§16.12 Products

Notes:: See Erdélyi et al. (1953a, §4.3). For (16.12.1) see Bailey (1928). For (16.12.2) see Clausen (1828). For (16.12.3) see Chaundy (1969, Chapter 12, Problem 12).
Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.12

16.12.1 $\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits\!\left(-;a;z\right)\mathop{{{}_{{0}}F_{{% 1}}}\/}\nolimits\!\left(-;b;z\right)=\mathop{{{}_{{2}}F_{{3}}}\/}\nolimits\!% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: TeX, pMML, png

16.12.2 $\left(\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop a+b+\frac{1}{2}};% z\right)\right)^{2}=\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({2a,2b,a+b% \atop a+b+\frac{1}{2},2a+2b};z\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: TeX, pMML, png

More generally,

16.12.3 $\left(\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};z\right)\right% )^{2}=\sum_{{k=0}}^{\infty}\frac{\left(2a\right)_{{k}}\left(2b\right)_{{k}}% \left(c-\frac{1}{2}\right)_{{k}}}{\left(c\right)_{{k}}\left(2c-1\right)_{{k}}k% !}\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left({-\frac{1}{2}k,\frac{1}{2}(1-k)% ,a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-% c};1\right)z^{k},$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: TeX, pMML, png

For further identities see Goursat (1883) and Erdélyi et al. (1953a, §4.3).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 16.11 Asymptotic Expansions 16.13 Appell Functions