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15 Hypergeometric FunctionProperties

§15.10 Hypergeometric Differential Equation

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Referenced by:
§15.17(i)
Permalink:
http://dlmf.nist.gov/15.10
See also:
Annotations for Ch.15
Contents
  1. §15.10(i) Fundamental Solutions
  2. §15.10(ii) Kummer’s 24 Solutions and Connection Formulas

§15.10(i) Fundamental Solutions

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Keywords:
fundamental solutions, hypergeometric differential equation, hypergeometric equation
A&S Ref:
15.5
Notes:
See Andrews et al. (1999, §2.3) and Olver (1997b, pp. 163–168). (15.10.9) is a corrected version of (2.3.20) in the first reference.
Referenced by:
§13.2(i), Erratum (V1.0.1) for Clarifications
Permalink:
http://dlmf.nist.gov/15.10.i
Clarification (effective with 1.0.1):
Item (c) was clarified by stating more explicitly the roles of the parameters a+n−1,b+n−1 and c.
See also:
Annotations for §15.10 and Ch.15
15.10.1 z⁢(1−z)⁢d2wdz2+(c−(a+b+1)⁢z)⁢dwdz−a⁢b⁢w=0.

This is the hypergeometric differential equation. It has regular singularities at z=0,1,∞, with corresponding exponent pairs {0,1−c}, {0,c−a−b}, {a,b}, respectively. When none of the exponent pairs differ by an integer, that is, when none of c, c−a−b, a−b is an integer, we have the following pairs f1⁡(z), f2⁡(z) of fundamental solutions. They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity.

Singularity z=0

Singularity z=1

15.10.4 f1⁡(z) =F⁡(a,ba+b+1−c;1−z),
f2⁡(z) =(1−z)c−a−b⁢F⁡(c−a,c−bc−a−b+1;1−z),
15.10.5 𝒲⁡{f1⁡(z),f2⁡(z)}=(a+b−c)⁢z−c⁢(1−z)c−a−b−1.

Singularity z=∞

15.10.6 f1⁡(z) =z−a⁢F⁡(a,a−c+1a−b+1;1z),
f2⁡(z) =z−b⁢F⁡(b,b−c+1b−a+1;1z),
15.10.7 𝒲⁡{f1⁡(z),f2⁡(z)}=(a−b)⁢z−c⁢(z−1)c−a−b−1.

(a) If c equals n=1,2,3,…, and a=1,2,…,n−1, then fundamental solutions in the neighborhood of z=0 are given by (15.10.2) with the interpretation (15.2.5) for f2⁡(z).

(b) If c equals n=1,2,3,…, and a≠1,2,…,n−1, then fundamental solutions in the neighborhood of z=0 are given by F⁡(a,b;n;z) and

15.10.8 F⁡(a,bn;z)⁢ln⁡z−∑k=1n−1(n−1)!⁢(k−1)!(n−k−1)!⁢(1−a)k⁢(1−b)k⁢(−z)−k+∑k=0∞(a)k⁢(b)k(n)k⁢k!⁢zk⁢(ψ⁡(a+k)+ψ⁡(b+k)−ψ⁡(1+k)−ψ⁡(n+k)),
a,b≠n−1,n−2,…,0,−1,−2,…,
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Symbols:
F⁡(a,b;c;z) or F⁡(a,bc;z): =F12⁡(a,b;c;z) Gauss’ hypergeometric function, (a)n: Pochhammer’s symbol (or shifted factorial), ψ⁡(z): psi (or digamma) function, !: factorial (as in n!), ln⁡z: principal branch of logarithm function, n: integer, z: complex variable, a: real or complex parameter, b: real or complex parameter, k: integer and m: integer
A&S Ref:
15.5.19. (Compared with (15.10.8) this result has a multiple of F⁡(a,b;n;z) added to the right-hand side. It also contains an error: the conditions on a and b should be a,b≠1,2,…,m.)
Referenced by:
(15.10.8)
Permalink:
http://dlmf.nist.gov/15.10.E8
Encodings:
TeX, pMML, png
See also:
Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

