International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 630458 | 21 pages | https://doi.org/10.1155/2010/630458

A Rademacher Type Formula for Partitions and Overpartitions

Academic Editor: Stéphane Louboutin
Received03 Jul 2009
Revised21 Jan 2010
Accepted28 Feb 2010
Published25 Mar 2010

Abstract

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of and the Zuckerman formula for the Fourier coefficients of is presented.

1. Background

1.1. Partitions

A partition of an integer is a representation of as a sum of positive integers, where the order of the summands (called parts) is considered irrelevant. It is customary to write the parts in nonincreasing order. For example, there are three partitions of the integer , namely, , , and . Let denote the number of partitions of , with the convention that , and let denote the generating function of , that is, let

Euler [1] was the first to systematically study partitions. He showed that

Euler also showed that

and since the exponents appearing on the right side of (1.3) are the pentagonal numbers, (1.3) is often called “Euler's pentagonal number theorem."

Although Euler's results can all be treated from the point of view of formal power series, the series and infinite products above (and indeed all the series and infinite products mentioned in this paper) converge absolutely when , which is important for analytic study of these series and products.

Hardy and Ramanujan were the first to study analytically and produced an incredibly accurate asymptotic formula [2, page 85, equation (1.74)], namely,

where

is an arbitrary constant, and here and throughout is an abbreviation for .

Later Rademacher [3] improved upon (1.4) by finding the following convergent series representation for :

Rademacher's method was used extensively by many practitioners, including Grosswald [4, 5], Haberzetle [6], Hagis [715], Hua [16], Iseki [1719], Lehner [20], Livingood [21], Niven [22], and Subramanyasastri [23] to study various restricted partitions functions.

Recently, Bringmann and Ono [24] have given exact formulas for the coefficients of all harmonic Maass forms of weight . The generating functions considered herein are weakly holomorphic modular forms of weight , and thus they are harmonic Maass forms of weight . Accordingly, the results of this present paper could be derived from the general theorem in [24]. However, here we opt to derive the results via classical method of Rademacher.

1.2. Overpartitions

Overpartitions were introduced by Corteel and Lovejoy in [25] and have been studied extensively by them and others including Bringmann, Chen, Fu, Goh, Hirschhorn, Hitczenko, Lascoux, Mahlburg, Robbins, Rødseth, Sellers, Yee, and Zhao [2543].

An overpartition of is a representation of as a sum of positive integers with summands in nonincreasing order, where the last occurrence of a given summand may or may not be overlined. Thus the eight overpartitions of are , , , , , , , .

Let denote the number of overpartitions of and let denote the generating function of . Elementary techniques are sufficient to show that

Note that

via an identity of Gauss (see equation (2.2.12), page 23 in [44, 45]), so that the reciprocal of the generating function for overpartitions is a series wherein a coefficient is nonzero if and only if the exponent of is a perfect square, just as the reciprocal of the generating function for partitions is a series wherein a coefficient is nonzero if and only if the exponent of is a pentagonal number.

Hardy and Ramanujan, writing more than 80 years before the coining of the term “overpartition," stated [2, page 109-110] that the function which we are calling “has no very simple arithmetical interpretation; but the series is none the less, as the direct reciprocal of a simple -function, of particular interest." They went on to state that

In fact, (1.9) was improved to the following Rademacher-type convergent series by Zuckerman [46, page 321, equation (8.53)]:

A simplified proof of (1.10) was given by Goldberg in his Ph.D. thesis [47].

1.3. Partitions Where No Odd Part Is Repeated

Let denote the number of partitions of where no odd part appears more than once. Let denote the generating function of , so we have

Via another identity of Gauss (see equation (2.2.13), page 23 in [44, 45]), it turns out that

so in this case the reciprocal of the generating function under consideration has nonzero coefficients at the exponents which are triangular (or equivalently, hexagonal) numbers.

The analogous Rademacher-type formula for is as follows:

Equation (1.13) is the case of Theorem 2.1 to be presented in the next section.

2. A Common Generalization

Let us define

where is a nonnegative integer. Thus,

Let denote the coefficient of in the expansion of , that is,

Notice that can be represented by several forms of equivalent infinite products, each of which has a natural combinatorial interpretation:

Thus, equals each of the following:

(i)the number of overpartitions of where nonoverlined parts are multiples of (by (2.4)); (ii)the number of partitions of where all parts are either odd or multiples of (by (2.5)), provided ; (iii)the number of partitions of where nonmultiples of are distinct (by (2.6)), provided .

Theorem 2.1. For ,

3. A Proof of Theorem 2.1

The method of proof is based on Rademacher's proof of (1.6) in [48] with the necessary modifications. Additional details of Rademacher's proof of (1.6) are provided in [49], [50, Chapter 14], and [51, Chapter 5].

