(Abriged) We present the results of millimeter observations and a suitable chemical and radiative transfer model of the AB Aur (HD 31293) circumstellar disk and surrounding envelope. The integral molecular content of this system is studied by observing CO, C$^{18}$O, CS, HCO$^+$, DCO$^+$, H$_2$CO, HCN, HNC, and SiO rotational lines with the IRAM 30-m antenna, while the disk is mapped in the HCO$^+$(1-0) transition with the Plateau de Bure interferometer. Using a flared disk model with a vertical temperature gradient and an isothermal spherical envelope model with a shadowed midplane and two unshielded cones together with a gas-grain chemical network, time-dependent abundances of observationally important molecules are calculated. Then a 2D non-LTE line radiative transfer code is applied to compute excitation temperatures of several rotational transitions of HCO$^+$, CO, C$^{18}$O, and CS molecules. We synthesize the HCO$^+$(1-0) interferometric map along with single-dish CO(2-1), C$^{18}$O(2-1), HCO$^+$(1-0), HCO$^+$(3-2), CS(2-1), and CS(5-4) spectra and compared them with the observations. Our disk model successfully reproduces observed interferometric HCO$^+$(1-0) data, thereby constraining the following disk properties: (1) the inclination angle $iota=17^{+6}_{-3}degr$, (2) the position angle $phi=80pm30degr$, (3) the size $R_mathrm{out}=400pm200$ AU, (4) the mass $M_mathrm{disk}=1.3cdot10^{-2} M_{sun}$ (with a factor of $sim7$ uncertainty), and (5) that the disk is in Keplerian rotation. Furthermore, indirect evidence for a local inhomogeneity of the envelope at $ga600$ AU is found...
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