Towards all-order Laurent expansion of generalized hypergeometric   functions around rational values of parameters
  We prove the following theorems:   1) The Laurent expansions in epsilon of the Gauss hypergeometric functions   2F1(I_1+a*epsilon, I_2+b*epsilon; I_3+p/q + c epsilon; z),   2F1(I_1+p/q+a*epsilon, I_2+p/q+b*epsilon; I_3+ p/q+c*epsilon;z),   2F1(I_1+p/q+a*epsilon, I_2+b*epsilon; I_3+p/q+c*epsilon;z), where I_1,I_2,I_3,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(I_1+p/q+a*epsilon, I_2+b*epsilon; I_3+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. 
Mikhail Yu. Kalmykov (Hamburg U., Inst. Theor. Phys. II & Dubna,   JINR), Bernd A. Kniehl (Hamburg U., Inst. Theor. Phys. II)