Towards all-order Laurent expansion of generalised hypergeometric functions about rational values of parameters

We prove the following theorems: (1) the Laurent expansions in ε   of the Gauss hypergeometric functions F12(I1+aε,I2+bε;I3+pq+cε;z), F12(I1+pq+aε,I2+pq+bε;I3+pq+cε;z) and F12(I1+pq+aε,I2+bε;I3+pq+cε;z), where I1,I2,I3,p,qI1,I2,I3,p,q are arbitrary integers, a,b,ca,b,c are arbitrary numbers and ε 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 F12(I1+pq+aε,I2+bε;I3+cε;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 ∑j=1∞Γ(j)Γ(1+j−pq)zjjcSa1(j−1)×⋯×Sap(j−1) and the multiple rational sums ∑j=1∞Γ(j+pq)Γ(1+j)zjjcSa1(j−1)×⋯×Sap(j−1), where Sa(j)=∑k=1j1ka is a harmonic series and c   is an arbitrary integer, are expressible in terms of multiple polylogarithms; (4) the generalised hypergeometric functions Fp−1p(A→+a→ε;B→+b→ε,pq+Bp−1;z) and Fp−1p(A→+a→ε,pq+Ap;B→+b→ε;z) are expressible in terms of multiple polylogarithms with coefficients that are ratios of polynomials with complex coefficients.