# Using the computation rules for the formal mkmk Borel

From Proposition 5 and the conditions in the statement above, we get that PF-573228 the coefficients Un(m,?)Un(m,?) of Uˆ(T,m,?) are well defined, belong to E(β,μ)E(β,μ) for all ?∈D(0,?0)? 0 ?∈D(0,?0)? 0 , all n≥1n≥1 and satisfy the following recursion relationequation(73)(n+1)Un+1(m,?)=?−1Q(im)∑n1+n2=n,n1≥1,n2≥11(2π)1/2∫−∞+∞Q1(i(m−m1))Un1(m−m1,?)Q2(im1)Un2(m1,?)dm1+∑l=1DRl(im)Q(im)(?Δl−dl+δl−1Πj=0δl−1(n+δl−dl−j))Un+δl−dl(m,?)+?−1Q(im)∑n1+n2=n,n1≥1,n2≥11(2π)1/2∫−∞+∞C0,n1(m−m1,?)R0(im1)Un2(m1,?)dm1+?−1(2π)1/2Q(im)∫−∞+∞C0,0(m−m1,?)R0(im1)Un(m1,?)dm1+?−1Q(im)Fn(m,?) for all n≥max1≤l≤D?dln≥max1≤l≤D?dl.  □
We make the additional assumption Fungi there exists an unbounded sectorSQ,RD= z∈C/ z ≥rQ,RD, arg(z)−dQ,RD ≤ηQ,RD with direction dQ,RD∈RdQ,RD∈R, aperture ηQ,RD>0ηQ,RD>0 for some radius rQ,RD>0rQ,RD>0 such thatequation(81)Q(im)RD(im)∈SQ,RD for all m∈Rm∈R. We factorize the polynomial Pm(τ)=Q(im)k−RD(im)kδDτ(δD−1)kPm(τ)=Q(im)k−RD(im)kδDτ(δD−1)k in the formequation(82)Pm(τ)=−RD(im)kδDΠl=0(δD−1)k−1(τ−ql(m)) whereequation(83)ql(m)=( Q(im) RD(im) kδD−1)1(δD−1)kexp?(−1(arg(Q(im)RD(im)kδD−1)1(δD−1)k+2πl(δD−1)k)) for all 0≤l≤(δD−1)k−10≤l≤(δD−1)k−1, all m∈Rm∈R.