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21: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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17.9.3_5 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=\frac{{(c/a,c/b;q)}_{\mathrm{\infty }}}{{(c,c/(ab);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,abz/c}{qab/c,0};q,q)+\frac{{(a,b,abz/c;q)}_{\mathrm{\infty }}}{{(c,ab/c,z;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{c/a,c/b,z}{qc/(ab),0};q,q),$$

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Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable

Proof sketch:

Derivable from (17.9.13) after replacing $a$ with $d/a$ and then taking $d\to 0$. Then replace $a$, $b$, $c$, $e$ with $abz/c$, $a$, $b$, $c$ respectively.

Referenced by:

§17.9(i), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/17.9.E3_5

Encodings:

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Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

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17.9.6 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};q,de/(abc))=\frac{{(e/a,de/(bc);q)}_{\mathrm{\infty }}}{{(e,de/(abc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,d/b,d/c}{d,de/(bc)};q,e/a),$$

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Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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17.9.13 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};q,\frac{de}{abc})=\frac{{(e/b,e/c;q)}_{\mathrm{\infty }}}{{(e,e/(bc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{d/a,b,c}{d,bcq/e};q,q)+\frac{{(d/a,b,c,de/(bc);q)}_{\mathrm{\infty }}}{{(d,e,bc/e,de/(abc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{e/b,e/c,de/(abc)}{de/(bc),eq/(bc)};q,q).$$

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Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Referenced by:

(17.9.3_5)

Permalink:

http://dlmf.nist.gov/17.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

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17.9.14 $${}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},a,b,c}{d,e,f};q,q)=\frac{{(e/a,f/a;q)}_n}{{(e,f;q)}_n}{a}^n{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},a,d/b,d/c}{d,a{q}^{1-n}/e,a{q}^{1-n}/f};q,q)=\frac{{(a,ef/(ab),ef/(ac);q)}_n}{{(e,f,ef/(abc);q)}_n}{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},e/a,f/a,ef/(abc)}{ef/(ab),ef/(ac),q^{1-n}/a};q,q).$$

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Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

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Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)

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22: Bibliography

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Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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Notes:

Corrections appeared in later printings up to the 10th Printing, December, 1972. Reproductions by other publishers, in whole or in part, have been available since 1965.

External Links:

FullText, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§10.1, §10.21(ix), §10.47(i), §10.68(iv), §10.70, 4th item, 2nd item, 4th item, 5th item, 1st item, 1st item, 4th item, 1st item, 1st item, Ch.10, §11.10(vi), 1st item, 1st item, Ch.11, 1st item, Ch.12, 3rd item, Ch.13, 1st item, Ch.14, Ch.15, §18.14(i), §18.3, §18.41(i), §18.41(ii), §18.5(iv), §18.8, §19.1, §19.14(i), §19.14(ii), §19.25(v), §19.36(iii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.38, Ch.19, §20.15, Ch.20, §22.1, §22.15(ii), Ch.22, §23.1, §23.20(i), §23.23, §23.23, §23.9, §23.9, §23.9, Ch.23, §24.2(iv), §24.20, Ch.24, 1st item, §26.1, §26.1, §26.2, §26.21, §26.3(i), §26.4(i), §26.8(i), §27.2(ii), §27.21, §28.1, Ch.28, §30.1, Ch.30, 1st item, Ch.33, §4.44, §4.46, §5.11(i), §5.22(ii), §5.22(iii), §5.7(i), Ch.5, 1st item, 1st item, 1st item, Ch.6, 1st item, 2nd item, 1st item, Ch.7, (8.17.24), §8.22(ii), 1st item, Ch.8, 1st item, 1st item, 1st item, §9.8(i), Ch.9, Profile Frank W. J. Olver, About the Project, Preface, Possible Errors in DLMF, Notations E, Notations E, Notations F, Notations F, Notations H, Notations H, Notations J, Notations K, Notations P, Notations P, Notations S, Notations Y, H. E. Fettis and J. C. Caslin (1973).

Encodings:

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

Encodings:

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Special value of the hypergeometric function ${}_3{}^{}F_2^{}$ and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.

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External Links:

ISSN 0019-5839, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

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Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.

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External Links:

ISSN 0025-5718, FullText, MathReview (Aleksandar Ivić), ZentralBlatt

Referenced by:

(25.11.19), (25.11.20), (25.11.6), §25.11(vi), §25.18(i), (25.4.5), (25.4.6), §25.4, (25.6.11), (25.6.12), (25.6.13), (25.6.14), §25.6(ii), §25.6.

Encodings:

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Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:

§24.4(iii).

Encodings:

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23: Bibliography K

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On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.15(iv).

Encodings:

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On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).

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Notes:

In Russian. English translation: Soviet Math. Dokl. 33(3), 650–653, (1986).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(i), §10.74(vi).

Encodings:

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On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25

External Links:

ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 5th item.

Encodings:

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Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).

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Notes:

In Russian. English translation: Soviet Phys. Dokl. 33(3),196–198, (1988).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi).

Encodings:

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The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

Encodings:

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24: Bibliography C

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A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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A recurrence formula for $\zeta (2n)$ . Proc. Amer. Math. Soc. 12 (6), pp. 991–992.

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External Links:

FullText, MathReview (R. D. James)

Referenced by:

§24.14(i).

Encodings:

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An extension of a Kummer’s quadratic transformation formula with an application. Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.

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External Links:

ISSN 1598-7264, MathReview (Petr Blaschke), ZentralBlatt

Referenced by:

§16.6, §16.6.

Encodings:

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Approximations and Complex Multiplication According to Ramanujan. In Ramanujan Revisited (Urbana-Champaign, Ill., 1987), G. E. Andrews, R. A. Askey, B. C. Bernd, K. G. Ramanathan, and R. A. Rankin (Eds.), pp. 375–472.

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External Links:

MathReview (Maurice Mignotte), ZentralBlatt

Referenced by:

§20.11(iii).

Encodings:

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An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.

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Notes:

Numerical integration and its complexity (Oberwolfach, 2001)

External Links:

ISSN 0885-064X, Document, Web Page, MathReview, ZentralBlatt

Referenced by:

§3.5(x).

Encodings:

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25: 1.9 Calculus of a Complex Variable

… ►A contour is simple if it contains no multiple points, that is, for every pair of distinct values $t_1,t_2$ of $t$, $z(t_1)\ne z(t_2)$. … ►

Cauchy’s Integral Formula

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Operations

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26: 27.13 Functions

… ►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. … ►A general formula states that … ►Explicit formulas for $r_k\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. Exact formulas for $r_k\left(n\right)$ have also been found for $k=3,5$, and $7$, and for all even $k\le 24$. …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …

27: 4.35 Identities

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§4.35(i) Addition Formulas

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§4.35(iii) Multiples of the Argument

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