Fractional powers of the generating function for the partition function

Congruences in fractional partition functions

“…In addition to the aforementioned observation, Chan and Wang (Theorem 1.2 of [4]) display an infinite family of congruences for fractional partition functions. To arrive at their results, Chan and Wang took advantage of the well-known formulas that produce the coefficients of (q; q) d ∞ , rooted in Euler's pentagonal number theorem and Jacobi's identity.…”

“…From this theorem, we conclude that for a given rational number α, whenever gcd(L, b) = 1, congruences modulo L are meaningful to study. The second result that we state is a technical lemma (Lemma 2.1 of [4]) which results from Frobenius endomorphism. This lemma allows us to move exponents through q-Pochhammer symbols, a crucial step in the proof of our main results.…”

“…Therefore, it is imperative to keep in mind that not all modulo are worth exploration. Chan and Wang address this issue in a recent paper (Theorem 1.1 of [4]) through proving that p α (n) are ℓ − integral for ℓ prime and ℓ ∤ b. In the rest of this paper, we let ℓ always refer to a prime.…”

Ramanujan Congruences for Fractional Partition Functions

“…As an extension of the theory of congruences for the partition function, Chan and Wang [4] recently explored congruences for the fractional partition functions. For rational α, which we write as α = a/b for coprime integers a, b with b ≥ 1 throughout this paper, the fractional partition function p α : Z → Q is defined for n ≥ 0 by the power series (1.1) P α (q) := (q; q) α ∞ =:…”

“…We set p α (n) := 0 for n < 0. Chan and Wang [4] showed (see Section 2.1) that the numbers p α (n) are ℓ-integral for any prime ℓ ∤ b. For this reason, it is fruitful to study congruences for p α modulo ℓ r for any prime ℓ ∤ b.…”