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We present a new methodology, suitable for implementation on computer, toperform the $\epsilon$-expansion of hypergeometric functions with linear$\epsilon$ dependent Pochhammer parameters in any number of variables. Ourapproach allows one to perform Taylor as well as Laurent series expansion ofmultivariable hypergeometric functions. Each of the coefficients of $\epsilon$in the series expansion is expressed as a linear combination of multivariablehypergeometric functions with the same domain of convergence as that of theoriginal hypergeometric function. We present illustrative examples ofhypergeometric functions in one, two and three variables which are typical ofFeynman integral calculus.

$\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus