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78 Chapter 2. On a conjecture about addition modulo 2 k − 1We finally deduce different expressions for P d as a finite sum.Corollary 2.6.22. For d ≥ 1,P d = 13 2d−1 2 F 1 (1 − d, 1 − d; 1; 4) = 1 ∑d−1( ) 2 d − 13 2d−1 4 j ,j= 1 3 d 2 F 1 (1 − d, d; 1; −1/3) = 1 ∑d−1( )( )d − 1 + j d − 13 d 3 −j .d − 1 j2.6.2 The limit f d (1, ∞, . . . , ∞)In the previous subsection, we studied the behavior of P t,k = f d (β 1 , . . . , β d ) as all the β i ’s go toinfinity. We will now fix a subset of them to 1 and let the other ones go to infinity. As was thecase in the previous subsection, the expression of f d given in Proposition 2.5.1 shows that suchlimits are well defined.Recall the distribution probability for ɛ ′ i = γ′ i + β i − δ i ′ given by Proposition 2.4.6.Proposition 2.4.6. For e i ≥ 0,⎧⎪⎨P (ɛ ′ i = e i ) =⎪⎩j=0j=02 −βi if e i = 0,2 −β i3− 2 (2ei −ei ) if 0 < e i < β i ,2 β i −2 −β i32 −ei if β i ≤ e i .Therefore, if we set β i = 1 and let α i go to infinity for some i ∈ {1, . . . , d}, Proposition 2.4.10shows that ɛ ′ i has a similar behavior to the one of γ′ i and δ′ i : its law converges towards the law of ageometrically distributed variable with parameter 1/2. Then we have a probabilistic interpretationid−i{ }} { { }} {{ }} {d−i{ }} {for the limit lim βj→∞,j>i f d ( 1, . . . , 1, β i+1 , . . . , β d ) which we denote by f d ( 1, . . . , 1, ∞, . . . , ∞).As in the previous subsection, let G 1 , . . . , G d and H 1 , . . . , H d be 2d independent geometricallydistributed variables with parameter 1/2 and X k denote the random variable X k = ∑ kj=1 G j −∑ kj=1 H j. Then[ ∑f d (1, . . . , 1, ∞, . . . , ∞) = lim} {{ } } {{ }P γ ′ < ∑ ]δ ′β j→∞,j>ii d−id d[ ∑= lim P ɛ ′ + ∑ γ ′ < i + ∑ ]δ ′β j→∞,j>ii d−id−i⎡⎤d∑ ∑d−i= P ⎣ G j < i + H j⎦⎡j=1= P ⎣X d−i +d∑j=i+1j=1⎤G j < i⎦ .The first few values of such expressions, computed using explicit expressions for f d , are given inTable 2.4.Using the above probabilistic interpretation, it is possible to express f d (1, ∞, . . . , ∞) withf d (∞, . . . , ∞) = P 0 d = 1/2(1 − P d), and so to compute it with the short closed-form expressionsof the previous subsection.i