or

15.10.9 F⁡(−m,bn;z)⁢ln⁡z−∑k=1n−1(n−1)!⁢(k−1)!(n−k−1)!⁢(m+1)k⁢(1−b)k⁢(−z)−k+∑k=0m(−m)k⁢(b)k(n)k⁢k!⁢zk⁢(ψ⁡(1+m−k)+ψ⁡(b+k)−ψ⁡(1+k)−ψ⁡(n+k))+(−1)m⁢m!⁢∑k=m+1∞(k−1−m)!⁢(b)k(n)k⁢k!⁢zk,
a=−m, m=0,1,2,…; b≠n−1,n−2,…,0,−1,−2,…,

or

15.10.10 F⁡(−m,−ℓn;z)⁢ln⁡z−∑k=1n−1(n−1)!⁢(k−1)!(n−k−1)!⁢(m+1)k⁢(ℓ+1)k⁢(−z)−k+∑k=0ℓ(−m)k⁢(−ℓ)k(n)k⁢k!⁢zk⁢(ψ⁡(1+m−k)+ψ⁡(1+ℓ−k)−ψ⁡(1+k)−ψ⁡(n+k))+(−1)ℓ⁢ℓ!⁢∑k=ℓ+1m(k−1−ℓ)!⁢(−m)k(n)k⁢k!⁢zk,
a=−m, m=0,1,2,…; b=−ℓ, ℓ=0,1,2,…,m.

Moreover, in (15.10.9) and (15.10.10) the symbols a and b are interchangeable.

(c) If the parameter c in the differential equation equals 2−n=0,−1,−2,…, then fundamental solutions in the neighborhood of z=0 are given by zn−1 times those in (a) and (b), with a and b replaced throughout by a+n−1 and b+n−1, respectively.

(d) If a+b+1−c equals n=1,2,3,…, or 2−n=0,−1,−2,…, then fundamental solutions in the neighborhood of z=1 are given by those in (a), (b), and (c) with z replaced by 1−z.

(e) Finally, if a−b+1 equals n=1,2,3,…, or 2−n=0,−1,−2,…, then fundamental solutions in the neighborhood of z=∞ are given by z−a times those in (a), (b), and (c) with b and z replaced by a−c+1 and 1/z, respectively.

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

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Keywords:
Kummer’s solutions, hypergeometric differential equation
A&S Ref:
15.5
Notes:
See Luke (1969a, Chapter III).
Referenced by:
§15.12(iii), §16.4(iii), §31.3(iii)
Permalink:
http://dlmf.nist.gov/15.10.ii
See also:
Annotations for §15.10 and Ch.15

The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions.

15.10.11 w1⁢(z) =F⁡(a,bc;z)=(1−z)c−a−b⁢F⁡(c−a,c−bc;z)=(1−z)−a⁢F⁡(a,c−bc;zz−1)=(1−z)−b⁢F⁡(c−a,bc;zz−1).
15.10.12 w2⁢(z) =z1−c⁢F⁡(a−c+1,b−c+12−c;z)
=z1−c⁢(1−z)c−a−b⁢F⁡(1−a,1−b2−c;z)
=z1−c⁢(1−z)c−a−1⁢F⁡(a−c+1,1−b2−c;zz−1)
=z1−c⁢(1−z)c−b−1⁢F⁡(1−a,b−c+12−c;zz−1).
15.10.13 w3⁢(z) =F⁡(a,ba+b−c+1;1−z)=z1−c⁢F⁡(a−c+1,b−c+1a+b−c+1;1−z)=z−a⁢F⁡(a,a−c+1a+b−c+1;1−1z)=z−b⁢F⁡(b,b−c+1a+b−c+1;1−1z).
15.10.14 w4⁢(z) =(1−z)c−a−b⁢F⁡(c−a,c−bc−a−b+1;1−z)
=z1−c⁢(1−z)c−a−b⁢F⁡(1−a,1−bc−a−b+1;1−z)
=za−c⁢(1−z)c−a−b⁢F⁡(1−a,c−ac−a−b+1;1−1z)
=zb−c⁢(1−z)c−a−b⁢F⁡(1−b,c−bc−a−b+1;1−1z).
15.10.15 w5⁢(z) =ea⁢π⁢i⁢z−a⁢F⁡(a,a−c+1a−b+1;1z)
=e(c−b)⁢π⁢i⁢zb−c⁢(1−z)c−a−b⁢F⁡(1−b,c−ba−b+1;1z)
=(1−z)−a⁢F⁡(a,c−ba−b+1;11−z)
=e(c−1)⁢π⁢i⁢z1−c⁢(1−z)c−a−1⁢F⁡(1−b,a−c+1a−b+1;11−z).
15.10.16 w6⁢(z) =eb⁢π⁢i⁢z−b⁢F⁡(b,b−c+1b−a+1;1z)
=e(c−a)⁢π⁢i⁢za−c⁢(1−z)c−a−b⁢F⁡(1−a,c−ab−a+1;1z)
=(1−z)−b⁢F⁡(b,c−ab−a+1;11−z)
=e(c−1)⁢π⁢i⁢z1−c⁢(1−z)c−b−1⁢F⁡(1−a,b−c+1b−a+1;11−z).

