Symbolic computation tools for Fox-H function

Motivation
Demonstration
Prerequisites
Usage
Documentation
Related references
How to contribute
Cite this work?
LICENSE

Motivation

It is often very tedious and error-prone to check the conditions for the Fox-H function. This repo is to provide some symbolic tools to help check the conditions for the Fox-H function.

The parametrization of the Fox-H function is given by the following diagram:

The well-posedness of the Fox-H function is given by the following diagram:

The codes are used to verify Theorem C.1 of arXiv:2206.10069.

Demonstration

Prerequisites

Wolfram Mathematica and Wolfram Script.
Python > 3.8.
lualatex for generating pdf files.
Some scripts are written in bash, run in Linux. For Mac OS or Windows, you need to modify the scripts accordingly.

Usage

The input file for the Fox H function can be either csv file or wls file.

If the input the file is csv file, it should contain four rows, corresponding four lists:
- Upper Front List
- Upper Rear List
- Lower Front List
- Lower Rear List
Example is here: test csv, where comment lines start with #.

# Comment line starts with #
{1, \[Alpha]^(-1)}, {1, 1}
{Ceil[\[Beta]], \[Beta]}, {1, 1}
{1/2, \[Alpha]/2}, {1, 1}, {3, 3}, {2, 2}
# The following is the Lower Rear List
{1, \[Alpha]/2}

Or you can directly write the wls file in the format given in test.wls. Here is one example in the compact form:

{
  (* Upper List *) {
    (* Upper Front list *) {{1, \[Alpha]^(-1)}, {1, 1}},
    (* Upper Rear List *)  {{Ceil[\[Beta]], \[Beta]}, {1, 1}}
  },
  (* Lower List *) {
    (* Lower Front List *) {{1/2, \[Alpha]/2}, {1, 1}, {3, 3}, {2, 2}},
    (* Lower Rear List *)  {{1, \[Alpha]/2}}
  }
}

You can use the python script parseArg py to convert the csv file to wls file.
You can use PrettyFoxH wls to convert the nested list of wls file to the pretty form with comment lines as above.

Use FoxH wls to compute all conditions:

Run
```
./FoxH.wls
```
to see the usage.
Run
```
./FoxH.wls --help
```
to see how to type Greek letters.
In action, run
```
./FoxH.wls FoxH32-21.wls
```

The results are stored in FoxH_Results mx. See the Mathematica Notebook Result_Handle nb for how to load the results.
To load the results from the Mathematica notebook, do the following <<FoxH_Results.mx, make sure

Quit[] (* start a fresh kernel *)

Documentation

Documentation Fox Parametration pdf (Under construction)
More examples are under development under in examples folder.

Related references

Conditions and conventions for the Fox H function follow from

Kilbas, Anatoly A., and Megumi Saigo. 2004. $H$-Transforms. Vol. 9. Analytical Methods and Special Functions. Chapman & Hall/CRC, Boca Raton, FL. https://doi.org/10.1201/9780203487372.

Related papers that use this code include:

Chen, Le, Yuhui Guo, and Jian Song. 2022. "Moments and Asymptotics for a Class of SPDEs with Space-Time White Noise." Preprint arXiv:2206.10069, to Appear in Trans. Amer. Math. Soc. https://www.arxiv.org/abs/2206.10069.
Chen, Le, Guannan Hu, Yaozhong Hu, and Jingyu Huang. 2017. "Space-Time Fractional Diffusions in Gaussian Noisy Environment." Stochastics 89 (1): 171--206. https://doi.org/10.1080/17442508.2016.1146282.
Chen, Le, Yaozhong Hu, and David Nualart. 2019. "Nonlinear Stochastic Time-Fractional Slow and Fast Diffusion Equations on $\mathbb{R}^d$." Stochastic Process. Appl. 129 (12): 5073--5112. https://doi.org/10.1016/j.spa.2019.01.003.

The original paper by Fox on this special function:

Fox, Charles. 1961. "The $G$ and $H$ Functions as Symmetrical Fourier Kernels." Trans. Amer. Math. Soc. 98: 395--429. https://doi.org/10.2307/1993339.

Bib entries can be found here. References are produced using the reference bank: SPDEs-Bib.

How to contribute

Contributions are what make the open-source community such an amazing place to learn, inspire, and create. Any contributions you make are greatly appreciated.

If you have a suggestion that would make this better, please fork the repo and create a pull request. You can also simply open an issue with the tag "enhancement". Don't forget to give the project a star! Thanks again!

Create your Feature Branch (git checkout -b feature/AmazingFeature)
Add more examples in the Examples folder, say new_example.wls
Test your example by running in the same folder ./Build.sh new_example.wls
If all looks good, go to the root folder, run ./Build.sh all
Check the results in the documentation folder for the final result ./documentation/FoxH-Parametration.pdf
Commit your Changes (git commit -m 'Add some AmazingFeature')
Push to the Branch (git push origin feature/AmazingFeature)
Open a Pull Request

Alternatively, see the issues section to report any bugs or file feature requests, or just send me an email (chenle02@gmail.com) for any other inquiries or further discussion.

Cite this work?

We hope that the resources have been supportive in your research endeavors. We are sincerely grateful for any form of acknowledgment you might extend. Should you wish to mention this work, a statement such as the one below could be included in your acknowledgments section or as a footnote:

  The author(s) would like to recognize the contribution of the GitHub
  repository chenle02/Fox-H_Symbolic_Tools curated by Le Chen, which has
  supported this research.

Or, if you prefer to directly cite this repository, please use the following BibTeX entry¹:

Le Chen and Guannan Hu (2023) “Some symbolic tools for the Fox $H$-function”. GitHub & Zenodo. doi: 10.5281/zenodo.10143785.

@misc{chen:23:some,
  author        = {Le Chen and Guannan Hu},
  title         = {Some symbolic tools for the Fox {$H$}-function},
  month         = {nov},
  year          = {2023},
  publisher     = {GitHub \& Zenodo},
  journal       = {GitHub repository},
  doi           = {10.5281/zenodo.10143785},
  url           = {https://doi.org/10.5281/zenodo.10143785}
}

Your support in recognizing the effort put into compiling and maintaining this repository is much appreciated.

LICENSE

MIT License

To properly show the entry, one may replace misc by book. ↩

chenle02/Fox-H_Symbolic_Tools