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2.6. Asymptotic behavior: β i → ∞ 67The difference in parenthesis is shown to be zero using Lemma 2.5.22, so that ∆ = 0.Lemma 2.5.22. For n ≥ k ≥ 0,n−k∑( ) [ ]k + l nB lk k + ll=0= k + 1n + 1[ ] n + 1k + 1Proof. Let us fix k ≥ 0. We first recall classical results about exponential generating functions.∑ z nB nn! = z1 − e −z ,n≥0∑[ n zn(− log(1 − z))k=k]n! k!n≥0We now form the exponential generating function of the coefficients of interest.(∑ ∑ n ( [ ] lB l−kk) ) n z nl n! = ∑ ∑( [ ] l n znB l−kk)l n! = ∑ ( ) l ∑[ ] n znB l−kkl n!n≥0 l=kl≥k n≥ll≥k n≥l= ∑ ( l (− log(1 − z)) ll−kk)Bl!l≥k(− log(1 − z))k ∑ (− log(1 − z)) l−k= B l−kk!(l − k)!whence the identity of the lemma.=(− log(1 − z))kk!l≥k∑l≥02.6 Asymptotic behavior: β i → ∞2.6.1 The limit f d (∞, . . . , ∞).B l(− log(1 − z)) ll!(− log(1 − z))k − log(1 − z)=k! 1 − e = k + 1 (− log(1 − z)) k+1log(1−z) z (k + 1)!= k + 1 ∑[ ] n znz k + 1 n! = ∑ [ ]k + 1 n + 1 zn,n + 1 k + 1 n!n≥0n≥0We denote the limit of f d when all the β i ’s go to infinity by f d (∞, . . . , ∞). The expression of f dgiven in Proposition 2.5.1 shows that this value is well defined and is nothing but the constantterm Pd 0 in that expression.In this subsection we give several expressions involving Gaussian hypergeometric series whichare defined as follows.Definition 2.6.1 (Gaussian hypergeometric series [1, Formula 15.1.1]). The Gaussian hypergeometricseries 2 F 1 (a, b; c; z) is2F 1 (a, b; c; z) =∞∑n=0(a) n (b) n(c) nz nn!,.