The (63)=20 connection formulas for the principal branches of Kummer’s solutions are:

15.10.17 w3⁢(z) =Γ⁡(1−c)⁢Γ⁡(a+b−c+1)Γ⁡(a−c+1)⁢Γ⁡(b−c+1)⁢w1⁢(z)+Γ⁡(c−1)⁢Γ⁡(a+b−c+1)Γ⁡(a)⁢Γ⁡(b)⁢w2⁢(z),
15.10.18 w4⁢(z) =Γ⁡(1−c)⁢Γ⁡(c−a−b+1)Γ⁡(1−a)⁢Γ⁡(1−b)⁢w1⁢(z)+Γ⁡(c−1)⁢Γ⁡(c−a−b+1)Γ⁡(c−a)⁢Γ⁡(c−b)⁢w2⁢(z),
15.10.19 w5⁢(z) =Γ⁡(1−c)⁢Γ⁡(a−b+1)Γ⁡(a−c+1)⁢Γ⁡(1−b)⁢w1⁢(z)+e(c−1)⁢π⁢i⁢Γ⁡(c−1)⁢Γ⁡(a−b+1)Γ⁡(a)⁢Γ⁡(c−b)⁢w2⁢(z),
15.10.20 w6⁢(z) =Γ⁡(1−c)⁢Γ⁡(b−a+1)Γ⁡(b−c+1)⁢Γ⁡(1−a)⁢w1⁢(z)+e(c−1)⁢π⁢i⁢Γ⁡(c−1)⁢Γ⁡(b−a+1)Γ⁡(b)⁢Γ⁡(c−a)⁢w2⁢(z).
15.10.21 w1⁢(z) =Γ⁡(c)⁢Γ⁡(c−a−b)Γ⁡(c−a)⁢Γ⁡(c−b)⁢w3⁢(z)+Γ⁡(c)⁢Γ⁡(a+b−c)Γ⁡(a)⁢Γ⁡(b)⁢w4⁢(z),
15.10.22 w2⁢(z) =Γ⁡(2−c)⁢Γ⁡(c−a−b)Γ⁡(1−a)⁢Γ⁡(1−b)⁢w3⁢(z)+Γ⁡(2−c)⁢Γ⁡(a+b−c)Γ⁡(a−c+1)⁢Γ⁡(b−c+1)⁢w4⁢(z),
15.10.23 w5⁢(z) =ea⁢π⁢i⁢Γ⁡(a−b+1)⁢Γ⁡(c−a−b)Γ⁡(1−b)⁢Γ⁡(c−b)⁢w3⁢(z)+e(c−b)⁢π⁢i⁢Γ⁡(a−b+1)⁢Γ⁡(a+b−c)Γ⁡(a)⁢Γ⁡(a−c+1)⁢w4⁢(z),
15.10.24 w6⁢(z) =eb⁢π⁢i⁢Γ⁡(b−a+1)⁢Γ⁡(c−a−b)Γ⁡(1−a)⁢Γ⁡(c−a)⁢w3⁢(z)+e(c−a)⁢π⁢i⁢Γ⁡(b−a+1)⁢Γ⁡(a+b−c)Γ⁡(b)⁢Γ⁡(b−c+1)⁢w4⁢(z).
15.10.25 w1⁢(z) =Γ⁡(c)⁢Γ⁡(b−a)Γ⁡(b)⁢Γ⁡(c−a)⁢w5⁢(z)+Γ⁡(c)⁢Γ⁡(a−b)Γ⁡(a)⁢Γ⁡(c−b)⁢w6⁢(z),
15.10.26 w2⁢(z) =e(1−c)⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(b−a)Γ⁡(1−a)⁢Γ⁡(b−c+1)⁢w5⁢(z)+e(1−c)⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(a−b)Γ⁡(1−b)⁢Γ⁡(a−c+1)⁢w6⁢(z),
15.10.27 w3⁢(z) =e−a⁢π⁢i⁢Γ⁡(a+b−c+1)⁢Γ⁡(b−a)Γ⁡(b)⁢Γ⁡(b−c+1)⁢w5⁢(z)+e−b⁢π⁢i⁢Γ⁡(a+b−c+1)⁢Γ⁡(a−b)Γ⁡(a)⁢Γ⁡(a−c+1)⁢w6⁢(z),
