Binomial coefficients involving infinite powers of primes

TL;DR

This paper investigates the p-adic limits of binomial coefficients involving infinite powers of primes, establishing a formula for their behavior as exponents tend to infinity, and introduces the concept of p-adic limits of factorial-related sequences.

Contribution

It introduces the p-adic limit of factorial sequences and derives a formula for the limits of binomial coefficients involving infinite prime powers.

Findings

Defined the p-adic limit z_k of (-1)^{pke} u((kp^e)!) as e approaches infinity.

Derived a formula for the p-adic limit of binomial coefficients of the form binom(a p^e + c, b p^e + d).

Established the well-defined nature of these p-adic limits in terms of the units u(n).

Abstract

If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to infinity is a well-defined p-adic integer, which we call z_k. In terms of these, we give a formula for the p-adic limit of binom{a p^e +c, b p^e +d) as e goes to infinity, which we call binom(a p^\infty +c, b p^\infty +d). Here a \ge b are positive integers, and c and d are integers.