We consider certain probability problems which are naturally related to
integer partitions. We show that the corresponding probabilities are values of classical
modular forms. Thanks to this connection, we then show that certain ratios of
probabilities are specializations of the Rogers-Ramanujan and Ramanujan- Selberg-
Gordon-Göllnitz continued fractions. One particular evaluation depends on a result
from Ramanujan's famous first letter to Hardy.

1. Introduction and Statement of Results

In a recent paper [1], Holroyd et al. defined probability models whose properties are linked to the number theoretic functions pk(n), which count the number of integer partitions of n which do not contain k consecutive integers among its summands. Their asymptotic results for each k lead to a nice application: the calculation of thresholds of certain two dimensional cellular automata (see [1, Theorem 4]).

In a subsequent paper [2], Andrews explained the deep relationship between the generating functions for pk(n) and mock theta functions and nonholomorphic modular forms. Already in the special case of p2(n), one finds an exotic generating function whose unusual description requires both a modular form and a Maass form. This description plays a key role in the recent work of Bringmann and Mahlburg [3], who make great use of the rich properties of modular forms and harmonic Maass forms to obtain asymptotic results which improve on [1, Theorem 3] in the special case of p2(n).

It does not come as a surprise that the seminal paper of Holroyd et al. [1] has inspired further work at the interface of number theory and probability theory. (In addition to [2, 3], we point out the recent paper by Andrews et al. [4]) Here we investigate another family of probability problems which also turn out to be related to the analytical properties of classical partition generating functions.

These problems are relatives of the following standard undergraduate problem. Toss a fair coin repeatedly until it lands heads up. If one flips n tails before the first head, what is the probability that n is even? Since the probability of flipping n tails before the first head is 1/2n+1, the solution is

12+123+125+⋯=11-1/2-11-1/4=23.

Instead of continuing until the first head, consider the situation where a coin is repeatedly flipped: once, then twice, then three times, and so on. What is the probability of the outcome that each nth turn, where n is odd, has at least one head?

More generally, let 0<p<1, and let {C1,C2,…} be a sequence of independent events where the probability of Cn is given by

ℙp(Cn):=1-pn.
For each pair of integers 0≤r<t, we let

A(r,t):={set of sequences where Cn occurs if n≢±r mod t}.
In the case where p=1/2, one can think of Cn as the event where at least one of n tosses of a coin is a head. Therefore, if p=1/2, the problem above asks for the probability of the outcome A(0,2).

Remark 1.1.

Without loss of generality, we shall always assume that 0≤r≤t/2. In most cases A(r,t) is defined by two arithmetic progressions modulo t. The only exceptions are for r=0, and for even t when r=t/2.

It is not difficult to show that the problem of computing

ℙp(A(r,t)):=“probability of A(r,t)”
involves partitions. Indeed, if p(r,t;n) denotes the number of partitions of n whose summands are congruent to ±r(mod t), then we shall easily see that the probabilities are computed using the generating functions

𝒫(r,t;q):=∑n=0∞p(r,t;n)qn={∏n=0∞1(1-qtn+r)(1-qtn+t-r),if0<r<t2,∏n=1∞1(1-qtn),ifr=0,∏n=0∞1(1-qtn+t/2),ifr=t2.
This critical observation is the bridge to a rich area of number theory, one involving class field theory, elliptic curves, and partitions. In these areas, modular forms play a central role, and so it is natural to investigate the number theoretic properties of the probabilities ℙr(A(r,t)) from this perspective.

Here we explain some elegant examples of results which can be obtained in this way. Throughout, we let τ∈ℍ, the upper-half of the complex plane, and we let qτ:=e2πiτ. The Dedekind eta-function is the weight 1/2 modular form (see [5]) defined by the infinite product

η(τ):=qτ1/24∏n=1∞(1-qτn).
We also require some further modular forms, the so-called Siegel functions. For u and v real numbers, let B2(u):=u2-u+1/6 be the second Bernoulli polynomial, let z:=u-vτ, and let qz:=e2πiz. Then the Siegel function gu,v(τ) is defined by the infinite product

gu,v(τ):=-qτB2(u)/2e2πiv(u-1)/2∏n=1∞(1-qzqτn)(1-qz-1qτn).
These functions are weight 0 modular forms (e.g., see [5]).

