Quasi-classical limit of Toda hierarchy and W-infinity symmetries

Previous results on quasi-classical limit of the KP hierarchy and its W-infinity symmetries are extended to the Toda hierarchy. The Planck constant ħ now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined. W _{1 + ∞} symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted to w _{1 + ∞} symmetries of the dispersionless hierarchy through their action on the tau function.