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21: Bibliography D

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An approximation for ${\int }_{\chi }^{\mathrm{\infty }}{e}^{-t^2/2}{t}^p𝑑t,$ $\chi >0,$ $p$ real. Math. Comp. 32 (141), pp. 271–275.

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External Links:

FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

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Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.

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External Links:

ISSN 0962-8444, FullText, MathReview (Alexander Tovbis), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

BibTeX

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22: Errata

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The second and the fourth lines containing $k^{\prime }/ik$ have both been replaced with $-i{k}^{\prime }/k$ to clarify the meaning.

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Scales were corrected in all figures. The interval $-8.4\le \frac{x-y}{\sqrt{2}}\le 8.4$ was replaced by $-12.0\le \frac{x-y}{\sqrt{2}}\le 12.0$ and $-12.7\le \frac{x+y}{\sqrt{2}}\le 4.2$ replaced by $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0$. All plots and interactive visualizations were regenerated to improve image quality.

(a) Density plot.	(b) 3D plot.

Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,0\right)|$.

(a) Density plot.	(b) 3D plot.

Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,1\right)|$.

(a) Density plot.	(b) 3D plot.

Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,2\right)|$.

(a) Density plot.	(b) 3D plot.

Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,3\right)|$.

Reported 2016-09-12 by Dan Piponi.

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9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

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9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

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The entry for $\mathrm{sn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime })$ has been corrected. The correct entry is $(1+\mathrm{i})({(1+k^{\prime })}^{1/2}-\mathrm{i}{(1-k^{\prime })}^{1/2})/(2k^{1/2})$. Originally the terms ${(1+k^{\prime })}^{1/2}$ and ${(1-k^{\prime })}^{1/2}$ were given incorrectly as ${(1+k)}^{1/2}$ and ${(1-k)}^{1/2}$.

Similarly, the entry for $\mathrm{dn}z$ at $z=\frac{3}{2}(K+\mathrm{i}{K}^{\prime })$ has been corrected. The correct entry is $(-1+\mathrm{i})k_{}^{\prime }{}_{}{}^{1/2}({(1+k)}^{1/2}+i{(1-k)}^{1/2})/2$. Originally the terms ${(1+k)}^{1/2}$ and ${(1-k)}^{1/2}$ were given incorrectly as ${(1+k^{\prime })}^{1/2}$ and ${(1-k^{\prime })}^{1/2}$

Reported 2014-02-28 by Svante Janson.

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23: Bibliography S

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Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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External Links:

ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

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Effective calculation of the incomplete gamma function for parameter values $\alpha =(2n+1)/2$, $n=0,\mathrm{\dots },5$ . Angew. Informatik 17, pp. 30–32.

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Notes:

Includes Fortran code.

External Links:

ZentralBlatt

Referenced by:

§8.28(ii).

Encodings:

BibTeX

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Computation of angular momentum coefficients using sets of generalized hypergeometric functions. Comput. Phys. Comm. 22 (2-3), pp. 297–302.

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External Links:

Document

Referenced by:

§34.13.

Encodings:

BibTeX

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A stable quotient-difference algorithm. Math. Comp. 34 (150), pp. 515–519.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§3.10(ii).

Encodings:

BibTeX

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On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc. 1, pp. 731–737.

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External Links:

ISSN 0002-9939, Document, Link, MathReview (A. C. Offord)

Referenced by:

§18.35(i).

Encodings:

BibTeX

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24: Bibliography I

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Quasi-linear Stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16 (1), pp. 363–386.

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External Links:

ISSN 0951-7715, Document, MathReview (Youmin Lu), ZentralBlatt

Referenced by:

§32.12(ii).

Encodings:

BibTeX

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25: Bibliography O

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On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

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On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Math. 169 (1), pp. 235–246.

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External Links:

ISSN 0377-0427, Document, MathReview (Jorge Mozo Fernández), ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

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26: 4.30 Elementary Properties

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Table 4.30.1: Hyperbolic functions: interrelations. …

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	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
$\sinh \theta$	$a$	${(a^2-1)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(1-a^2)}^{1/2}$	${(a^2-1)}^{-1/2}$
$\cosh \theta$	${(1+a^2)}^{1/2}$	$a$	${(1-a^2)}^{-1/2}$	$a^{-1}{(1+a^2)}^{1/2}$	$a^{-1}$	$a{(a^2-1)}^{-1/2}$
$\tanh \theta$	$a{(1+a^2)}^{-1/2}$	$a^{-1}{(a^2-1)}^{1/2}$	$a$	${(1+a^2)}^{-1/2}$	${(1-a^2)}^{1/2}$	$a^{-1}$
$\mathrm{csch}\theta$	$a^{-1}$	${(a^2-1)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	$a{(1-a^2)}^{-1/2}$	${(a^2-1)}^{1/2}$
$\mathrm{sech}\theta$	${(1+a^2)}^{-1/2}$	$a^{-1}$	${(1-a^2)}^{1/2}$	$a{(1+a^2)}^{-1/2}$	$a$	$a^{-1}{(a^2-1)}^{1/2}$
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27: 17.10 Transformations of ${}_r{}^{}\psi _r^{}$ Functions

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17.10.1 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,b}{c,d};q,z)=\frac{{(az,d/a,c/b,dq/(abz);q)}_{\mathrm{\infty }}}{{(z,d,q/b,cd/(abz);q)}_{\mathrm{\infty }}}{}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{a,abz/d}{az,c};q,\frac{d}{a}),$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: 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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17.10.3 $${}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},c,d,e,f,a{q}^{-n},q^{-n}}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},a{q}^{n+1}};q,\frac{a^2{q}^{2n+2}}{cdef})=\frac{{(aq,q/a,aq/(cd),aq/(ef);q)}_n}{{(q/c,q/d,aq/e,aq/f;q)}_n}{}_4{}^{}\psi _4^{}(\genfrac{}{}{0pt}{}{e,f,a{q}^{n+1}/(cd),q^{-n}}{aq/c,aq/d,q^{n+1},ef/(a{q}^n)};q,q),$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-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/17.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.4 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{e,f}{aq/c,aq/d};q,\frac{aq}{ef})=\frac{{(q/c,q/d,aq/e,aq/f;q)}_{\mathrm{\infty }}}{{(aq,q/a,aq/(cd),aq/(ef);q)}_{\mathrm{\infty }}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(1-a{q}^{2n}){(c,d,e,f;q)}_n}{(1-a){(aq/c,aq/d,aq/e,aq/f;q)}_n}{\left(\frac{q{a}^3}{cdef}\right)}^n{q}^{n^2}.$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-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/17.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.5 $$\frac{{(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q)}_{\mathrm{\infty }}}{{(fa,ga,f/a,g/a,q{a}^2,q/a^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\psi _8^{}(\genfrac{}{}{0pt}{}{qa,-qa,ba,ca,da,ea,fa,ga}{a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^2}{bcdefg})=\frac{{(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf/e;q)}_{\mathrm{\infty }}}{{(fa,q/(fa),aq/f,f/a,g/f,fg,q{f}^2;q)}_{\mathrm{\infty }}}{}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{f^2,qf,-qf,fb,fc,fd,fe,fg}{f,-f,fq/b,fq/c,fq/d,fq/e,fq/g};q,\frac{q^2}{bcdefg})+\mathrm{idem}(f;g).$$

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Symbols:

$\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)$: $\mathrm{idem}$ function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-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/17.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

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17.10.6 $$\frac{{(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q)}_{\mathrm{\infty }}}{{(ag,ah,ak,g/a,h/a,k/a,q{a}^2,q/a^2;q)}_{\mathrm{\infty }}}{}_{10}{}^{}\psi _{10}^{}(\genfrac{}{}{0pt}{}{qa,-qa,ba,ca,da,ea,fa,ga,ha,ka}{a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^2}{bcdefghk})=\frac{{(q,q/(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q)}_{\mathrm{\infty }}}{{(gh,gk,h/g,ag,q/(ag),g/a,aq/g,q{g}^2;q)}_{\mathrm{\infty }}}{}_{10}{}^{}\varphi _9^{}(\genfrac{}{}{0pt}{}{g^2,qg,-qg,gb,gc,gd,ge,gf,gh,gk}{g,-g,qg/b,qg/c,qg/d,qg/e,qg/f,qg/h,qg/k};q,\frac{q^2}{bcdefghk})+\mathrm{idem}(g;h,k).$$

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Symbols:

$\mathrm{idem}({\chi }_1;{\chi }_2,\mathrm{\dots },{\chi }_n)$: $\mathrm{idem}$ function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $k$: nonnegative integer and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.10, §17.10 and Ch.17

28: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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17.8.4 $${}_2{}^{}\psi _2^{}(b,c;aq/b,aq/c;q,-aq/(bc))=\frac{{(aq/(bc);q)}_{\mathrm{\infty }}{(a{q}^2/b^2,a{q}^2/c^2,q^2,aq,q/a;q^2)}_{\mathrm{\infty }}}{{(aq/b,aq/c,q/b,q/c,-aq/(bc);q)}_{\mathrm{\infty }}},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-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/17.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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17.8.5 $${}_3{}^{}\psi _3^{}(\genfrac{}{}{0pt}{}{b,c,d}{q/b,q/c,q/d};q,\frac{q}{bcd})=\frac{{(q,q/(bc),q/(bd),q/(cd);q)}_{\mathrm{\infty }}}{{(q/b,q/c,q/d,q/(bcd);q)}_{\mathrm{\infty }}},$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-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/17.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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17.8.6 $${}_4{}^{}\psi _4^{}(\genfrac{}{}{0pt}{}{-q{a}^{\frac{1}{2}},b,c,d}{-a^{\frac{1}{2}},aq/b,aq/c,aq/d};q,\frac{q{a}^{\frac{3}{2}}}{bcd})=\frac{{(aq,aq/(bc),aq/(bd),aq/(cd),q{a}^{\frac{1}{2}}/b,q{a}^{\frac{1}{2}}/c,q{a}^{\frac{1}{2}}/d,q,q/a;q)}_{\mathrm{\infty }}}{{(aq/b,aq/c,aq/d,q/b,q/c,q/d,q{a}^{\frac{1}{2}},q{a}^{-\frac{1}{2}},q{a}^{\frac{3}{2}}/(bcd);q)}_{\mathrm{\infty }}},$$

