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⇑ 15 Hypergeometric Function
⇑ Properties
⇐ 15.3 Graphics15.5 Derivatives and Contiguous Functions ⇒

§15.4 Special Cases

Contents

§15.4(i) Elementary Functions

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Notes:
For (15.4.1)–(15.4.6) see Temme (1996b, §5.1). For (15.4.7) use (15.8.27), (15.4.6), and (5.5.5). For (15.4.9) use (15.8.28), (15.4.6), and (5.5.5). For (15.4.11) use (15.8.1) and (15.4.7). For (15.4.13) use (15.8.1) and (15.4.11). For (15.4.15) use (15.8.1) and (15.4.9). For (15.4.17) and (15.4.18) use (15.8.15) and (15.4.6). For (15.4.19) use (15.5.11) and (15.4.6).
A&S Ref:
15.1
Keywords:
elementary functions, hypergeometric function
Permalink:
http://dlmf.nist.gov/15.4.i

The following results hold for principal branches when |z|<1, and by analytic continuation elsewhere. Exceptions are (15.4.8) and (15.4.10), that hold for |z|<\ifrac{\pi}{4}, and (15.4.12), (15.4.14), and (15.4.16), that hold for |z|<\ifrac{\pi}{2}.

compare §15.2(ii).

15.4.7\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};z^{2}\right)=\tfrac{1}{2}\left((1+z)^{{-2a}}+(1-z)^{{-2a}}\right),
15.4.8\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};-{\mathop{\tan\/}\nolimits^{{2}}}z\right)=(\mathop{\cos\/}\nolimits z)^{{2a}}\mathop{\cos\/}\nolimits\!\left(2az\right).
15.4.9\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};z^{2}\right)=\frac{1}{(2-4a)z}\left((1+z)^{{1-2a}}-(1-z)^{{1-2a}}\right),
15.4.10\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};-{\mathop{\tan\/}\nolimits^{{2}}}z\right)=(\mathop{\cos\/}\nolimits z)^{{2a}}\frac{\mathop{\sin\/}\nolimits\!\left((1-2a)z\right)}{(1-2a)\mathop{\sin\/}\nolimits z}.
15.4.11\mathop{F\/}\nolimits\!\left(-a,a;\tfrac{1}{2};-z^{2}\right)=\tfrac{1}{2}\left(\left(\sqrt{1+z^{2}}+z\right)^{{2a}}+\left(\sqrt{1+z^{2}}-z\right)^{{2a}}\right),
15.4.12\mathop{F\/}\nolimits\!\left(-a,a;\tfrac{1}{2};{\mathop{\sin\/}\nolimits^{{2}}}z\right)=\mathop{\cos\/}\nolimits\!\left(2az\right).
15.4.13\mathop{F\/}\nolimits\!\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{1+z^{2}}}\left(\left(\sqrt{1+z^{2}}+z\right)^{{2a-1}}+\left(\sqrt{1+z^{2}}-z\right)^{{2a-1}}\right),
15.4.14\mathop{F\/}\nolimits\!\left(a,1-a;\tfrac{1}{2};{\mathop{\sin\/}\nolimits^{{2}}}z\right)=\frac{\mathop{\cos\/}\nolimits\!\left((2a-1)z\right)}{\mathop{\cos\/}\nolimits z}.
15.4.15\mathop{F\/}\nolimits\!\left(a,1-a;\tfrac{3}{2};-z^{2}\right)=\frac{1}{(2-4a)z}\left(\left(\sqrt{1+z^{2}}+z\right)^{{1-2a}}-\left(\sqrt{1+z^{2}}-z\right)^{{1-2a}}\right),
15.4.16\mathop{F\/}\nolimits\!\left(a,1-a;\tfrac{3}{2};{\mathop{\sin\/}\nolimits^{{2}}}z\right)=\frac{\mathop{\sin\/}\nolimits\!\left((2a-1)z\right)}{(2a-1)\mathop{\sin\/}\nolimits z}.
15.4.17\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;1+2a;z\right)=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{{-2a}},
15.4.18\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;2a;z\right)=\frac{1}{\sqrt{1-z}}\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{{1-2a}}.
15.4.19\mathop{F\/}\nolimits\!\left(a+1,b;a;z\right)=\left(1-(1-(\ifrac{b}{a}))z\right)(1-z)^{{-1-b}}.

For an extensive list of elementary representations see Prudnikov et al. (1990, pp. 468–488).

§15.4(ii) Argument Unity

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Notes:
See Andrews et al. (1999, §2.2). For (15.4.21) use (15.8.10). For (15.4.22) and (15.4.23) use (15.8.4), (5.5.3), and (5.5.1). For (15.4.25) see Dougall (1907) or Andrews et al. (1999, §2.8).
Keywords:
gamma function, hypergeometric function
Referenced by:
§15.2(i)
Permalink:
http://dlmf.nist.gov/15.4.ii

Dougall’s Bilateral Sum

This is a generalization of (15.4.20). If a,b are not integers and \realpart{(c+d-a-b)}>1, then

15.4.25\sum _{{n=-\infty}}^{{\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+n\right)\mathop{\Gamma\/}\nolimits\!\left(b+n\right)}{\mathop{\Gamma\/}\nolimits\!\left(c+n\right)\mathop{\Gamma\/}\nolimits\!\left(d+n\right)}=\frac{\pi^{2}}{\mathop{\sin\/}\nolimits\!\left(\pi a\right)\mathop{\sin\/}\nolimits\!\left(\pi b\right)}\*\frac{\mathop{\Gamma\/}\nolimits\!\left(c+d-a-b-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(d-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)\mathop{\Gamma\/}\nolimits\!\left(d-b\right)}.

§15.4(iii) Other Arguments

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Notes:
For (15.4.26) use (15.8.24) and (5.5.5). For (15.4.27) use (15.2.1) and (5.7.7). For (15.4.28) use (15.8.1), (15.4.26), and (5.5.5). For (15.4.29) use (15.8.1), (15.5.15), (15.4.26), (5.5.5), and (5.5.1). For (15.4.30) use (15.8.1) and (15.4.28). For (15.4.31) use (15.8.1), Erdélyi et al. (1953a, Eq. (2.11.41)), (15.4.20), and (5.5.5). For (15.4.32) use (15.8.30), (15.4.28), (5.5.5), and (5.5.6). (Note that Erdélyi et al. (1953a, Eq. (2.8.54)) contains an error.) For (15.4.33) let z\to-\infty in (15.8.32) and use (5.5.5).
A&S Ref:
15.1
Keywords:
gamma function, hypergeometric function, psi function
Permalink:
http://dlmf.nist.gov/15.4.iii
15.4.26\mathop{F\/}\nolimits\!\left(a,b;a-b+1;-1\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a-b+1\right)}.
15.4.27\mathop{F\/}\nolimits\!\left(1,a;a+1;-1\right)=\tfrac{1}{2}a\left(\mathop{\psi\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)-\mathop{\psi\/}\nolimits\!\left(\tfrac{1}{2}a\right)\right).
15.4.28\mathop{F\/}\nolimits\!\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2};\tfrac{1}{2}\right)=\sqrt{\pi}\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}.
15.4.29\mathop{F\/}\nolimits\!\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+1;\tfrac{1}{2}\right)=\frac{2\sqrt{\pi}}{a-b}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{2}b+1\right)\*\left(\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}-\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}b\right)}\right).
15.4.30\mathop{F\/}\nolimits\!\left(a,1-a;b;\tfrac{1}{2}\right)=\frac{2^{{1-b}}\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(b\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{2}b\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2}\right)}.
15.4.31\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{3}{2}-2a;-\tfrac{1}{3}\right)=\left(\frac{8}{9}\right)^{{-2a}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{4}{3}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}-2a\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{4}{3}-2a\right)}.
15.4.32\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{5}{6}+\tfrac{2}{3}a;\tfrac{1}{9}\right)=\sqrt{\pi}\left(\frac{3}{4}\right)^{a}\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{5}{6}+\tfrac{2}{3}a\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+\tfrac{1}{3}a\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{5}{6}+\tfrac{1}{3}a\right)}.
15.4.33\mathop{F\/}\nolimits\!\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;e^{{\ifrac{i\pi}{3}}}\right)=\sqrt{\pi}e^{{\ifrac{i\pi a}{2}}}\left(\frac{16}{27}\right)^{{(3a+1)/6}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{5}{6}+a\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{2}{3}+a\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{2}{3}\right)}.
© 2010–2012 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.5; Release date 2012-10-01 Math-Image version (See MathML) . A Printed Companion is available. ⇐ 15.3 Graphics15.5 Derivatives and Contiguous Functions ⇒