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15 Hypergeometric FunctionProperties

§15.4 Special Cases

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Referenced by:
§15.2(ii)
Permalink:
http://dlmf.nist.gov/15.4
See also:
Annotations for Ch.15
Contents
  1. §15.4(i) Elementary Functions
  2. §15.4(ii) Argument Unity
  3. §15.4(iii) Other Arguments

§15.4(i) Elementary Functions

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Keywords:
elementary functions, hypergeometric function, relations to other functions, special cases
A&S Ref:
15.1
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).
Referenced by:
Erratum (V1.0.16) for Subsections 15.4(i), 15.4(ii)
Permalink:
http://dlmf.nist.gov/15.4.i
Clarification (effective with 1.0.16):
Information was added after (15.4.6) and (15.4.19) specifying that special care is required under certain circumstances.

Suggested 2017-01-01 by Louis Klauder

See also:
Annotations for §15.4 and Ch.15

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|<π/4, and (15.4.12), (15.4.14), and (15.4.16), that hold for |z|<π/2.

15.4.1 F⁡(1,1;2;z) =−z−1⁢ln⁡(1−z),
15.4.2 F⁡(12,1;32;z2) =12⁢z⁢ln⁡(1+z1−z),
15.4.3 F⁡(12,1;32;−z2) =z−1⁢arctan⁡z,
15.4.4 F⁡(12,12;32;z2) =z−1⁢arcsin⁡z,
15.4.5 F⁡(12,12;32;−z2) =z−1⁢ln⁡(z+1+z2).
15.4.6 F⁡(a,b;a;z) =(1−z)−b,
F⁡(a,b;b;z) =(1−z)−a,
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Symbols:
F⁡(a,b;c;z) or F⁡(a,bc;z): =F12⁡(a,b;c;z) Gauss’ hypergeometric function, z: complex variable, a: real or complex parameter and b: real or complex parameter
Notes:
See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.
Referenced by:
§10.22(iv), §15.2(ii), §15.4(i), (18.12.3_5), Erratum (V1.0.16) for Subsections 15.4(i), 15.4(ii)
Permalink:
http://dlmf.nist.gov/15.4.E6
Encodings:
TeX, TeX, pMML, pMML, png, png
Addition (effective with 1.0.16):
This equation was expanded by adding the formula F⁡(a,b;a;z)=(1−z)−b.
See also:
Annotations for §15.4(i), §15.4 and Ch.15

where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when the third parameter is a nonpositive integer. See the final paragraph in §15.2(ii).

15.4.7 F⁡(a,12+a;12;z2)=12⁢((1+z)−2⁢a+(1−z)−2⁢a),
15.4.8 F⁡(a,12+a;12;−tan2⁡z)=(cos⁡z)2⁢a⁢cos⁡(2⁢a⁢z).
15.4.9 F⁡(a,12+a;32;z2)=1(2−4⁢a)⁢z⁢((1+z)1−2⁢a−(1−z)1−2⁢a),
15.4.10 F⁡(a,12+a;32;−tan2⁡z)=(cos⁡z)2⁢a⁢sin⁡((1−2⁢a)⁢z)(1−2⁢a)⁢sin⁡z.
15.4.11 F⁡(−a,a;12;−z2)=12⁢((1+z2+z)2⁢a+(1+z2−z)2⁢a),
15.4.12 F⁡(−a,a;12;sin2⁡z)=cos⁡(2⁢a⁢z).
15.4.13 F⁡(a,1−a;12;−z2)=12⁢1+z2⁢((1+z2+z)2⁢a−1+(1+z2−z)2⁢a−1),
15.4.14 F⁡(a,1−a;12;sin2⁡z)=cos⁡((2⁢a−1)⁢z)cos⁡z.
15.4.15 F⁡(a,1−a;32;−z2)=1(2−4⁢a)⁢z⁢((1+z2+z)1−2⁢a−(1+z2−z)1−2⁢a),
15.4.16 F⁡(a,1−a;32;sin2⁡z)=sin⁡((2⁢a−1)⁢z)(2⁢a−1)⁢sin⁡z.
15.4.17 F⁡(a,12+a;1+2⁢a;z)=(12+12⁢1−z)−2⁢a,
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Symbols:
F⁡(a,b;c;z) or F⁡(a,bc;z): =F12⁡(a,b;c;z) Gauss’ hypergeometric function, z: complex variable and a: real or complex parameter
Notes:
See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.
Referenced by:
§15.4(i), §15.4(i)
Permalink:
http://dlmf.nist.gov/15.4.E17
Encodings:
TeX, pMML, png
See also:
Annotations for §15.4(i), §15.4 and Ch.15
15.4.18 F⁡(a,12+a;2⁢a;z)=11−z⁢(12+12⁢1−z)1−2⁢a,
ⓘ
Symbols:
F⁡(a,b;c;z) or F⁡(a,bc;z): =F12⁡(a,b;c;z) Gauss’ hypergeometric function, z: complex variable and a: real or complex parameter
Notes:
See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.
Referenced by:
§15.4(i), §15.4(i)
Permalink:
http://dlmf.nist.gov/15.4.E18
Encodings:
TeX, pMML, png
See also:
Annotations for §15.4(i), §15.4 and Ch.15
15.4.19 F⁡(a+1,b;a;z)=(1−(1−(b/a))⁢z)⁢(1−z)−1−b.
ⓘ
Symbols:
F⁡(a,b;c;z) or F⁡(a,bc;z): =F12⁡(a,b;c;z) Gauss’ hypergeometric function, z: complex variable, a: real or complex parameter and b: real or complex parameter
Notes:
See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.
Referenced by:
§15.4(i), §15.4(i)
Permalink:
http://dlmf.nist.gov/15.4.E19
Encodings:
TeX, pMML, png
See also:
Annotations for §15.4(i), §15.4 and Ch.15

