This historical question recalls Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation Chebyshev used the factorial ratio sequence $$ ...
In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
EDIT: Context for this investigation can be found in one of my other MO posts, "Pythagorean Theorem for Right-Corner Hyperbolic Simplices?" -- I'm investigating a function that has led me to this ...