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11: 15.10 Hypergeometric Differential Equation

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15.10.3 $$𝒲\left\{f_1(z),f_2(z)\right\}=(1-c)z^{-c}{(1-z)}^{c-a-b-1}.$$

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Symbols:

$𝒲$: Wronskian, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $f_1(z)$, $f_2(z)$: fundamental solutions

Permalink:

http://dlmf.nist.gov/15.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

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15.10.5 $$𝒲\left\{f_1(z),f_2(z)\right\}=(a+b-c)z^{-c}{(1-z)}^{c-a-b-1}.$$

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Symbols:

$𝒲$: Wronskian, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $f_1(z)$, $f_2(z)$: fundamental solutions

Permalink:

http://dlmf.nist.gov/15.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

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15.10.7 $$𝒲\left\{f_1(z),f_2(z)\right\}=(a-b)z^{-c}{(z-1)}^{c-a-b-1}.$$

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Symbols:

$𝒲$: Wronskian, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $f_1(z)$, $f_2(z)$: fundamental solutions

Permalink:

http://dlmf.nist.gov/15.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10(i), §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. … ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …