On the characteristic function of the asymmetric Student’s t -distribution and an integral involving the sine function

We obtain a new closed-form formula for the characteristic function of the asymmetric Student’s t -distribution. As part of our analysis, we derive a new closed-form formula for the integral ∫ 0 ∞ sin ( a x ) / ( b 2 + x 2 ) n d x , for a , b > 0 , n ∈ Z + , expressed in terms of the exponential integral function. As a consequence of our integral formula, we deduce a closed-form formula for the limit lim ν → n { I ν − 1 / 2 ( x ) − L 1 / 2 − ν ( x ) } / sin ( π ν ) , for n ∈ Z + , x > 0 .