A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that the deformation parameter q2=e-iℏ. The first variable r runs over non-zero real numbers; the range of the second one depends on the sign of r: ϱ=0 for r>0 and ϱ=±1 for r<0. After the holomorphic continuation the function satisfies the functional equation [Formula: see text] The name "exponential function" is justified by the formula: [Formula: see text] where R, S are selfadjoint operators satisfying certain commutation relations and [R+S] is a selfadjoint extension of the sum R+S determined by operators ρ and σ appearing in the formula. This formula will be used in a forthcoming paper to construct a unitary operator W satisfying the pentagonal equation of Baaj and Skandalis.