implement Meijer G function

[rwst]

This function would allow to express the result from differentiation of several special functions with respect to the order parameter, which at the moment gives an error:

sage: diff(exp_integral_e(n,x),n)
NotImplementedError: The derivative of this function is only implemented for 
sage: gamma_inc(n,x).diff(n)
D[0](gamma)(n, x)

sage: sympy.expint(n, x).diff(n)
-x**(n - 1)*meijerg(((), (1, 1)), ((0, 0, -n + 1), ()), x)
sage: sympy.uppergamma(n, x).diff(n)
meijerg(((), (1, 1)), ((0, 0, n), ()), x) + log(x)*uppergamma(n, x)
sage: sympy.lowergamma(n, x).diff(n)
-meijerg(((), (1, 1)), ((0, 0, n), ()), x) - log(x)*uppergamma(n, x) + gamma(n)*polygamma(0, n)
sage: meijerg?
Object `meijerg` not found.

sage: laplace(cos(-1/t), t, s, algorithm='sympy')
...
AttributeError: Unable to convert SymPy result (=meijerg(((), ()), ((-1/2, 0, 1/2), (0,)), s**2/16)/4) into Sage

Mpmath has numerics too.

https://en.wikipedia.org/wiki/Meijer_G-function

See also https://ask.sagemath.org/question/60006/meijer-g-function-in-sage/

CC: @slel

Component: symbolics

Keywords: meijerg

Issue created by migration from https://trac.sagemath.org/ticket/17970

[rwst]

Description changed:

--- 
+++ 
@@ -2,7 +2,8 @@
 
 ```
 sage: diff(exp_integral_e(n,x),n)
-NotImplementedError: The derivative of this function is only implemented for sage: gamma_inc(n,x).diff(n)
+NotImplementedError: The derivative of this function is only implemented for 
+sage: gamma_inc(n,x).diff(n)
 D[0](gamma)(n, x)
 
 sage: sympy.expint(n, x).diff(n)

[rwst]

Description changed:

--- 
+++ 
@@ -14,6 +14,10 @@
 -meijerg(((), (1, 1)), ((0, 0, n), ()), x) - log(x)*uppergamma(n, x) + gamma(n)*polygamma(0, n)
 sage: meijerg?
 Object `meijerg` not found.
+
+sage: laplace(cos(-1/t), t, s, algorithm='sympy')
+...
+AttributeError: Unable to convert SymPy result (=meijerg(((), ()), ((-1/2, 0, 1/2), (0,)), s**2/16)/4) into Sage
 ```
 
 Mpmath has numerics too.

[videlec]

comment:3

update milestone 8.3 -> 8.4

[sagetrac-tmonteil]

Description changed:

--- 
+++ 
@@ -23,3 +23,5 @@
 Mpmath has numerics too.
 
 https://en.wikipedia.org/wiki/Meijer_G-function
+
+See also https://ask.sagemath.org/question/60006/meijer-g-function-in-sage/

[slel]

comment:5

The Meijer G function is mentioned in questions,
answers, comments or discussions at:

• Ask Sage question 60006: Meijer G function in Sage

• Ask Sage question 48908: Avoid scientific notation in the Mathematica interface

• Ask Sage question 43392: Force 'sympy' backend for simplifying symbolic integrals

• Ask Sage question 42637: Error using algorithm="sympy" on an integral

• Ask Sage question 10724: Symbolic integration

• Ask Sage question 8634: Implement and plot meijerg

• sage-support, 2018-04, simple?--from_meijer in jupyter

• sage-support, 2014-10, Formal methods are not human one

• sage-devel, 2017-01, GSoC 2017 kickoff

• sage-devel, 2013-05, 3 GSoC projects for Sage

• sage-devel, 2009-08, ANN: mpmath 0.13 released

• sage-devel, 2010-08, test suites for Sage

• sage-devel, 2010-05, sage vs others by NIST

• sage-devel, 2008-08, GiNaC as the symbolic manipulation engine in Sage

[slel]

Changed keywords from none to meijerg

[sheerluck]

comment:6

sage: integrate(x * bessel_Y(1, x), (x, 0 ,1), algorithm="sympy")
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
/usr/lib/python3.10/site-packages/sage/interfaces/sympy.py in _sympysage_function_by_name(fname)
    303         else:
    304             # the function defined in sympy is not known in sage
--> 305             raise AttributeError(fname)
    306     return func
    307 

AttributeError: meijerg