Laurent expansion about rational values of parameters
Multiloop calculations
Two-loop sunset
Gauss hypergeometric functions
Multiple polylogarithms
Loops
Integrals
Master integrals
Decay
Quarks
Neutrinos
Beschreibung:
We prove the following theorems: (1) the Laurent expansions in a of the Gauss hypergeometric functions F-2(1) (l(1) + a epsilon. l(2) + b epsilon; l(3) + p/q + c epsilon: z), F-2(1) (l(1) + p/q + a epsilon. l(2) + p/q +b epsilon; l(3) + p/q + c epsilon: z) and F-2(1) (l(1) + p/q + a epsilon. l(2) + p/q +b epsilon; l(3) + p/q + c epsilon: z), where l(1), l(2), l(3), p, q are arbitrary integers, a, b, c are arbitrary numbers and epsilon is an infnitesimal 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 hypergcometric function F-2(1) (l(1) + p/q + a epsilon. l(2) + p/q +b epsilon; l(3) + p/q + 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 Sigma(infinity)(j=1)Gamma(j)/Gamma(l + j - p/q) z(j)/j(c)S(a1) (j -1) x ... x S-ap ( j - 1) and the multiple rational sums Sigma(infinity)(j=1)Gamma(j + p/q)/Gamma(l + j - p/q) z(j)/j(c)S(a1) (j -1) x ... x S-ap ( j - 1), where Sa(j) = Sigma(j)(k=1)1/k(a) is a harmonic series and c is an arbitrary integer, are expressible in terms of multiple polylogarithms; (4) the generalised hypergeometric functions pFp-1((A) over bar + (a) over bar epsilon: (B) over bar + (b) over bar epsilon. p/q + Bp-1: z) and pFp-1((A) over bar + (a) over bar epsilon. p/q + A(p): (B) over bar + (b) over bar epsilon: z) are expressible in terms of multiple polylogarithms with coefficients that are ratios of polynomials with complex coefficients. (c) 2008 Elsevier B.V. All rights reserved.