What is "extended polynomials"
It turns out the definition is straightforward as seen in [1]. And as an example, exponentiation is not definable in simply-type $\lambda$-calculus.


  The class of extended polynomials is the smallest class of func- tions
  over N which contains:
  
  
  the constant functions: 0 and 1, 
  projections,
  addition,
  multiplication,
  the function ifzero(n, m, p) = if n = 0 then m else p
  





Definable functions in the simply typed lambda-calculus by M. Zakrzewski (2007)