F. H. Jackson q-analog

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1: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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$q$-Analog of Bailey’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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$q$-Analog of Gauss’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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F. H. Jackson’s Terminating $q$-Analog of Dixon’s Sum

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$q$-Analog of Dixon’s ${}_3{}^{}F_2^{}\left(1\right)$ Sum

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F. H. Jackson’s $q$-Analog of Dougall’s ${}_7{}^{}F_6^{}\left(1\right)$ Sum

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2: 17.1 Special Notation

… ►The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${}_r{}^{}\varphi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, and the $q$-analogs of the Appell functions ${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$, ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$, ${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$, and ${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$. … ►Fine (1988) uses $F(a,b;t:q)$ for a particular specialization of a ${}_2{}^{}\varphi _1^{}$ function.

3: Bibliography M

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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 $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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External Links:

ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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External Links:

ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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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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4: Errata

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The title of the paragraph which was previously “Gasper’s $q$-Analog of Clausen’s Formula” has been changed to “Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)”.

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The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog of (17.7.8)” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Watson’s ${}_3{}^{}F_2^{}$ Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Whipple’s ${}_3{}^{}F_2^{}$ Sum (16.4.7)”.

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Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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Pochhammer symbol representations for the functions $F_k(\nu )$ and $G_k(\nu )$ were inserted.

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Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${}_2{}^{}F_1^{}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

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5: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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F. H. Jackson’s Transformations

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Watson’s $q$-Analog of Whipple’s Theorem

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

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6: 33.8 Continued Fractions

… ►If we denote $u=F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $p+\mathrm{i}q=H_{\mathrm{\ell }}^{+}{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{+}$, then ►

$F_{\mathrm{\ell }}=\pm {(q^{-1}{(u-p)}^2+q)}^{-1/2},$

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$F_{\mathrm{\ell }}^{\prime }=u{F}_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}=q^{-1}(u-p)F_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}^{\prime }=q^{-1}(up-p^2-q^2)F_{\mathrm{\ell }}.$

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7: 31.7 Relations to Other Functions

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31.7.1 $${}_2{}^{}F_1^{}(\alpha ,\beta ;\gamma ;z)=H\mathrm{\ell }(1,\alpha \beta ;\alpha ,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(0,0;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma ;z)=H\mathrm{\ell }(a,a\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma ;z).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. … ►

31.7.2 $$H\mathrm{\ell }(2,\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta -2\gamma +1;z)={}_2{}^{}F_1^{}(\frac{1}{2}\alpha ,\frac{1}{2}\beta ;\gamma ;1-{(1-z)}^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

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31.7.3 $$H\mathrm{\ell }(4,\alpha \beta ;\alpha ,\beta ,\frac{1}{2},\frac{2}{3}(\alpha +\beta );z)={}_2{}^{}F_1^{}(\frac{1}{3}\alpha ,\frac{1}{3}\beta ;\frac{1}{2};1-{(1-z)}^2(1-\frac{1}{4}z)),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

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31.7.4 $$H\mathrm{\ell }(\frac{1}{2}+\mathrm{i}\frac{\sqrt{3}}{2},\alpha \beta (\frac{1}{2}+\mathrm{i}\frac{\sqrt{3}}{6});\alpha ,\beta ,\frac{1}{3}(\alpha +\beta +1),\frac{1}{3}(\alpha +\beta +1);z)={}_2{}^{}F_1^{}(\frac{1}{3}\alpha ,\frac{1}{3}\beta ;\frac{1}{3}(\alpha +\beta +1);1-{\left(1-\left(\frac{3}{2}-\mathrm{i}\frac{\sqrt{3}}{2}\right)z\right)}^3).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

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8: 10.16 Relations to Other Functions

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$$H_{\frac{1}{2}}^{(1)}\left(z\right)=-\mathrm{i}{H}_{-\frac{1}{2}}^{(1)}\left(z\right)=-\mathrm{i}{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}z},$$

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10.16.9 $$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +1\right)}{}_0{}^{}F_1^{}(-;\nu +1;-\frac{1}{4}{z}^2).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.16

Permalink:

http://dlmf.nist.gov/10.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

►For ${}_0{}^{}F_1^{}$ see (16.2.1). ►With $𝐅$ as in §15.2(i), and with $z$ and $\nu$ fixed, ►

10.16.10 $$J_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim𝐅(\lambda ,\mu ;\nu +1;-z^2/(4\lambda \mu )),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.70 (modified)

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

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9: 16.11 Asymptotic Expansions

… ►For subsequent use we define two formal infinite series, $E_{p,q}(z)$ and $H_{p,q}(z)$, as follows: … ►It may be observed that $H_{p,q}(z)$ represents the sum of the residues of the poles of the integrand in (16.5.1) at $s=-a_j,-a_j-1,\mathrm{\dots }$, $j=1,\mathrm{\dots },p$, provided that these poles are all simple, that is, no two of the $a_j$ differ by an integer. (If this condition is violated, then the definition of $H_{p,q}(z)$ has to be modified so that the residues are those associated with the multiple poles. … ►The formal series (16.11.2) for $H_{q+1,q}(z)$ converges if $\left|z\right|>1$, and … ►Asymptotic expansions for the polynomials ${}_{p+2}{}^{}F_q^{}(-r,r+a_0,𝐚;𝐛;z)$ as $r\to \mathrm{\infty }$ through integer values are given in Fields and Luke (1963b, a) and Fields (1965).

10: 33.2 Definitions and Basic Properties

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§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

►The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by … ►

§33.2(iii) Irregular Solutions $G_{\mathrm{\ell }}(\eta ,\rho ),H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$

… ► $H_{\mathrm{\ell }}^{+}(\eta ,\rho )$ and $H_{\mathrm{\ell }}^{-}(\eta ,\rho )$ are complex conjugates, and their real and imaginary parts are given by … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . …