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

TL;DR

This paper derives a new connection formula for the $q$-confluent hypergeometric function, unifying existing formulas and extending to classical Kummer functions via a limiting process.

Contribution

It presents a novel connection formula for the $q$-confluent hypergeometric function and links it to classical Kummer functions through a limit, enhancing understanding of these special functions.

Findings

Derived a connection formula for ${}_2 ext{ extphi}_1(a,b;0;q,x)$

Unified connection formulas for $q$-hypergeometric functions in matrix form

Connected $q$-hypergeometric functions to classical Kummer functions via $q o 1^{-}$ limit

Abstract

We show a connection formula for the $q$-confluent hypergeometric functions ${}_2\varphi_1(a,b;0;q,x)$. Combining our connection formula with Zhang's connection formula for ${}_2\varphi_0(a,b;-;q,x)$, 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.