Stoichiometric ratio (SR) and gas-phase stoichiometric ratio (SRgas) are fundamental burning conditions needed to consider combustion phenomena. For oxy-fuel combustion, large quantities of CO2 and H2O are included in the feed gas. CO2 and H2O are not combustible species, but C and H are combustible elements. Two types of stoichiometric ratios can be defined, depending on whether or not C, H, and O from CO2 and H2O in the feed gas are included in evaluating the stoichiometric ratio. We examined which definition was easier to use when considering the reaction mechanism. We measured NOx and the intermediate species, hydrocarbons, NH3, HCN, and H2S, under the fuel-rich conditions and using a staged high-temperature drop-tube furnace. Burning temperatures were 1573-1873 K. We also measured NOx emission and unburned carbon in ash at the furnace exit of staged combustion. SR-all (or SRgas-all) was obtained by an evaluation including C, H, and O from CO2 and H2O in the feed gas. SRgas-all was easy to obtain by considering the reaction mechanisms for intermediate species, such as hydrocarbons, NH3, HCN, and H2S, in the fuel-rich zone. When the SRgas-all values were the same, the differences in concentrations of these intermediate species between air and oxy-fuel combustions were small. SR (or SRgas) was obtained by an evaluation excluding C, H, and O from CO2 and H2O in the feed gas. The association with SR (or SRgas) was stronger for NOx concentration or conversion. NOx conversion ratio at the furnace exit for oxy-fuel combustion was almost the same as that for air combustion at the furnace exit, when the SR value in the burner combustion zone was the same.