implement harmonic number function H(n,m)

[rwst]

H(n,m) belongs to the holonomic repertoir and should be available in Sage, both numerically and symbolically. For some info, see https://groups.google.com/forum/#!topic/sage-devel/uaA0yU7dZHU and https://en.wikipedia.org/wiki/Harmonic_number

This is actually a defect because we leave a Maxima result undefined:

sage: sum(1/k^3,k,1,n)
gen_harmonic_number(3, n)
sage: gen_harmonic_number?
Object `gen_harmonic_number` not found.

CC: @eviatarbach @dkrenn

Component: symbolics

Keywords: special, log

Author: Ralf Stephan, Armin Straub

Branch/Commit: 9042104

Reviewer: Ralf Stephan, Armin Straub

Issue created by migration from https://trac.sagemath.org/ticket/16671

[fredrik-johansson]

comment:1

For exact values, FLINT has arith_harmonic_number()

[rwst]

comment:2

Replying to @fredrik-johansson:

For exact values, FLINT has arith_harmonic_number()

Unfortunately, neither FLINT nor mpmath implements the generalized H(m,n) returned by Maxima in the code example.

[rwst]

Description changed:

--- 
+++ 
@@ -1 +1,11 @@
 H_n belongs to the holonomic repertoir and should be available in Sage, both numerically and symbolically. For some info, see https://groups.google.com/forum/#!topic/sage-devel/uaA0yU7dZHU and https://en.wikipedia.org/wiki/Harmonic_number
+
+This is actually a defect because we leave a Maxima result undefined:
+
+```
+sage: sum(1/k^3,k,1,n)
+gen_harmonic_number(3, n)
+sage: gen_harmonic_number?
+Object `gen_harmonic_number` not found.
+```
+

[rwst]

Description changed:

--- 
+++ 
@@ -1,4 +1,4 @@
-H_n belongs to the holonomic repertoir and should be available in Sage, both numerically and symbolically. For some info, see https://groups.google.com/forum/#!topic/sage-devel/uaA0yU7dZHU and https://en.wikipedia.org/wiki/Harmonic_number
+H(n,m) belongs to the holonomic repertoir and should be available in Sage, both numerically and symbolically. For some info, see https://groups.google.com/forum/#!topic/sage-devel/uaA0yU7dZHU and https://en.wikipedia.org/wiki/Harmonic_number
 
 This is actually a defect because we leave a Maxima result undefined:
 

[rwst]

Branch: u/rws/implement_harmonic_number_function_h_n_m_

[rwst]

Commit: 5310ca1

[rwst]

Author: Ralf Stephan

[rwst]

comment:5

It is already implemented, I'm just waiting for https://groups.google.com/forum/?hl=en#!topic/sage-devel/SM_FLL33t2g as to argument order of the generalized version.

---

New commits:

5310ca1

16671: implement harmonic number function H(n,m) (modulo argument order)