Context : The growing body of spectral observations of the extremely metal-poor ( EMP ) stars in the Galactic Halo provides constraints on theoretical studies of the chemical and stellar evolution of the early Universe .

Aims : To calculate yields for EMP stars for use in chemical evolution calculations and to test whether such models can account for some of the recent abundance observations of EMP stars , in particular the highly C-rich EMP ( CEMP ) halo stars .

Methods : We modify an existing 1D stellar structure code to include time-dependent mixing in a diffusion approximation .
Using this code and a post-processing nucleosynthesis code we calculate the structural evolution and nucleosynthesis of a grid of models covering the metallicity range : -6.5 \leq [ Fe/H ] \leq - 3.0 ( plus Z=0 ) , and mass range : 0.85 \leq M \leq 3.0 M _ { \odot } , amounting to 20 stars in total .

Results : Many of the models experience violent nuclear burning episodes not seen at higher metallicities .
We refer to these events as ‘ Dual Flashes ’ since they are characterised by nearly simultaneous peaks in both hydrogen and helium burning .
These events have been reported by previous studies .
Some of the material processed by the Dual Flashes is dredged up causing significant surface pollution with a distinct chemical composition .
We have calculated the entire evolution of the Z=0 and EMP models , from the ZAMS to the end of the TPAGB , including extensive nucleosynthesis .
In this paper , the first of a series describing and analysing this large data set , we present the resulting stellar yields .
Although subject to many uncertainties these are , as far as we are aware , the only yields currently available in this mass and metallicity range .
We also analyse the yields in terms of C and N , comparing them to the observed CEMP abundances .
At the lowest metallicities ( [ Fe/H ] \lesssim - 4.0 ) we find the yields to contain \sim 1 to 2 dex too much carbon , in agreement with all previous studies .
At higher metallicities ( [ Fe/H ] \sim - 3.0 ) , where the observed data set is much larger , all our models produce yields with [ C/Fe ] values consistent with those observed in the most C-rich CEMPs .
However it is only the low-mass models that undergo the Dual Shell Flash ( which occurs at the start of the TPAGB ) that can best reproduce the C and N observations .
Normal Third Dredge-Up can not reproduce the observations because at these metallicities intermediate mass models ( M \gtrsim 2 M _ { \odot } ) suffer HBB which converts the C to N thus lowering [ C/N ] well below the observations , whilst if TDU were to occur in the low-mass ( M \leq 1 M _ { \odot } ) models ( we do not find it to occur in our models ) , the yields would be expected to be C-rich only , which is at odds with the ‘ dual pollution ’ of C and N generally observed in the CEMPs .
Interestingly events similar to the EMP Dual Flashes have been proposed to explain objects similarly containing a dual pollution of C and N – the ‘ Blue Hook ’ stars and the ‘ Born Again AGB ’ stars .
We also find that the proportion of CEMP stars should continue to increase at lower metallicities , based on the results that some of the low mass EMP models already have polluted surfaces by the HB phase , and that there are more C-producing evolutionary episodes at these metallicities .
Finally we note that there is a need for multidimensional fluid dynamics calculations of the Dual Flash events , to ascertain whether the overproduction of C and N at ultra-low metallicities found by all studies is an artifact of the 1D treatment .

Conclusions :