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 FoxHToMeijerG (1.0.0) current version: 1.0.1 »

Source Notebook

Express FoxH in terms of MeijerG whenever applicable

Contributed by: Jan Mangaldan

ResourceFunction["FoxHToMeijerG"][expr]

converts FoxH expressions in expr to a form involving MeijerG.

ResourceFunction["FoxHToMeijerG"][expr,f]

wraps f around any MeijerG head in the result.

Details

The Fox H-function can be converted to an equivalent Meijer G-function representation whenever the parameters αj and βj are rational numbers.
If any of the αj and βj are nonpositive, the function is first converted to an equivalent form where the αj and βj are positive.

Examples

Basic Examples (2) 

A Fox H-function:

In[1]:=
expr = FoxH[{{{0, 3/2}}, {}}, {{{-(1/2), 1/2}, {-(1/2), 1}}, {}}, z]
Out[1]=

Convert to an equivalent form involving Meijer G-functions:

In[2]:=
ResourceFunction["FoxHToMeijerG"][expr]
Out[2]=

Scope (3) 

FoxHToMeijerG works on inactivated FoxH expressions:

In[3]:=
ResourceFunction["FoxHToMeijerG"][ Inactive[FoxH][{{}, {{1, 1/4}}}, {{{1/8, 1/4}, {3/8, 1/4}, {5/8, 1/ 4}, {7/8, 1/4}}, {{0, 1/4}}}, x]]
Out[3]=

Convert a FoxH expression with nonpositive αj and βj parameters:

In[4]:=
ResourceFunction["FoxHToMeijerG"][ FoxH[{{{0, 3/2}, {1/2, 0}}, {{1, -(3/2)}}}, {{{1/2, 3/2}, {-(1/2), 1/ 2}}, {{1, 1}}}, z]]
Out[4]=

The following automatically evaluates to a form that is free from Meijer G-functions, due to auto-evaluation of MeijerG:

In[5]:=
ResourceFunction["FoxHToMeijerG"][ FoxH[{{{1, 1/2}, {1/2, 1/2}}, {}}, {{{1/2, 1/2}}, {{0, 1}}}, z]]
Out[5]=

Wrap the MeijerG heads in Inactive to prevent evaluation:

In[6]:=
ResourceFunction["FoxHToMeijerG"][ FoxH[{{{1, 1/2}, {1/2, 1/2}}, {}}, {{{1/2, 1/2}}, {{0, 1}}}, z], Inactive]
Out[6]=

Properties and Relations (2) 

Use FoxHToMeijerG along with the resource function MeijerGToHypergeometricPFQ to get expressions in terms of hypergeometric functions:

In[7]:=
ResourceFunction["MeijerGToHypergeometricPFQ"][ ResourceFunction["FoxHToMeijerG"][ FoxH[{{}, {{1/3, 2/3}}}, {{{0, 1/3}, {1/3, 1/3}}, {{1/3, 2/3}}}, z], Inactive], Inactive]
Out[7]=

Activate and simplify the result:

In[8]:=
Activate[%] // FullSimplify
Out[8]=

Possible Issues (1) 

FoxH expressions with irrational parameters are left unchanged:

In[9]:=
ResourceFunction["FoxHToMeijerG"][ FoxH[{{{1, 1/2}, {1/2, Sqrt[2]}}, {{}}}, {{}, {{1, 1}}}, z]]
Out[9]=

Version History

  • 1.0.1 – 28 March 2022
  • 1.0.0 – 04 October 2021

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