Prove the following power series representation for the tail probabilities for Student’s t distribution:                      Q(t | k) = ((k + 1)/2) (k/2) √π u−k/2 ∞ j=0 cj (k + 2j )uj , where...

Prove the following power series representation for the tail probabilities for Student’s t distribution:

                     Q(t | k) = ((k + 1)/2) (k/2) √π u−k/2 ∞ j=0 cj (k + 2j )uj ,

where

                      c0 = 1, cj = 2j − 1 2j cj−1, and u = 1 + t 2 k .

(Hint: Change variables y = 1/ 1 + t 2/k and then expand 1/ √1 − v in a power series, as in (7.6.1).)