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A Connection Formula for the $q$-Confluent Hypergeometric Function
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: $q$-Borel–Laplace transformation; $q$-difference equation; connection problem; $q$-confluent hypergeometric function. 
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     author = {Takeshi Morita},
     title = {A {Connection} {Formula} for the $q${-Confluent} {Hypergeometric} {Function}},
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}
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Takeshi Morita. A Connection Formula for the $q$-Confluent Hypergeometric Function. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a49/

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