-3500, Document, GAMS (module 588; package TOMS)
Referenced by:: 2nd item.
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

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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

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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

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12: 4.48 Software

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Bailey (1993). Fortran.

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

13: 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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J. D. Secada (1999) Numerical evaluation of the Hankel transform. Comput. Phys. Comm. 116 (2-3), pp. 278–294.

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External Links:: ISSN 0010-4655, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:: MathReview (J. Kuntzmann), ZentralBlatt
Referenced by:: §17.18.
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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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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14: 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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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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17.7.14

{{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,q^{-n}% \atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq^{n+1}};q,q\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(cd);q\right)_{n}}{\left(aq/b,aq/c,aq/d,aq% /(bcd);q\right)_{n}},

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Symbols:: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -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
Referenced by:: §17.7(iii)
Permalink:: http://dlmf.nist.gov/17.7.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

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15: Bibliography M

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J. C. Mason (1993) Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. In Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), Vol. 49, pp. 169–178.

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External Links:: ISSN 0377-0427, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Erratum (V1.0.28) for Table 18.3.1.
Encodings:: BibTeX

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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. 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 (1994) A

q

-analog of a Whipple’s transformation for hypergeometric series in

U(n)

. Adv. Math. 108 (1), pp. 1–76.

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External Links:: ISSN 0001-8708, Document, MathReview, ZentralBlatt
Referenced by:: §17.15.
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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16: 5.24 Software

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§32.7 Bäcklund Transformations

… ►Then the transformations … ►The quadratic transformation …The quartic transformation … ►

Bailey transformation of very-well-poised 8ϕ7

11—20 of 342 matching pages

11: Bibliography

12: 4.48 Software

13: Bibliography S

14: 17.7 Special Cases of Higher ϕ s r 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

15: Bibliography M

16: 5.24 Software

17: 17.6 ϕ 1 2 Function

Bailey–Daum q -Kummer Sum

Heine’s First Transformation

Heine’s Third Transformation

Fine’s Second Transformation

Fine’s Third Transformation

18: 3.9 Acceleration of Convergence

§3.9(i) Sequence Transformations

§3.9(ii) Euler’s Transformation of Series

§3.9(iv) Shanks’ Transformation

§3.9(vi) Applications and Further Transformations

19: 17.8 Special Cases of ψ r r Functions

Bailey’s Bilateral Summations

20: 32.7 Bäcklund Transformations

§32.7 Bäcklund Transformations

14: 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

17: 17.6 ${{}_{2}\phi_{1}}$ Function

Bailey–Daum $q$ -Kummer Sum

19: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions