User:Tohline/Appendix/Mathematics/ToroidalConfusion
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Confusion Regarding Whipple Formulae
May, 2018 (J.E.Tohline): I am trying to figure out what the correct relationship is between halfinteger degree, associated Legendre functions of the first and second kinds. In order to illustrate my current confusion, here I will restrict my presentation to expressions that give in terms of .
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Published Expressions
From equation (34) of H. S. Cohl, J. E. Tohline, A. R. P. Rau, & H. M. Srivastiva (2000, Astronomische Nachrichten, 321, no. 5, 363  372) I find:
Expression #1  




From Howard Cohl's online overview of toroidal functions, I find:
Expression #2  




Copying the Whipple's formula from §14.19 of DLMF,
Expression #3  




So far, this gives me three similar but not identical formulae for the same function mapping! As per equation (8) in (yet another source!) A. Gil, J. Segura, & N. M. Temme (2000, JCP, 161, 204  217), the relationship is:




Gil, Segura, & Temme (2000): eq. (8) 
where: 

This expression from Gil et al. (2000) means, for example, that by identifying with , we have , and,









That is, we have,
Expression #4  




which matches the above expression #1 drawn from Cohl et al. (2000), but which appears not to match either of the other two "published" (online) formulae, expressions #2 or #3.
Specific Application
I stumbled into this dilemma when I tried to explicitly demonstrate how can be derived from where, from §8.13 of M. Abramowitz & I. A. Stegun (1995), we find,



Abramowitz & Stegun (1995), eq. (8.13.4) 
and,



Abramowitz & Stegun (1995), eq. (8.13.1) 
When I used the Whipple formula as defined in §14.19 of DLMF (expression #3 reprinted above), the function mapping gave me the wrong result; I was off by a factor of . But, as demonstrated below, the Whipple formula provided by Cohl et al. (2000) and by Gil et al. (2000) — that is, expressions #1 and #4, above — does give the correct result.
Demonstration that can be derived from 

Copying equation (34) from Cohl et al. (2000), we begin with,
then setting , we have,
Step #1: Associate … . Then,
Step #2: Now making the association … , and drawing on eq. (8.13.1) from Abramowitz & Stegun (1995), we can write,
Step #3: Again, making the association … , means,
This, indeed, matches eq. (8.13.4) from Abramowitz & Stegun (1995). 
Cohl's Response to My (May 2018) Email Query
Proper Interpretation of DLMF Expression
Most of the confusion expressed above stems from the DLMF's use of bold fonts, such as the function on the lefthand side of expression #3, above — that is, the Whipple formula from §14.19 of DLMF,



What has been missing in my discussion is an appreciation of the following relationship between bold and plaintext function names,
After making the substitutions, and , the Whipple formula displayed above as expression #3 becomes,









which matches expression #2, above. But it does not appear to match expressions #1 or #4.
The standard "Euler reflection formula for gamma functions" is usually presented in the form,
If we make the association,
with and both being either zero or a positive integer, then, this Euler reflection formula becomes,

However, in our situation the socalled "Euler reflection formula for gamma functions" gives the relation,



Hence, we may also write,






which matches expressions #1 and #4. So everything appears to be in agreement! Hooray!
Derivation From Scratch
Whenever he deals with these types of relations, Cohl usually begins with,
Expression #5  




Making the pair of substitutions,











we also have,




















in which case,



Now, since,



if we make the substitution,






we also know that,



Hence, we can write,



Finally, another relation states that, for ,



So, we obtain,






This matches expressions #2 and #3, above.
Index Values of Zero
Setting gives the following soughtfor relationship:






Joel's Additional Manipulations
From §14.19.6 of DLMF, we find the following summation expression:
Then, if we again employ the DLMF relationship between bold and plaintext function names, namely,
where we have made the substitution, , the Sums expression becomes,






When dealing with DysonWong tori, we will set , in which case the Sums expression becomes,



But this can be rewritten in the form,



See Also
© 2014  2020 by Joel E. Tohline 