In (15.4.17), (15.4.18) and (15.4.19) when the third entry is a nonpositive integer one has to use the limit interpretation (15.2.6), rather than (15.2.5). Compare the final paragraph in §15.2(ii).

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

§15.4(ii) Argument Unity

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Keywords:
argument ±1, gamma function, hypergeometric function, relations to hypergeometric function, relations to other functions, special cases
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).
Referenced by:
§15.2(i), Erratum (V1.0.16) for Subsections 15.4(i), 15.4(ii)
Permalink:
http://dlmf.nist.gov/15.4.ii
See also:
Annotations for §15.4 and Ch.15

If ℜ⁡(c−a−b)>0, then

15.4.20 F⁡(a,b;c;1)=Γ⁡(c)⁢Γ⁡(c−a−b)Γ⁡(c−a)⁢Γ⁡(c−b).

If c=a+b, then

15.4.21 limz→1−F⁡(a,b;a+b;z)−ln⁡(1−z)=Γ⁡(a+b)Γ⁡(a)⁢Γ⁡(b).

If ℜ⁡(c−a−b)=0 and c≠a+b, then

15.4.22 limz→1−(1−z)a+b−c⁢(F⁡(a,b;c;z)−Γ⁡(c)⁢Γ⁡(c−a−b)Γ⁡(c−a)⁢Γ⁡(c−b))=Γ⁡(c)⁢Γ⁡(a+b−c)Γ⁡(a)⁢Γ⁡(b).

If ℜ⁡(c−a−b)<0, then

15.4.23 limz→1−F⁡(a,b;c;z)(1−z)c−a−b=Γ⁡(c)⁢Γ⁡(a+b−c)Γ⁡(a)⁢Γ⁡(b).

Chu–Vandermonde Identity

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Keywords:
Chu–Vandermonde identity, hypergeometric function
See also:
Annotations for §15.4(ii), §15.4 and Ch.15

Dougall’s Bilateral Sum

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Keywords:
Dougall’s bilateral sum
See also:
Annotations for §15.4(ii), §15.4 and Ch.15

This is a generalization of (15.4.20). If a,b are not integers and ℜ⁡(c+d−a−b)>1, then

15.4.25 ∑n=−∞∞Γ⁡(a+n)⁢Γ⁡(b+n)Γ⁡(c+n)⁢Γ⁡(d+n)=π2sin⁡(π⁢a)⁢sin⁡(π⁢b)⁢Γ⁡(c+d−a−b−1)Γ⁡(c−a)⁢Γ⁡(d−a)⁢Γ⁡(c−b)⁢Γ⁡(d−b).