For fixed integers 0≤r≤t/2, we first establish that ℙp(A(r,t)) is essentially a value of a single quotient of modular forms.

Theorem 1.2.

If 0<p<1 and τp:=-log(p)·i/2π, then
ℙp(A(r,t))={qτp(t-1)/24·η(τp)η(tτp),ifr=0,qτp-(t+2)/481-qτpt/2·η(τp)η(tτp)η((t/2)τp),ifr=t2,-qτp-(2t+1)/24e-πi(r/t)1-qτpr·η(τp)g0,-r/t(tτp),otherwise.

One of the main results in the work of Holroyd et al. (see [1, Theorem 2]) was the asymptotic behavior of their probabilities as p→1. In Section 4 we obtain the analogous results for logℙp(A(r,t)).

Theorem 1.3.

As p→1, one has
-logℙp(A(r,t))~{π26(1-p)(1-1t),ifr=0,t2,π26(1-p)(1-2t),otherwise.

To fully appreciate the utility of Theorem 1.2, it is important to note that the relevant values of Dedekind's eta-function and the Siegel functions can be reformulated in terms of the real-analytic Eisenstein series (see [5])

Eu,v(τ,s):=∑(m,n)∈ℤ2(m,n)≠(0,0)e2πi(mu+nv)ys|mτ+n|2s,τ=x+iy∈ℍ,Re(s)>1.
One merely makes use of the Kronecker Limit Formulas (e.g., see [5, Part 4]). These limit formulas are prominent in algebraic number theory for they explicitly relate such values of modular forms to values of zeta-functions of number fields.

We give two situations where one plainly sees the utility of these observations. Firstly, one can ask for a more precise limiting behavior than is dictated by Theorem 1.3. For example, consider the limiting behavior of the quotient L(p):=ℙp(A(2,5))/ℙp(A(1,5)), as p→1. Table 1 is very suggestive.

Values of L(p).

p

ℙp(A(2,5))

ℙp(A(1,5))

L(p)

0.3

0.692⋯

0.883⋯

0.61607⋯

0.4

0.576⋯

0.776⋯

0.61778⋯

⋮

⋮

⋮

⋮

0.97

6.43⋯×10-14

1.03⋯×10-13

0.61803⋯

0.98

5.65⋯×10-21

9.11⋯×10-21

0.61803⋯

0.99

3.11⋯×10-42

5.03⋯×10-42

0.61803⋯

Theorem 1.4.

The following limits are true:
limp→1ℙp(A(2,5))ℙp(A(1,5))=-1+52,limp→1ℙp(A(3,8))ℙp(A(1,8))=-1+2.

Remark 1.5.

Notice that (1/2)(-1+5)=-1+ϕ, where ϕ is the golden ratioϕ:=12(1+5)=1.61803….

Theorem 1.4 is a consequence of our second application which concerns the problem of obtaining algebraic formulas for all of these ratios, not just the limiting values. As a function of p, one may compute the ratios of these probabilities in terms of continued fractions. To ease notation, we let

a1b1+a2b2+a3b3+⋯
denote the continued fraction

a1b1+a2b2+a3b3+⋯.
The following continued fractions are well known:

11+11+11+⋯=11+11+11+⋯=-1+52,12+12+12+⋯=12+12+12+⋯=-1+2.
Theorem 1.4 is the limit of the following exact formulas.

Theorem 1.6.

If 0<p<1 and τp:=-log(p)·(i/2π), then one has that
ℙp(A(2,5))ℙp(A(1,5))=(11+qτp1+qτp21+qτp31+⋯),ℙp(A(3,8))ℙp(A(1,8))=(11+qτp+qτp21+qτp3+qτp41+qτp5+qτp61+qτp7+⋯).

Theorem 1.6 can also be used to obtain many further beautiful expressions, not just those pertaining to the limit as p→1. For example, we obtain the following simple corollary.

Corollary 1.7.

For p1=1/e2π and p2=1/eπ, one has that
ℙp1(A(2,5))ℙp1(A(1,5))=e-2π/5(-ϕ+12(5+5)),ℙp2(A(3,8))ℙp2(A(1,8))=e-π/2(4+22-3+22).

Remark 1.8.

The evaluation of ℙp(A(2,5))/ℙp(A,1,5)) when p=1/e2π follows from a formula in Ramanujan's famous first letter to Hardy dated January 16, 1913.

Theorem 1.6 and Corollary 1.7 follow from the fact that the Fourier expansions of the relevant modular forms in Theorem 1.2 are realized as the q-continued fractions of Rogers-Ramanujan and Ramanujan-Selberg-Gordon-Göllnitz. We shall explain these results in Section 5.

We chose to focus on two particularly simple examples of ratios, namely,

ℙp(A(2,5))ℙp(A(1,5)),ℙp(A(3,8))ℙp(A(1,8)).
One can more generally consider ratios of the form

ℙp(A(r1,t))ℙp(A(r2,t)).
It is not the case that all such probabilities, for fixed r1 and r2, can be described by a single continued fraction. However, one may generalize these two cases, thanks to Theorem 1.2, by making use of the so-called Selberg relations [6] (see also [7]) which extend the notion of a continued fraction. Arguing in this way, one may obtain Theorem 1.6 in its greatest generality. Although we presented Theorem 1.4 as the limiting behavior of the continued fractions in Theorem 1.6, we stress that its conclusion also follows from the calculation of the explicit Fourier expansions at cusps of the modular forms in Theorem 1.2. In this way one may also obtain Theorem 1.4 in generality. Turning to the problem of obtaining explicit formulas such as those in Corollary 1.7, we have the theory of complex multiplication at our disposal. In general, one expects to obtain beautiful evaluations as algebraic numbers whenever τp is an algebraic integer in an imaginary quadratic extension of the field of rational numbers. We leave these generalizations to the reader.

2. Combinatorial Considerations

Here we give a lemma which expresses the probability ℙp(A(r,t)) in terms of the infinite products in (1.5).

Lemma 2.1.

If 0<p<1 and τp:=-log(p)·(i/2π), then
ℙp(A(r,t))=𝒫(r,t;qτp)·∏n=1∞(1-qτpn)·

Proof.

By the Borel-Cantelli lemma, with probability 1 at most finitely many of the Cn's will fail to occur. Now, for each pair of integers 0≤r≤t/2, let S(r,t) be the countable set of binary strings a1a2a3a4⋯∈{0,1}ℕ in which an=1 if n≢±rmodt, and an=0 for at most finitely many n satisfying n≡±rmodt (with the appropriate modifications on the arithmetic progressions modulo t for r=0, and for t even with r=t/2). Then the event A(r,t) can be written as the countable disjoint union
A(r,t)=⋃a1a2a3⋯∈Sr,t⋂n:an=1Cn∩⋂n:an=0Cnc.
Using this, we find that
ℙp(A(r,t))=∑a1a2a3⋯∈Sr,tℙp(⋂n:an=1Cn∩⋂n:an=0Cnc)=∑a1a2a3⋯∈Sr,t∏n:an=1(1-qτpn)∏n:an=0qτpn=∏i=1∞(1-qτpn)∑a1a2a3⋯∈Sr,t∏n:an=0qτpn1-qτpn=∏n=1∞(1-qτpn)·∑a1a2a3⋯∈Sr,t∏n:an=0(qτpn+qτp2n+qτp3n+⋯)=∏n=1∞(1-qτpn)·𝒫(r,t;qτp).

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.2</xref>

We first note that qτp=e2πiτp=p, and by (1.6) we have

∏n=1∞(1-qτpn)=qτp-1/24η(τp).

Now we prove each of the three relevant cases in turn. First, suppose that r=0. By Lemma 2.1 we need to show

qτp-1/24η(τp)∏n=1∞(1-qτptn)-1=qτp(t-1)/24η(τp)η(tτp).
This follows from the identity

Next, suppose that t is even and r=t/2. By Lemma 2.1 we need to show

qτp-1/24η(τp)∏n=0∞(1-qτptn+(t/2))-1=qτp-(t+2)/481-qτpt/2η(τp)η(tτp)η((t/2)τp).
Observe that

∏n=1∞(1-qτptn+(t/2))-1=∏n=1∞(1-q(t/2)τp2n+1)-1=∏n=1∞(1-q(t/2)τp2n)(1-q(t/2)τpn)-1=∏n=1∞(1-qtτpn)(1-q(t/2)τpn)-1.
Then (3.4) follows from the identities

