Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters

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We prove the following theorems: 1) The Laurent expansions in epsilon of the Gauss hypergeometric functions 2F1(I₁+a*epsilon, I₂+b*epsilon; I₃+p/q + c epsilon; z), 2F1(I₁+p/q+a*epsilon, I₂+p/q+b*epsilon; I₃+ p/q+c*epsilon;z), 2F1(I₁+p/q+a*epsilon, I₂+b*epsilon; I₃+p/q+c*epsilon;z), where I₁,I₂,I₃,p,q are arbitrary integers, a,b,c are arbitrary numbers and epsilon is an infinitesimal parameter, are expressible in terms of multiple polylogarithms of q-roots of unity with coefficients that are ratios of polynomials; 2) The Laurent expansion of the Gauss hypergeometric function 2F1(I₁+p/q+a*epsilon, I₂+b*epsilon; I₃+c*epsilon;z) is expressible in terms of multiple polylogarithms of q-roots of unity times powers of logarithm with coefficients that are ratios of polynomials; 3) The multiple inverse rational sums (see Eq. (2)) and the multiple rational sums (see Eq. (3)) are expressible in terms of multiple polylogarithms; 4) The generalized hypergeometric functions (see Eq. (4)) are expressible in terms of multiple polylogarithms with coefficients that are ratios of polynomials.