A stroll along the gamma

We provide the first in-depth study of the Laguerre interpolation scheme between an arbitrary probability measure and the gamma distribution. We propose new explicit representations for the Laguerre semigroup as well as a new intertwining relation. We use these results to prove a local De Bruijn identity which hold under minimal conditions. We obtain a new proof of the logarithmic Sobolev inequality for the gamma law with α≥1/2 as well as a new type of HSI inequality linking relative entropy, Stein discrepancy and standardized Fisher information for the gamma law with α≥1/2.