A Connection Formula for the q-Confluent Hypergeometric Function

Abstract

We show a connection formula for the $q$-confluent hypergeometric functions ${}_2varphi_1(a,b;0;q,x)$. Combining our connection formula with Zhang's connection formula for ${}_2varphi_0(a,b;-;q,x)$, we obtain the connection formula for the $q$-confluent hypergeometric equation in the matrix form. Also we obtain the connection formula of Kummer's confluent hypergeometric functions by taking the limit $qo 1^{-}$ of our connection formula.

Keywords

• q-Borel-Laplace transformation
• q-difference equation
• connection problem
• q-confluent hypergeometric function