§15.16 Products

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)).
Keywords:: hypergeometric function
Permalink:: http://dlmf.nist.gov/15.16

15.16.1 $\mathop{F\/}\nolimits\!\left({a,b\atop c-\frac{1}{2}};z\right)\mathop{F\/}\nolimits\!\left({c-a,c-b\atop c+\frac{1}{2}};z\right)=\sum _{{s=0}}^{{\infty}}\frac{\left(c\right)_{{s}}}{\left(c+\frac{1}{2}\right)_{{s}}}A_{s}z^{s},$ $|z|<1$ ,

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $s$ : 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

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

15.16.2 $(1-z)^{{a+b-c}}\mathop{F\/}\nolimits\!\left(2a,2b;2c-1;z\right)=\sum _{{s=0}}^{{\infty}}A_{s}z^{s},$ $|z|<1$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

Also,

15.16.3 $\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right)\mathop{F\/}\nolimits\!\left({a,b\atop c};\zeta\right)=\sum _{{s=0}}^{{\infty}}\frac{\left(a\right)_{{s}}\left(b\right)_{{s}}\left(c-a\right)_{{s}}\left(c-b\right)_{{s}}}{\left(c\right)_{{s}}\left(c\right)_{{2s}}s!}\left(z\zeta\right)^{s}\mathop{F\/}\nolimits\!\left({a+s,b+s\atop c+2s};z+\zeta-z\zeta\right),$ $|z|<1$ , $|\zeta|<1$ , $|z+\zeta-z\zeta|<1$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $s$ : 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

15.16.4 $\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right)\mathop{F\/}\nolimits\!\left({-a,-b\atop-c};z\right)+\frac{ab(a-c)(b-c)}{c^{2}(1-c^{2})}z^{2}\mathop{F\/}\nolimits\!\left({1+a,1+b\atop 2+c};z\right)\mathop{F\/}\nolimits\!\left({1-a,1-b\atop 2-c};z\right)=1.$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

¶ Generalized Legendre’s Relation

Keywords:: Legendre’s relation for the hypergeometric function

15.16.5 $\mathop{F\/}\nolimits\!\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)\mathop{F\/}\nolimits\!\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)+\mathop{F\/}\nolimits\!\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)\mathop{F\/}\nolimits\!\left({-\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)-\mathop{F\/}\nolimits\!\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)\mathop{F\/}\nolimits\!\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+\lambda+\mu\right)\mathop{\Gamma\/}\nolimits\!\left(1+\nu+\mu\right)}{\mathop{\Gamma\/}\nolimits\!\left(\lambda+\mu+\nu+\frac{3}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+\nu\right)},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , $|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)|<\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/15.16.E5
Encodings:: TeX, pMML, png

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