I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fifth presentation, about "The Appell doubly hypergeometric functions", describes a very recent project I've been working at Maple, i.e. the very first complete computational implementation of the Appell doubly hypergeometric functions. This work appeared in Maple 2017. These functions have a tremendous potential in that, at the same time, they have a myriad of properties, and include as particular cases most of the existing mathematical language, and so they have obvious applications in integration, differential equations, and applied mathematics all around. I think these will be the functions of this XXI century, analogously to what happened with hypergeometric functions in the previous century.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

The four double-hypergeometric Appell functions,

a complete implementation in a computer algebra system


Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft


The four multi-parameter Appell functions, AppellF1 , AppellF2 , AppellF3  and AppellF4  are doubly hypergeometric functions that include as particular cases the 2F1 hypergeometric  and some cases of the MeijerG  function, and with them most of the known functions of mathematical physics. Appell functions have been popping up with increasing frequency in applications in quantum mechanics, molecular physics, and general relativity. In this talk, a full implementation of these functions in the Maple computer algebra system, including, for the first time, their numerical evaluation over the whole complex plane, is presented, with details about the symbolic and numerical strategies used.

Appell Functions (symbolic)



The main references:


P. Appel, J.Kamke de Feriet, "Fonctions hypergeometriques et Hyperspheriques", 1926


H. Srivastava, P.W. Karlsson, "Multiple Gaussian Hypergeometric Series", 1985


24 papers in the literature, ranging from 1882 to 2015


Definition and Symmetries


Polynomial and Singular Cases


Single Power Series with Hypergeometric Coefficients


Analytic Extension from the Appell Series to the Appell Functions


Euler-Type and Contiguity Identities


Appell Differential Equations


Putting all together


Problem: some formulas in the literature are wrong or miss the conditions indicating when are they valid (exchange with the Mathematics director of the DLMF - NIST)


Appell Functions (numeric)






Compute these Appell functions over the whole complex plane


Considering that this is a research problem, implement different methods and flexible optional arguments to allow for:

a) comparison between methods (both performance and correctness),

b) investigation of a single method in different circumstances.


Develop a computational structure that can be reused with other special functions (abstract code and provide the main options), and that could also be translated to C (so: only one numerical implementation, not 100 special function numerical implementations)

Limitation: the Maple original evalf command does not accept optional arguments


The cost of numerically evaluating an Appell function



If it is a special hypergeometric case, then between 1 to 2 hypergeometric functions


Next simplest case (series/recurrence below) 3 to 4 hypergeometric functions plus adding somewhat large formulas that involve only arithmetic operations up to 20,000 times (frequently less than 100 times)


Next simplest case: the formulas themselves are power series with hypergeometric function coefficients; these cases frequently converge rapidly but may involve the numerical evaluation of up to hundreds of hypergeometric functions to get the value of a single Appell function.


Strategy for the numerical evaluation of Appell functions (or other functions ...)



The numerical evaluation flows orderly according to:

1) check whether it is a singular case

2) check whether it is a special value

3) compute the value using a series derived from a recurrence related to the underlying ODE

4) perform an sum using an infinite sum formula, checking for convergence

5) perform the numerical integration of the ODE underlying the given Appell function

6) perform a sequence of concatenated Taylor series expansions





Numerical integration of an underlying differential equation (ODEs and dsolve/numeric)


Concatenated Taylor series expansions covering the whole complex plane




Improvements in the numerical evaluation of hypergeometric functions


Evalf: an organized structure to implement the numerical evaluation of special functions in general


To be done



Download Appell_Functions.mw   
Download Appell_Functions.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

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