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Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-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/17.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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17.8.7 $${}_6{}^{}\psi _6^{}(\genfrac{}{}{0pt}{}{q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},b,c,d,e}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{q{a}^2}{bcde})=\frac{{(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q)}_{\mathrm{\infty }}}{{(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,q{a}^2/(bcde);q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-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/17.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §17.8, §17.8 and Ch.17

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17.8.8 $${}_2{}^{}\psi _2^{}(\genfrac{}{}{0pt}{}{b^2,b^2/c}{q,cq};q^2,c{q}^2/b^2)=\frac{1}{2}\frac{{(q^2,q{b}^2,q/b^2,cq/b^2;q^2)}_{\mathrm{\infty }}}{{(cq,c{q}^2/b^2,q^2/b^2,c/b^2;q^2)}_{\mathrm{\infty }}}\left(\frac{{(c\sqrt{q}/b;q)}_{\mathrm{\infty }}}{{(b\sqrt{q};q)}_{\mathrm{\infty }}}+\frac{{(-c\sqrt{q}/b;q)}_{\mathrm{\infty }}}{{(-b\sqrt{q};q)}_{\mathrm{\infty }}}\right),$$ .

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: 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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29: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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17.9.2 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},b}{c};q,z)=\frac{{(c/b;q)}_n}{{(c;q)}_n}{b}^n{}_3{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},b,q/z}{b{q}^{1-n}/c};q,z/c),$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Referenced by:

Erratum (V1.0.14) for Equation (17.9.2)

Permalink:

http://dlmf.nist.gov/17.9.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.14):

The entry $q/c$ in the first row of ${}_3{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},b,q/c}{b{q}^{1-n}/c};q,z/c)$ was replaced by $q/z$.

Suggested 2016-08-30 by Xinrong Ma

See also:

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

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17.9.12 $${}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{a,b,c}{d,e};q,\frac{de}{abc})=\frac{{(e/b,e/c,cq/a,q/d;q)}_{\mathrm{\infty }}}{{(e,cq/d,q/a,e/(bc);q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{c,d/a,cq/e}{cq/a,bcq/e};q,\frac{bq}{d})-\frac{{(q/d,eq/d,b,c,d/a,de/(bcq),bc{q}^2/(de);q)}_{\mathrm{\infty }}}{{(d/q,e,bq/d,cq/d,q/a,e/(bc),bcq/e;q)}_{\mathrm{\infty }}}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{aq/d,bq/d,cq/d}{q^2/d,eq/d};q,\frac{de}{abc}),$$

ⓘ

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.E12

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

ⓘ

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

ⓘ

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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17.9.16 $${}_8{}^{}\varphi _7^{}(\genfrac{}{}{0pt}{}{a,q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}},b,c,d,e,f}{a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,\frac{a^2{q}^2}{bcdef})=\frac{{(aq,aq/(de),aq/(df),aq/(ef);q)}_{\mathrm{\infty }}}{{(aq/d,aq/e,aq/f,aq/(def);q)}_{\mathrm{\infty }}}{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{aq/(bc),d,e,f}{aq/b,aq/c,def/a};q,q)+\frac{{(aq,aq/(bc),d,e,f,a^2{q}^2/(bdef),a^2{q}^2/(cdef);q)}_{\mathrm{\infty }}}{{(aq/b,aq/c,aq/d,aq/e,aq/f,a^2{q}^2/(bcdef),def/(aq);q)}_{\mathrm{\infty }}}{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{aq/(de),aq/(df),aq/(ef),a^2{q}^2/(bcdef)}{a^2{q}^2/(bdef),a^2{q}^2/(cdef),a{q}^2/(def)};q,q).$$

ⓘ

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.E16

Encodings:

pMML, png, TeX

See also:

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

…

30: 21.1 Special Notation

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$g,h$	positive integers.
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${\mathbb{Z}}^{g\times h}$	set of all $g\times h$ matrices with integer elements.
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$S^g$	set of $g$-dimensional vectors with elements in $S$.
$|S|$	number of elements of the set $S$.
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$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)
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… ►The function $\mathrm{\Theta }(\mathit{\varphi }|𝐁)=\theta \left(\mathit{\varphi }/(2\pi \mathrm{i})\right|𝐁/(2\pi \mathrm{i}))$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).