doi:10.1007/s00009-022-02155-7engMahillo, AlejandroMiana, Pedro J.Catalan Generating Functions for Generators of Uni-parametric Families of OperatorsART-2022-130677In this paper we study solutions of the quadratic equation AY2−Y+I=0 where A is the generator of a one parameter family of operator (C0-semigroup or cosine functions) on a Banach space X with growth bound w0≤14. In the case of C0-semigroups, we show that a solution, which we call Catalan generating function of A, C(A), is given by the following Bochner integral, C(A)x:=∫∞0c(t)T(t)xdt,x∈X, where c is the Catalan kernel, c(t):=12π∫∞14e−λt4λ−1−−−−−√λdλ,t>0. Similar (and more complicated) results hold for cosine functions. We study algebraic properties of the Catalan kernel c as an element in Banach algebras L1ω(R+), endowed with the usual convolution product, ∗ and with the cosine convolution product, ∗c. The Hille–Phillips functional calculus allows to transfer these properties to C0-semigroups and cosine functions. In particular, we obtain a spectral mapping theorem for C(A). Finally, we present some examples, applications and conjectures to illustrate our results.2022http://zaguan.unizar.es/record/11991610.1007/s00009-022-02155-7http://zaguan.unizar.es/record/119916oai:zaguan.unizar.es:119916info:eu-repo/grantAgreement/ES/DGA/E48-20Rinfo:eu-repo/grantAgreement/ES/MCYTS-DGI-FEDER/PID2019-105979GB-I00Mediterranean Journal of Mathematics 19, 5 (2022), 238 [27 pp.]byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess