  The single-particle inclusive fragmentation function and the particle
multiplicity are observables of fundamental importance in studying properties
of quantum chromodynamics at colliders. It is well-known that at high energies,
the multiplicity distribution satisfies KNO scaling in which all moments are
proportional to powers of the mean multiplicity. We prove that, under weak
assumptions, the leading dependence of the fragmentation function on
multiplicity is itself a kind of KNO scaling in which all moments are inversely
proportional to powers of the mean multiplicity. This scaling with multiplicity
additionally accounts for the dominant dependence on collision energy in the
fragmentation function. The proof relies crucially on properties of the
fragmentation function conditioned on the total multiplicity and application of
the Stieltjes moment problem. In the process, we construct a novel basis of the
fragmentation function expressed as an overall exponential suppression times a
series of Laguerre polynomials. We study this scaling of the fragmentation
function in experimental electron-position collision data and observe that
residual scale violations are significantly reduced.
