A Connection Formula for the $q$-Confluent Hypergeometric Function

Abstract

We show a connection formula for the $q$-confluent hypergeometric functions ${}_2{\phi }_1(a,b;0;q,x)$. Combining our connection formula with Zhang's connection formula for ${}_2{\phi }_0(a,b;-;q,x)$, 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 $q\to 1^{-}$ of our connection formula.

References in corpus (2)

• A Connection Formula of the Hahn-Exton $q$-Bessel Function
• A unified approach to $q$-special functions of the Laplace type

Cited by in corpus (5)

• Asymptotic formulae of two divergent bilateral basic hypergeometric series
• The Stokes phenomenon for the $q$-difference equation satisfied by the basic hypergeometric series ${}_3{\phi }_1(a_1,a_2,a_3;b_1;q,x)$
• On local solutions of the Ramanujan equation and their connection formulae
• The Stokes phenomenon for the Ramanujan's $q$-difference equation and its higher order extension
• A connection formula of a divergent bilateral basic hypergeometric function