@carter
#HyperGeometric
3. Can the equation provide a road map to ${}_p{F}_q\left(\right)$?
4. Is this whole process just a silly extension of DLMF 16.3.1 ?

5/$\omega$ (?)

@carter
#HyperGeometric
Luke- Questions
1. Luke says that ν, μ, λ need to be positive integers. Can they be extended by fractional derivatives?
2. Can the equation be extended by the standard Analytic continued to the real line (except for poles and such) which Luke excludes; but provides guidance in other parts of the book? Of course, the extensions aren't identical to the original but satisfy the defining differential equation.

4/$\omega$

@carter
#HyperGeometric
${}_2{F}_1(1,1,2;-z)=z^{-1}\cdot ln(1+z)$
Where I have kept Luke's notation since it is not too confusing.
Except that we need the conversion
${}_2{F}_1(1,1,2;z)=-z^{-1}\cdot ln(1-z)$
per DLMF 15.4.1 (I didn't want to “into the weeds to early”)

3/ω

@carter
#HyperGeometric
1.1 As mentioned, here is the equation.
$${}_2{F}_1\left(\frac{\nu +1,\nu +\mu +1}{\nu +\mu +\lambda +2}|z\right)=\frac{{(-1)}^{\mu +1}\cdot (\nu +\mu +\lambda +1)!}{\lambda !\cdot \nu !\cdot (\nu +\mu )!\cdot (\mu +\lambda )!}$$
$$\cdot \frac{d^{\nu +\mu }}{d{z}^{\nu +\mu }}[{(1-z)}^{\mu +\lambda }\cdot \frac{d^{\lambda }}{d{z}^{\lambda }}\{z^{-1}\cdot ln(1-z)\}]$$
and
2/$\omega$

@carter
#HyperGeometric
A note on Luke-3.4-15
Abstract.
This is just an exploration of Luke's “The Special Functions and Their Applications” Vol1, section 3.4 equation 15.
This a conversion of
Section 3.1 (15)
to a certain constrained set of ${}_2{F}_1()$ to ${}_2{F}_1(1,1,2;-z)=z^{-1}\cdot ln(1+z)$ .
The questions are about extending this in various ways.

Y. L. Luke,The Special Functions and Their Applications, Academic Press, New York (1969).
1/$\omega$