§15.4(iii) Other Arguments

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Keywords:
argument a fraction, argument ±1, arguments e±i⁢π/3, gamma function, hypergeometric function, psi function, relation to hypergeometric function, relations to hypergeometric function, relations to other functions, special cases
A&S Ref:
15.1
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→−∞ in (15.8.32) and use (5.5.5).
Permalink:
http://dlmf.nist.gov/15.4.iii
Addition (effective with 1.1.2):
Equation (15.4.34) and some explanatory text immediately after it were added.
Clarification (effective with 1.0.16):
Information was added after (15.4.33) specifying special care is required under certain circumstances.
See also:
Annotations for §15.4 and Ch.15
15.4.26 F⁡(a,b;a−b+1;−1)=Γ⁡(a−b+1)⁢Γ⁡(12⁢a+1)Γ⁡(a+1)⁢Γ⁡(12⁢a−b+1).
15.4.27 F⁡(1,a;a+1;−1)=2−a⁢F⁡(a,a;a+1;12)=12⁢a⁢(ψ⁡(12⁢a+12)−ψ⁡(12⁢a)).
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Symbols:
F⁡(a,b;c;z) or F⁡(a,bc;z): =F12⁡(a,b;c;z) Gauss’ hypergeometric function, ψ⁡(z): psi (or digamma) function and a: real or complex parameter
Proof sketch:
Intermediate equality follows from (15.8.1).
Referenced by:
§15.4(iii), Erratum (V1.1.2) for Additions
Permalink:
http://dlmf.nist.gov/15.4.E27
Encodings:
TeX, pMML, png
Addition (effective with 1.1.2):
Intermediate equality was added.
See also:
Annotations for §15.4(iii), §15.4 and Ch.15
15.4.28 F⁡(a,b;12⁢a+12⁢b+12;12)=π⁢Γ⁡(12⁢a+12⁢b+12)Γ⁡(12⁢a+12)⁢Γ⁡(12⁢b+12).
15.4.29 F⁡(a,b;12⁢a+12⁢b+1;12)=2⁢πa−b⁢Γ⁡(12⁢a+12⁢b+1)⁢(1Γ⁡(12⁢a)⁢Γ⁡(12⁢b+12)−1Γ⁡(12⁢a+12)⁢Γ⁡(12⁢b)).
15.4.30 F⁡(a,1−a;b;12)=21−b⁢π⁢Γ⁡(b)Γ⁡(12⁢a+12⁢b)⁢Γ⁡(12⁢b−12⁢a+12).
15.4.31 F⁡(a,12+a;32−2⁢a;−13)=(89)−2⁢a⁢Γ⁡(43)⁢Γ⁡(32−2⁢a)Γ⁡(32)⁢Γ⁡(43−2⁢a).
15.4.32 F⁡(a,12+a;56+23⁢a;19)=π⁢(34)a⁢Γ⁡(56+23⁢a)Γ⁡(12+13⁢a)⁢Γ⁡(56+13⁢a).
15.4.33 F⁡(3⁢a,13+a;23+2⁢a;ei⁢π/3)=π⁢ei⁢π⁢a/2⁢(1627)(3⁢a+1)/6⁢Γ⁡(56+a)Γ⁡(23+a)⁢Γ⁡(23),
15.4.34 F⁡(3⁢a,a;2⁢a;ei⁢π/3)=π⁢ei⁢π⁢a/2⁢22⁢a⁢Γ⁡(12+a)3(3⁢a+1)/2⁢(1Γ⁡(13+a)⁢Γ⁡(23)+1Γ⁡(23+a)⁢Γ⁡(13)),
ⓘ
Symbols:
Γ⁡(z): gamma function, F⁡(a,b;c;z) or F⁡(a,bc;z): =F12⁡(a,b;c;z) Gauss’ hypergeometric function, π: the ratio of the circumference of a circle to its diameter, e: base of natural logarithm, i: imaginary unit and a: real or complex parameter
Proof sketch:
From §15.5(ii) one obtains that w⁢(a)=e−i⁢π⁢a/2⁢2−2⁢a⁢3(3⁢a+1)/2⁢F⁡(3⁢a,a;2⁢a;ei⁢π/3)⁢Γ⁡(13+a)/Γ⁡(12+a) is a solution of the difference equation (3⁢a+5)⁢w⁢(a+2)−6⁢(a+1)⁢w⁢(a+1)+(3⁢a+1)⁢w⁢(a)=0. Two linearly independent solutions of this difference equation are w1⁢(a)=1 and w2⁢(a)=Γ⁡(13+a)/Γ⁡(23+a). Combining (15.6.1) with Laplace’s method (see §2.4(iii)) we obtain that w⁢(a)∼π⁢(1Γ⁡(23)+a−1/3Γ⁡(13)), as a→∞. Hence, w⁢(a)=π⁢(1Γ⁡(23)+w2⁢(a)Γ⁡(13)).
Notes:
See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.
Referenced by:
§15.4(iii), §15.4(iii), Erratum (V1.1.2) for Additions
Permalink:
http://dlmf.nist.gov/15.4.E34
Encodings:
TeX, pMML, png
Addition (effective with 1.1.2):
This equation was added.
See also:
Annotations for §15.4(iii), §15.4 and Ch.15

where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when in (15.4.33) a=−13,−43,−73,…, and in (15.4.34) a=0,−1,−2,…. Compare the final paragraph in §15.2(ii).

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