15.10.28 w4⁢(z) =e(b−c)⁢π⁢i⁢Γ⁡(c−a−b+1)⁢Γ⁡(b−a)Γ⁡(1−a)⁢Γ⁡(c−a)⁢w5⁢(z)+e(a−c)⁢π⁢i⁢Γ⁡(c−a−b+1)⁢Γ⁡(a−b)Γ⁡(1−b)⁢Γ⁡(c−b)⁢w6⁢(z).
15.10.29 w1⁢(z) =eb⁢π⁢i⁢Γ⁡(c)⁢Γ⁡(a−c+1)Γ⁡(a+b−c+1)⁢Γ⁡(c−b)⁢w3⁢(z)+e(b−c)⁢π⁢i⁢Γ⁡(c)⁢Γ⁡(a−c+1)Γ⁡(b)⁢Γ⁡(a−b+1)⁢w5⁢(z),
15.10.30 w1⁢(z) =ea⁢π⁢i⁢Γ⁡(c)⁢Γ⁡(b−c+1)Γ⁡(a+b−c+1)⁢Γ⁡(c−a)⁢w3⁢(z)+e(a−c)⁢π⁢i⁢Γ⁡(c)⁢Γ⁡(b−c+1)Γ⁡(a)⁢Γ⁡(b−a+1)⁢w6⁢(z),
15.10.31 w2⁢(z) =e(b−c+1)⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(a)Γ⁡(a+b−c+1)⁢Γ⁡(1−b)⁢w3⁢(z)+e(b−c)⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(a)Γ⁡(a−b+1)⁢Γ⁡(b−c+1)⁢w5⁢(z),
15.10.32 w2⁢(z) =e(a−c+1)⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(b)Γ⁡(a+b−c+1)⁢Γ⁡(1−a)⁢w3⁢(z)+e(a−c)⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(b)Γ⁡(b−a+1)⁢Γ⁡(a−c+1)⁢w6⁢(z).
15.10.33 w1⁢(z) =e(c−a)⁢π⁢i⁢Γ⁡(c)⁢Γ⁡(1−b)Γ⁡(a)⁢Γ⁡(c−a−b+1)⁢w4⁢(z)+e−a⁢π⁢i⁢Γ⁡(c)⁢Γ⁡(1−b)Γ⁡(a−b+1)⁢Γ⁡(c−a)⁢w5⁢(z),
15.10.34 w1⁢(z) =e(c−b)⁢π⁢i⁢Γ⁡(c)⁢Γ⁡(1−a)Γ⁡(b)⁢Γ⁡(c−a−b+1)⁢w4⁢(z)+e−b⁢π⁢i⁢Γ⁡(c)⁢Γ⁡(1−a)Γ⁡(b−a+1)⁢Γ⁡(c−b)⁢w6⁢(z),
15.10.35 w2⁢(z) =e(1−a)⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(c−b)Γ⁡(a−c+1)⁢Γ⁡(c−a−b+1)⁢w4⁢(z)+exp⁡e−a⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(c−b)Γ⁡(a−b+1)⁢Γ⁡(1−a)⁢w5⁢(z),
15.10.36 w2⁢(z) =e(1−b)⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(c−a)Γ⁡(b−c+1)⁢Γ⁡(c−a−b+1)⁢w4⁢(z)+e−b⁢π⁢i⁢Γ⁡(2−c)⁢Γ⁡(c−a)Γ⁡(b−a+1)⁢Γ⁡(1−b)⁢w6⁢(z).
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