Finally, suppose that 0<r<t/2. By Lemma 2.1 we need to show

qτp-1/24η(τp)∏n=0∞(1-qτptn+r)-1(1-qτptn+t-r)-1=-qτp-(2t+1)/24e-πi(r/t)1-qτprη(τp)g0,-r/t(tτp).
Observe that

∏n=0∞(1-qτptn+r)(1-qτptn+t-r)=(1-qτpr)(1-qτpt-r)∏n=1∞(1-qτptn+r)(1-qτptn+t-r)=(1-qτpr)(1-qτpt-r)∏n=1∞(1-qrτpqτptn)(1-qrτp-1qτpt(n+1))=(1-qτpr)(1-qrτp-1qtτp)(1-qrτp-1qtτp)∏n=1∞(1-qrτpqtτpn)(1-qrτp-1qtτpn)=(1-qτpr)∏n=1∞(1-qrτpqtτpn)(1-qrτp-1qtτpn)
It follows from the definition of the Siegel function gu,v(τ) that

∏n=1∞(1-qrτpqtτpn)(1-qrτp-1qtτpn)=-qtτp1/12eπi(r/t)g0,-r/t(tτp),
from which we obtain

∏n=0∞(1-qτptn+r)-1(1-qτptn+t-r)-1=-qτp-t/12e-πi(r/t)(1-qτpr)-1g0,-r/t(tτp)-1.
Substitute the preceding identity into the left-hand side of (3.7) to obtain the result.

4. Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.3</xref>

Recall that the partition generating function is defined by

G(x):=∏n=1∞(1-xn)-1,0<x<1.
We now prove each of the relevant cases in turn. First, suppose that r=0. Then 𝒫(0,t;qτp)=G(qτpt), and by Lemma 2.1,

-logℙp(A(0,t))=logG(qτp)-logG(qτpt).
Make the change of variables x=e-w. Then a straightforward modification of the analysis in [8, pages 19–21] shows that, for each integer α≥1,

logG(e-αw)=π26αw+12log(1-e-αw)+Cα+O(w)asw→0
for some constant Cα. It follows from (4.3) that

logG(qτp)-logG(qτpt)~π26(1-p)(1-1t)asp→1.

Next, suppose that t is even and r=t/2. Then

𝒫(t2,t;qτp)=11-qτpt/2∏n=1∞(1-q(t/2)τpn)-1(1-qtτpn)=G(qτpt/2)G(qτpt)-11-qτpt/2,
and by Lemma 2.1 and (4.3),

5. Proof of Theorem <xref ref-type="statement" rid="thm1.4">1.6</xref> and Corollary <xref ref-type="statement" rid="coro1.5">1.7</xref>

Here we prove Theorem 1.6 and Corollary 1.7. These results will follow from well-known identities of Rogers-Ramanujan, Selberg, and Gordon-Göllnitz (e.g., see [9]). We require the celebrated Rogers-Ramanujan continued fraction

R(q)=q1/5(11+q1+q21+q31+⋯),
and the Ramanujan-Selberg-Gordon-Göllnitz continued fraction

H(q)=q1/2(11+q+q21+q3+q41+q5+q61+q7+⋯).
It turns out that these q-continued fractions satisfy the following identities (e.g., see [10]):

R(q)=q1/5∏n=1∞(1-q5n-1)(1-q5n-4)(1-q5n-2)(1-q5n-3)=q1/5𝒫(2,5;q)𝒫(1,5;q),H(q)=q1/2∏n=1∞(1-q8n-1)(1-q8n-7)(1-q8n-3)(1-q8n-5)=q1/2𝒫(3,8;q)𝒫(1,8;q).
Theorem 1.6 now follows easily from Theorem 1.2.

To prove Corollary 1.7, notice that if p=1/e2π (resp., p=1/eπ), then τp=i (resp., τp=i/2). In particular, we have that qτp=e-2π (resp., qτp=e-π). Corollary 1.7 now follows from the famous evaluations (e.g., see [11] and [12, page xxvii] which is Ramanujan's first letter to Hardy)

R(e-2π)=-ϕ+12(5+5),H(e-π)=4+22-3+22.

Acknowledgment

Ken Ono thanks the support of the NSF, the Hilldale Foundation and the Manasse family.

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