Summary: We define a function in terms of quotients of the \(p\)-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the \(p\)-adic setting. We prove, for primes \(p > 3\), that the trace of Frobenius of any elliptic curve over \(\mathbb F_p\), whose \(j\)-invariant does not equal 0 or \(1728\), is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over \(\mathbb F_p\) when \(p \equiv 1\pmod{12}\).