Unimodal sequences and quantum and mock modular forms

Abstract

We show that the rank generating function U(t; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U(-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F(q). As a result, we obtain a new representation for a certain generating function for L-values. The series U(i; q) = U(-i; q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions.

1. Introduction and Statement of Results

A sequence of integers is a strongly unimodal sequence of size n if it satisfies

for some k and a₁ + …+a_s = n. Let u(n) be the number of such sequences. The rank of such a sequence is s - 2k + 1, the number of terms after the maximal term minus the number of terms that precede it.

By letting t (respectively, t^-1) keep track of the terms after (resp., before) a maximal term, we find that u(m,n), the number of size n and rank m sequences, satisfies^†

[1.1]

where (x; q)_n≔(1 - x)(1 - xq)(1 - xq²)⋯(1 - xq^n-1) for n≥1 and (x; q)₀≔1.

Example:

The strongly unimodal sequences of size 5 are: {5}, {1,4}, {4,1}, {1,3,1}, {2,3}, {3,2}, and so u(5) = 6. Respectively, their ranks are 0,-1,1,0,-1,1.

The q-series U(-1; q), the generating function for the number of size n sequences with even rank minus the number with odd rank, is intimately related to Kontsevich’s strange function^‡

[1.2]

It is strange because it does not converge on any open subset of , but is well-defined at all roots of unity. Zagier (1) proved that this function satisfies the even “stranger” identity

[1.3]

where . Neither side of this identity makes sense simultaneously. Indeed, the right-hand side^§ converges in the unit disk |q| < 1, but nowhere on the unit circle. The identity means that F(q) at roots of unity agrees with the radial limit of the right-hand side.

We prove that U(-1; q), which converges in |q| < 1, also gives F(q^-1) at roots of unity.

Theorem 1.1.

If q is a root of unity, then F(q^-1) = U(-1; q).

Example:

Here are two examples: U(-1; -1) = F(-1) = 3 and U(-1; i) = F(-i) = 8 + 3i.

Remark:

Theorem 1.1 is analogous to the result of Cohen (2, 3) that σ(q) = -σ^∗(q^-1) for roots of unity q, for the well-known q-series σ(q) and σ^∗(q) that Andrews et al. (4) defined in their work on partition ranks.

Zagier (1) used Eq. 1.3 to obtain the following identity:

[1.4]

where Glaisher’s T_n numbers (see Eq. 2.3 and A002439 in ref. 5) are the “algebraic factors” of L(χ₁₂,2n + 2). As a companion to Theorem 1.1, we use U(-1; q) to give these same L-values.

Theorem 1.2.

As a power series in t, we have that

These results are related to the next theorem, which gives a new quantum modular form. Following Zagier^¶ (3), a weight k quantum modular form is a complex-valued function f on , or possibly for some finite set S, such that for all , the function

satisfies a “suitable” property of continuity or analyticity. The ϵ(γ) are roots of unity, such as those in the theory of half-integral weight modular forms when . We prove that

[1.5]

is a weight quantum modular form. Because and , it suffices to consider . The following theorem establishes the desired relationship on the larger domain , where is the upper-half of the complex plane.

Theorem 1.3.

If , then

where is the principal branch and

Here, is Dedekind’s eta-function. Moreover, taking η(x) = 0 for , is a C^∞ function that is real analytic everywhere except at x = 0, and h⁽ⁿ⁾(0) = (-πi/12)ⁿ·T_n, where T_n is the nth Glaisher number.

Remark:

Zagier (1) proved that is a quantum modular form. Theorem 1.3 gives a dual quantum modular form, one whose domain naturally extends beyond to include . This is somewhat analogous to the situation for σ(q) and σ^∗(q) discussed above. Zagier constructed a quantum modular form from these q-series in example 1 of ref. 3.

Remark:

Theorem 1.3 implies that Φ(z)≔η(z)ϕ(z) behaves analogously to a weight 2 modular form for for with a suitable error function. Namely, Φ(z + 1) = Φ(z) and ; see also theorem 1.1 of ref. 6.

It turns out that U(1; q) and U( ± i; q) also possess deep properties. We have that U(1; q) (7) is a mixed mock modular form, and U( ± i; q) is a mock theta function (see refs. 8–10). We use these facts to study congruences for certain enumerative functions.

Theorem 1.4.

If 3 < ℓ≢23 (mod 24) is prime, δ(ℓ)≔(ℓ² - 1)/24 and ℓ∤k, then for all n

Example:

If ℓ = 7, then Theorem 1.4 gives u(49n + a) ≡ 0 (mod 2) for a∈{5,12,19,26,33,40}.

The nature of Theorem 1.4 suggests the existence of a Hecke-type identity for U(-1; q) analogous to those obtained for σ(q) and σ^∗(q) in ref. 4. Here we obtain such an identity.

Theorem 1.5.

We have that

These congruences appear to have refinements modulo 4. In analogy with the theory of partition ranks (11–13), we suspect that ranks also “explain” these congruences. Namely, let u(a,b; n) be the number of size n strongly unimodal sequences with rank ≡ a (mod b).

Conjecture 1.6.

