"I have tried some of the posted in the file exchange, but they all tend to give different results for large arguments compared to hypergeom."
The functions posted in FEX are numerical where as the hypergeom function is symbolical. There will be a difference in the precision of the values obtained.
"is there any way to use the hypergeom function without having the symbolic math toolbox?"
No, atleast not symbolically.
Why do you not want to use the symbolic math toolbox? Or do you not have access to it?
