When we are talking about expansion the one of the question is in terms of what function we want to get result. There are numerical approach:

Z.W. Huang, J. Liu, Comput. Phys. Commun.184(2013) 1973 ;

D. Greynat, J. Sesma, Comput. Phys. Commun.185(2014) 472 ;

Article Epsilon expansion of Appell and Kamp\'e de F\'eriet functions

There is universal for values of parameters approach:

Ancarani and G. Gasaneo:

. U. Ancarani and G. Gasaneo,Derivatives of any order of the Gaussian hypergeometricfunction2F1(a,b,c;z)with respect to the parametersa,bandc, Journal of Physics A:Mathematical and Theoretical42, 395208 (2009).

L. U. Ancarani and G. Gasaneo,Derivatives of any order of the hypergeometric functionpFq(a1,...,ap;b1,...,bq;z)with respect to the parametersaiandbi, Journal of Physics A:Mathematical and Theoretical43, 085210 (2010).

L. U. Ancarani, J. A. D. Punta, and G. Gasaneo,

Derivatives of Horn hypergeomet-ric functions with respect to their parameters, J. Math. Phys.58, 073504 (2017)

V. Sahai and A. Verma,Derivatives of Appell functions with respect to respect to parameters,Journal of Inequalities and Special Functions6, 1 (2015)

It is unversal, in application to expansion around any small parameter, but produces new extented set of Horn-type hypergeometric functions. Another limitation is that this method is not applicable to some multiple hypergeometric funcions (when one of summation include negative values).

There is my approach, based on the iterative solution of Pfaff system of differential equations:

Article When epsilon-expansion of hypergeometric functions is expres...

This technique is applicable to the expansion of hypergeometric functions of a few variable in terms of multiple polylogarithms (relevant for physik), but it is not algoritmically closed since at the some moment problem of expansion is reduce to problem of finding transformation which convert the given set of algebraic functions to set of rational functions.

Nevertheless, I have successfully applied this technique to a few function where expansion was not able to construct using another technique.

Unfortunately, the further extension and application of this project has not get financial support from external sources

and at the present moment it is frozen.

I do not know a general method.

When we are talking about expansion the one of the question is in terms of what function we want to get result. There are numerical approach:

Z.W. Huang, J. Liu, Comput. Phys. Commun.184(2013) 1973 ;

D. Greynat, J. Sesma, Comput. Phys. Commun.185(2014) 472 ;

Article Epsilon expansion of Appell and Kamp\'e de F\'eriet functions

There is universal for values of parameters approach:

Ancarani and G. Gasaneo:

. U. Ancarani and G. Gasaneo,Derivatives of any order of the Gaussian hypergeometricfunction2F1(a,b,c;z)with respect to the parametersa,bandc, Journal of Physics A:Mathematical and Theoretical42, 395208 (2009).

L. U. Ancarani and G. Gasaneo,Derivatives of any order of the hypergeometric functionpFq(a1,...,ap;b1,...,bq;z)with respect to the parametersaiandbi, Journal of Physics A:Mathematical and Theoretical43, 085210 (2010).

L. U. Ancarani, J. A. D. Punta, and G. Gasaneo,

Derivatives of Horn hypergeomet-ric functions with respect to their parameters, J. Math. Phys.58, 073504 (2017)

V. Sahai and A. Verma,Derivatives of Appell functions with respect to respect to parameters,Journal of Inequalities and Special Functions6, 1 (2015)

It is unversal, in application to expansion around any small parameter, but produces new extented set of Horn-type hypergeometric functions. Another limitation is that this method is not applicable to some multiple hypergeometric funcions (when one of summation include negative values).

There is my approach, based on the iterative solution of Pfaff system of differential equations:

Article When epsilon-expansion of hypergeometric functions is expres...

This technique is applicable to the expansion of hypergeometric functions of a few variable in terms of multiple polylogarithms (relevant for physik), but it is not algoritmically closed since at the some moment problem of expansion is reduce to problem of finding transformation which convert the given set of algebraic functions to set of rational functions.

Nevertheless, I have successfully applied this technique to a few function where expansion was not able to construct using another technique.

Unfortunately, the further extension and application of this project has not get financial support from external sources

and at the present moment it is frozen.

spline interpolation and quasi-interpolation