If ℓ ≡ 7,11,13,17 (mod 24) is prime and , then for all n we have

[1.6]

Moreover, for a∈{0,1,2,3} we have u(a,4; ℓ²n + kℓ - δ(ℓ)) ≡ 0 (mod 2) and

[1.7]

We have that u(1,4; n) = u(3,4; n), and so the truth of Eq. 1.7 is a proposed explanation of Eq. 1.6. Therefore, it is natural to study U( ± 1; q) and the third-order mock theta function (14–16):

Using this mock theta function, we are able to obtain the following related congruences.

Theorem 1.7.

If (Q,6) = 1, then there are arithmetic progressions An + B such that

Example:

For Q = 5, the cusp form in the proof of Theorem 1.7 is annihilated by T(11²), and so if

[note. a(n) = 0 if n≢23 (mod 24)], then for every n ≡ 23,47 (mod 120) we have that

Because and a(n/121) = 0 when 11||n, this gives congruences such as

2. Quantum Properties of U(-1; q)

Here we prove the quantum properties of U(-1; q). We first prove Theorem 1.1 relating the values of Kontsevich’s F(q) and U(-1; q) at roots of unity. We then prove Theorem 1.2 giving a new representation of Zagier’s L-value generating function, and we conclude with a proof of Theorem 1.3.

2.1 Proof of Theorem 1.1:

For ξ a fixed kth root of unity, define the polynomial

We have the identity

[2.1]

Define the functions u_a(X) for a≥1 by

Hence, for a = k we have

[2.2]

Then we have

By Eq. 2.1, we have

Letting X = 1 gives u_a+1(1) - u_a(1) = ξ^a+1(1 - ξ)²⋯(1 - ξ^a)². Induction and Eq. 2.2 give

2.2 Proof of Theorem 1.2:

By the results of Andrews et al. (6) (see equation 9.2 and propositions 9.2 and 9.3) with q = e^-2πz, we have

for any positive N where . Because we have and for any positive N, we have

for any N. The Glaisher’s T-numbers are given by

[2.3]

We also have the identity

Combining these identities and then setting t = 2πz completes the proof.

2.3 Proof of Theorem 1.3:

Define G(z)≔(e^2πiz; e^2πiz)_∞U(-1; e^2πiz). Theorem 1.1 of ref. 6 gives

[2.4]

Note that using , we have

[2.5]

Similarly, we have

[2.6]

Combining Eqs. 2.4–2.6 gives

Dividing by η(z) and using its modular transformation property give the result for .

For , note that (e^2πix; e^2πix)_∞ = 0. Moreover, Zagier, in the discussion after the theorem of section 6 of ref. 1, explains how the integral is real analytic for real x.

3. Congruence Properties and the Hecke-Type Identity

We first prove Theorem 1.4 on the parity of u(n), and we then prove Theorem 1.5 giving the Hecke-type identity for U(-1; q). We then conclude this section with the proof of Theorem 1.7.

3.1 Proof of Theorem 1.4:

By theorem 1 of ref. 14 (see Eq. 1.2), we have that

If spt(n) is the smallest parts partition function of Andrews, then by theorem 4 of ref. 17 we have:

We have used the elementary fact that

[3.1]

We have U(-1; q) ≡ S(q) (mod 2), and so the theorem follows from theorem 1.2 in ref. 18^||.

3.2 Proof of Theorem 1.5:

We prove Theorem 1.5 using the method of Bailey pairs. As usual, we let (a)_n≔(a; q)_n. Two sequences (α_n,β_n) form a Bailey pair for a if

The following Bailey pair is central to the proof of Theorem 1.5.

Lemma 3.1.

If β_n = 1 and α₀ = 1 and for n > 0

then (α_n,β_n) is a Bailey pair with respect to 1.

Proof:

We apply theorem 8 of ref. 19 with β_n = 1 for all n. By letting b,c,d → 0, and then letting a = 1, one obtains the lemma. Some care is required for the j = 0 and j = 1 terms.

The following is Bailey’s Lemma (for example, see ref. 19).

Lemma 3.2.

(Bailey’s Lemma). If α_n and β_n form a Bailey pair relative to a, then

Proof of Theorem 1.5:

By Lemma 3.2 with ρ₁ = x, ρ₂ = x^-1 and a = 1, Lemma 3.1 gives

Dividing by (1 - x)(1 - x^-1) and collecting the n = 0 terms give

[3.2]

To simplify the α_n, we have that

which in turn implies that

Thus, α₀ = 1, and for n≥1 we have

We note that

Now, insert these facts in Eq. 3.2, let x → 1, and use the identity .

3.3 Proof of Theorem 1.7:

We give a sketch because it is analogous to theorem 1.5 of ref. 12 and theorem 1 of ref. 20. We have

where Ψ(q) is one of Ramanujan’s third-order mock theta functions. We have that q^-1Ψ(q²⁴) is the holomorphic part of a weight 1/2 harmonic Maass form whose shadow is a unary theta function. Using quadratic and trivial twists modulo Q, one obtains a weight 1/2 weakly holomorphic modular form. By work of Treneer (21), one obtains weakly holomorphic forms of half-integer weight that are congruent to cusp forms modulo Q. By the Shimura correspondence, we obtain even integer weight cusp forms, which by lemma 3.30 of ref. 22 are annihilated modulo Q by infinitely many Hecke operators T(p). Because the Shimura correspondence is Hecke equivariant, it follows that infinitely many half-integral weight Hecke operators T(p²) annihilate these cusp forms modulo Q. The proof follows from the formula for the action of these operators.