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Because of its large dissociation energy and the relatively high abundance of its constituent atoms, carbon monoxide in its several isotopic forms is observed in a variety of astrophysical sources. Since many of these sources are quite hot (T $>$\ 7000 K), the molecular emissions can be observed from highly excited vibrational levels that may or may not be in LTE. Improvements in the electric dipole moment function (EDMF) by Chackerian, et al. [Can. J. Phys., 62, 1579 (1984)] and recent advances in calculating wave functions by Goorvitch and Galant [J. Quant. Spectrosc. Rad. Transf., 47, 391 (1992)] make it worthwhile to recalculate the rovibrational transition moments for all the transitions with $v \leq$\ 20, $J \leq$\ 150 and $\Delta v$\ = +1, +2, and +3 for the $^{12}$C$^{16}$O and $^{13}$C$^{16}$O $X\ ^{1} \Sigma^{+}$\ states. These calculated transition moments combined with highly accurate term values and transition frequencies reported by Farrenq, et al. [ J. Molec. Spectrosc., 149, 1579 (1991)] allow the calculation of the Einstein A values, $gf$\ values, and line strengths. We observe that there is intensity enhancement for the P transitions at the expense of the R transitions for $\Delta v$\ = +1. This enhancement can be as large as 40\% at the highest $J$. For the $\Delta v$ = +2 and +3 transitions, the R transitions are enhanced by as much as a factor of 2.75 and 10 at the highest J, respectively. The P transitions exhibit minor decreases. The expectation value of the transition moments are given as a polynomial expansion in terms of the parameter $m$\ which is defined in terms of the lower state angular momentum quantum number $J$. These results are derived using orthogonal polynomials which are tested for being statistically significant.
