Bhargava’s Exponential Functions and Bernoulli Numbers Associated to the Set of Prime Numbers

Bhargava associates to each infinite subset E of ℤ $$\mathbb Z$$ a sequence of positive integers called the factorials of E. Such a sequence has many properties of the classical factorials, and Bhargava asked if the corresponding generalizations of functions defined by means of factorials could have interesting properties. In this paper we consider the generalization expE(x) of the exponential function and the generalization BE,n of the Bernoulli numbers. We are particularly interested in the case where the subset E is the set ℙ $$\mathbb P$$ of prime numbers. We prove in particular that, for every rational r ≠ − 2 of the form ± 1 d $$\frac {\pm 1}{d}$$ or ± 2 d $$\frac {\pm 2}{d}$$ , exp ℙ ( r ) $$\exp _{\mathbb P}(r)$$ is irrational and that the Bernoulli polynomials without constant term B ℙ , n ( X ) − B ℙ , n ( 0 ) $$B_{\mathbb P,n}(X)-B_{\mathbb P,n}(0)$$ are integer polynomials. By the way, we answer a question from Mingarelli by showing that the sequence of factorials of the set ℙ ∪ 2 ℙ $$\mathbb P\cup 2\mathbb P$$ has infinitely many pairs of equal consecutive terms.