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DLMF: §15.16 Products ‣ Properties ‣ Chapter 15 Hypergeometric Function

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15 Hypergeometric Function Properties 15.15 Sums 15.17 Mathematical Applications

§15.16 Products

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Keywords:: hypergeometric function, products, series expansions
Notes:: See Burchnall and Chaundy (1940, 1948) and Elliott (1903). For (15.16.4) use (15.8.2) and (15.8.4), combined with (15.8.1) to show that the left-hand side of (15.16.4) is an entire function of $z$ . Then apply Liouville’s theorem (§1.9(iii)).
Permalink:: http://dlmf.nist.gov/15.16
See also:: Annotations for Ch.15

15.16.1		$F\left({a,b\atop c-\frac{1}{2}};z\right)F\left({c-a,c-b\atop c+\frac{1}{2}};z% \right)=\sum_{s=0}^{\infty}\frac{{\left(c\right)_{s}}}{{\left(c+\frac{1}{2}% \right)_{s}}}A_{s}z^{s},$
$\|z\|<1$ ,
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $A_{s}$ : coefficients Permalink: http://dlmf.nist.gov/15.16.E1 Encodings: TeX, pMML, png See also: Annotations for §15.16 and Ch.15

where $A_{0}=1$ and $A_{s}$ , $s=1,2,\dots$ , are defined by the generating function

15.16.2		$(1-z)^{a+b-c}F\left(2a,2b;2c-1;z\right)=\sum_{s=0}^{\infty}A_{s}z^{s},$
$\|z\|<1$ .
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $A_{s}$ : coefficients Permalink: http://dlmf.nist.gov/15.16.E2 Encodings: TeX, pMML, png See also: Annotations for §15.16 and Ch.15

Also,

15.16.3		$F\left({a,b\atop c};z\right)F\left({a,b\atop c};\zeta\right)=\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}{\left(c-a\right)_{s}}{% \left(c-b\right)_{s}}}{{\left(c\right)_{s}}{\left(c\right)_{2s}}s!}\left(z% \zeta\right)^{s}F\left({a+s,b+s\atop c+2s};z+\zeta-z\zeta\right),$
$\|z\|<1$ , $\|\zeta\|<1$ , $\|z+\zeta-z\zeta\|<1$ .
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.16.E3 Encodings: TeX, pMML, png See also: Annotations for §15.16 and Ch.15

15.16.4		$F\left({a,b\atop c};z\right)F\left({-a,-b\atop-c};z\right)+\frac{ab(a-c)(b-c)}% {c^{2}(1-c^{2})}z^{2}F\left({1+a,1+b\atop 2+c};z\right)F\left({1-a,1-b\atop 2-% c};z\right)=1.$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.16 Permalink: http://dlmf.nist.gov/15.16.E4 Encodings: TeX, pMML, png See also: Annotations for §15.16 and Ch.15

Generalized Legendre’s Relation

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Keywords:: Legendre’s relation for the hypergeometric function, generalized
See also:: Annotations for §15.16 and Ch.15

15.16.5		$F\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F% \left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)+F\left({% \frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({-\frac% {1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)-F\left({\frac{1}{2}+% \lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({\frac{1}{2}-% \lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)=\frac{\Gamma\left(1+\lambda% +\mu\right)\Gamma\left(1+\nu+\mu\right)}{\Gamma\left(\lambda+\mu+\nu+\frac{3}{% 2}\right)\Gamma\left(\frac{1}{2}+\nu\right)},$
$\|\operatorname{ph}z\|<\pi$ , $\|\operatorname{ph}\left(1-z\right)\|<\pi$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase and $z$ : complex variable Permalink: http://dlmf.nist.gov/15.16.E5 Encodings: TeX, pMML, png See also: Annotations for §15.16, §15.16 and Ch.15

For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).

15.15 Sums 15.17 Mathematical Applications

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