Distinction between associated Legendre and Ferrers functions

[mjungmath]

As pointed out in #25034 and https://groups.google.com/g/sage-devel/c/IDtiGF6HB28/m/QWwnAeLJBAAJ there is currently a conflict of conventions.

The current implementation is in line with [1], but as mentioned above this source seems to be flawed and a distinction to Ferrers functions needs to be made (cf. [2] and [3]).

More precisely, we want to implement solutions of the ODE [4]. There are solutions on (-1,1) called "Ferrers functions" [2] and solutions on (1,oo) which can be extended to the complex plane without the slit (-oo,1] [3]. This distinction has not been made yet but yields to different formulas in different cases. The current formulas favor the Ferrers functions.

Here are a few references to consider:

Depends on #25034

CC: @sagetrac-jcwomack @slel @egourgoulhon

Component: misc

Issue created by migration from https://trac.sagemath.org/ticket/31637

[mjungmath]

Description changed:

--- 
+++ 
@@ -1,6 +1,6 @@
 As pointed out in #25034 and https://groups.google.com/g/sage-devel/c/IDtiGF6HB28/m/QWwnAeLJBAAJ there is currently a conflict of conventions.

-The implementation is in line https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html now, but as mentioned above this source seems to be flawed and a distinction has to be made.
+The current implementation is in line https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html now, but as mentioned above this source seems to be flawed and a distinction to Ferrers functions has to be made.

 There are a few references to consider:

[mjungmath]

Description changed:

--- 
+++ 
@@ -1,8 +1,8 @@
 As pointed out in #25034 and https://groups.google.com/g/sage-devel/c/IDtiGF6HB28/m/QWwnAeLJBAAJ there is currently a conflict of conventions.

-The current implementation is in line https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html now, but as mentioned above this source seems to be flawed and a distinction to Ferrers functions has to be made.
+The current implementation is in line with https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html, but as mentioned above this source seems to be flawed and a distinction to Ferrers functions needs to be made (cf. https://dlmf.nist.gov/14.21#i.p1 and https://dlmf.nist.gov/14.3#E1).

-There are a few references to consider:
+Here are a few references to consider:

 - https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html (probably flawed?)
 - https://dlmf.nist.gov/14.3#E1

[mjungmath]

Description changed:

--- 
+++ 
@@ -1,11 +1,11 @@
 As pointed out in #25034 and https://groups.google.com/g/sage-devel/c/IDtiGF6HB28/m/QWwnAeLJBAAJ there is currently a conflict of conventions.

-The current implementation is in line with https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html, but as mentioned above this source seems to be flawed and a distinction to Ferrers functions needs to be made (cf. https://dlmf.nist.gov/14.21#i.p1 and https://dlmf.nist.gov/14.3#E1).
+The current implementation is in line with [1], but as mentioned above this source seems to be flawed and a distinction to Ferrers functions needs to be made (cf. [2] and [3]).

 Here are a few references to consider:

-- https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html (probably flawed?)
-- https://dlmf.nist.gov/14.3#E1
-- https://dlmf.nist.gov/14.21#i.p1
-- https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
-- http://www.math.ubc.ca/~cbm/aands/page_332.htm
+- [1]: https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html (probably flawed?)
+- [2]: https://dlmf.nist.gov/14.3#E1
+- [3]: https://dlmf.nist.gov/14.21#i.p1
+- [4]: https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
+- [5]: http://www.math.ubc.ca/~cbm/aands/page_332.htm

[mjungmath]

Description changed:

--- 
+++ 
@@ -1,11 +1,14 @@
 As pointed out in #25034 and https://groups.google.com/g/sage-devel/c/IDtiGF6HB28/m/QWwnAeLJBAAJ there is currently a conflict of conventions.

 The current implementation is in line with [1], but as mentioned above this source seems to be flawed and a distinction to Ferrers functions needs to be made (cf. [2] and [3]).
+
+More precisely, we want to implement solutions of the ODE [4]. There are solutions on (-1,1) called "Ferrers functions" [2] and solutions on (1,oo) which can be extended to the complex plane [3]. This distinction has not been made yet but yields to different formulas in different cases.

 Here are a few references to consider:

 - [1]: https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html (probably flawed?)
 - [2]: https://dlmf.nist.gov/14.3#E1
 - [3]: https://dlmf.nist.gov/14.21#i.p1
-- [4]: https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
-- [5]: http://www.math.ubc.ca/~cbm/aands/page_332.htm
+- [4]: https://dlmf.nist.gov/14.2#E2
+- [5]: https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
+- [6]: http://www.math.ubc.ca/~cbm/aands/page_332.htm

[mjungmath]

Description changed:

--- 
+++ 
@@ -2,7 +2,7 @@

 The current implementation is in line with [1], but as mentioned above this source seems to be flawed and a distinction to Ferrers functions needs to be made (cf. [2] and [3]).

-More precisely, we want to implement solutions of the ODE [4]. There are solutions on (-1,1) called "Ferrers functions" [2] and solutions on (1,oo) which can be extended to the complex plane [3]. This distinction has not been made yet but yields to different formulas in different cases.
+More precisely, we want to implement solutions of the ODE [4]. There are solutions on (-1,1) called "Ferrers functions" [2] and solutions on (1,oo), which can be extended to the complex plane [3]. This distinction has not been made yet but yields to different formulas in different cases. The current formulas favor the Ferrers functions.

 Here are a few references to consider:

[mjungmath]

Description changed:

--- 
+++ 
@@ -2,7 +2,7 @@

 The current implementation is in line with [1], but as mentioned above this source seems to be flawed and a distinction to Ferrers functions needs to be made (cf. [2] and [3]).

-More precisely, we want to implement solutions of the ODE [4]. There are solutions on (-1,1) called "Ferrers functions" [2] and solutions on (1,oo), which can be extended to the complex plane [3]. This distinction has not been made yet but yields to different formulas in different cases. The current formulas favor the Ferrers functions.
+More precisely, we want to implement solutions of the ODE [4]. There are solutions on (-1,1) called "Ferrers functions" [2] and solutions on (1,oo) which can be extended to the complex plane without the slit `(-oo,1]` [3]. This distinction has not been made yet but yields to different formulas in different cases. The current formulas favor the Ferrers functions.

 Here are a few references to consider:

[mjungmath]

comment:7

I see three options here:

1. We introduce a flag switching back and forth between Ferrers functions and Legendre functions (like it is done in Mathematica).
2. We introduce an environment variable making that distinction (like in Maple).
3. We separate Ferrers functions and Legendre functions completely; for example ferrers_func and gen_legendre_P.

I advocate the latter option since it leads to less confusion for the end-user (and developer).

Other than that, implementing connection formulas to Legendre (Ferrers) functions of second kind are also desirable.

[mjungmath]

Dependencies: #25034