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2.6. Asymptotic behavior: β i → ∞ 77so thatP [X d = k] = 1 4 d 12 k 1(d − 1)!⎛k−d⎝h (d−1)d,k+1−d (1/4) − ∑j=0(= 1 ∑ d−11 1 (d − 1)!j=0 (−1)j( d−1+kk+j4 d 2 k (d − 1)!= 2kd−13 2d−1∑( d − 1 + j(−1) d−1 d − 1k−d−j=0∑( )( d − 1 + k k − d + j(−1) j k + j jj=0+ (−1) d 2kk−d∑4 dj=0( d − 1 + jd − 1⎞( ) d − 1 + j(z d−1−k+j ) (d−1) (1/4) ⎠d − 1)( k−d+j)(1/4)d−1−k−j(3/4) 2d−1j⎞) (k − 1 − j)!(k − d − j)! (1/4)−k+j ⎠)4 j)( ) k − 1 − j4 −j .d − 1To conclude this subsection, let us mention that, using the expression of P [X d = k] as aGaussian hypergeometric series, most of the above expressions can be directly deduced from theexpression of P [X d = k] using linear transformations.Proposition 2.6.21. For d ≥ 1 and 0 ≤ k,( )P [X d = k] =4d−1 d − 1 + k2 k 3 2d−1 2F 1 (k + 1 − d, 1 − d; k + 1; 1/4)d − 1{2k ∑ d−1−k( d−1−k)( d−1+k)3=2d−1 j=0 j j+k 4jif 0 ≤ k ≤ d − 1∑4 d−1 d−12 k 3 2d−1 j=0 (−1)j( )(d−1+k k−d+j)k+j k−d 4−jif d − 1 < k( )P [X d = k] =2k d − 1 + k3 d+k 2F 1 (k + 1 − d, k + d; k + 1; −1/3)d − 1{2k ∑ d−1−k( d−1+k+j)( d−1−k)3=d+k j=0 d−1 j 3−jif 0 ≤ k ≤ d − 1∑2 k ∞3 d+k j=0 (−1)j( )(d−1+k+j k−d+j)d−1 k−d 3−jif d − 1 < kP [X d = k] = 1 ( ) d − 1 + k2 k 3 d 2F 1 (d, 1 − d; k + 1; −1/3)d − 1= 1 ∑d−1( )( )d − 1 + j d − 1 + k2 k 3 d 3 −j .d − 1 k + jj=0;;Proof. The first expression comes from Euler’s transformation [1, Formula 15.3.3]:2F 1 (a, b; c; z) = (1 − z) c−a−b 2F 1 (c − a, c − b; c; z) .The second one from Pfaff’s transformation [1, Formula 15.3.5]:2F 1 (a, b; c; z) = (1 − z) −b 2F 1 (c − a, b; c; z/(z − 1)) .The third one from the other Pfaff’s transformation [1, Formula 15.3.4]:2F 1 (a, b; c; z) = (1 − z) −a 2F 1 (a, c − b; c; z/(z − 1)) .