Of fundamental importance is the path of integration to be used. In [48], Rademacher improved upon his original proof of (1.6) given in [3], by altering his path of integration from a carefully chosen circle to a more complicated path based on Ford circles, which in turn led to considerable simplifications later in the proof.

3.1. Farey Fractions

The sequence of proper Farey fractions of order is the set of all with and , arranged in increasing order. Thus, for example,

For a given , let , , , and be such that is the immediate predecessor of and is the immediate successor of in . It will be convenient to view each cyclically, that is, to view as the immediate successor of .

3.2. Ford Circles and the Rademacher Path

Let and be integers with and . The Ford circle [52] is the circle in of radius centered at the point

The upper   of the Ford circle is those points of from the initial point

to the terminal point

traversed in the clockwise direction.

Note that we have

Every Ford circle is in the upper half plane. For , and are either tangent or do not intersect.

The Rademacher path of order is the path in the upper half of the -plane from to consisting of

traversed left to right and clockwise. In particular, we consider the left half of the Ford circle and the corresponding upper arc  to be translated to the right by 1 unit. This is legal given then periodicity of the function which is to be integrated over .

3.3. Set Up the Integral

Let and be fixed, with .

Since

Cauchy's residue theorem implies that

where is any simply closed contour enclosing the origin and inside the unit circle. We introduce the change of variable

so that the unit disk in the -plane maps to the infinitely tall, unit-wide strip in the -plane where and . The contour is then taken to be the preimage of under the map .

Better yet, let us replace with in (3.7) to express the integration in the -plane:

3.4. Another Change of Variable

Next, we change variables again, taking

so that

Thus (in the -plane) maps to the clockwise-oriented circle (in the -plane) centered at with radius .

So we now have

where (resp., ) is the image of (see (3.2)) (resp. [see 3.2]) under the transformation (3.11).

So the transformation (3.11) maps the upper arc  of in the -plane to the arc on which initiates at

and terminates at

3.5. Exploiting a Modular Transformation

From the theory of modular forms, we have the transformation formula [2, page 93, Lemma ]:

where is the principal branch, , and is a solution to the congruence

From (3.15), we deduce the analogous transformation for .

The transformation formula is a piecewise defined function with cases corresponding to , where :

where is divisible by and is a solution to the congruence , and is the Kronecker -function.

Notice that in particular, for , (3.17) simplifies to

Since the case was established by Zuckerman, and the case by Rademacher, we will proceed with the assumption that .

Apply (3.17) to (3.12) to obtain

3.6. Normalization

Next, introduce a normalization (this is not strictly necessary, but it will allow us in the sequel to quote various useful results directly from the literature):

where

Let us now rewrite (3.21) as where

3.7. Estimation

It will turn out that as , only for and ultimately make a contribution. Note that all the integrations in the -plane occur on arcs and chords of the circle of radius centered at the point . So, inside and on , and .

3.7.1. Estimation of for

The regularity of the integrand allows us to alter the path of integration from the arc connecting and to the directed segment. By [51, page 104, Theorem ], the length of the path of integration does not exceed , and on the segment connecting to , . Thus, the absolute value of the integrand is

Thus, for ,

for a constant (recalling that and are fixed).

3.7.2. Estimation of for

We have the absolute value of the integrand:

for a constant . So, for ,

for a constant .

3.7.3. Estimation of

Let be defined by

Again, the regularity of the integrand allows us to alter the path of integration from the arc connecting and to the directed segment.

With this in mind, we estimate the absolute value of the integrand:

for a constant . So,

for a constant .

3.7.4. Estimation of for

Again, the regularity of the integrand allows us to alter the path of integration from the arc connecting and to the directed segment.

With this in mind, for a constant . So,

for a constant , when .

3.7.5. Combining the Estimates

One has

Thus, we may revise (3.21) to

3.8. Evaluation of for and

Write as

where

and and have the same integrand as (3.36) (and analogously for ).

3.8.1. Estimation of and

We note that the length of the arc of integration in is less than , and on this . [50, page 272]. Also, on [50, page 271, equation (120.2)]. Further, [50, page 271, equation (119.6)]. The absolute value of the integrand is thus so that

By the same reasoning,

We may therefore revise (3.35) to and upon letting tend to infinity, obtain

3.9. The Final Form

We may now introduce the change of variable (where the first summation in (3.41) is the term separated out for clarity), which allows the integral to be evaluated in terms of , the Bessel function of the first kind of order with purely imaginary argument [53, page 372, 17.7] when we bear in mind that a “bent” path of integration is allowable according to the remark preceding Equation () on page 177 of [54]. See also [51, page 109]. The final form of the formula is then obtained by using the fact that Bessel functions of half-odd integer order can be expressed in terms of elementary functions.

We therefore have, for ,

which, after application of the formula [51, page 110] is equivalent to Theorem 2.1.

Acknowledgments

The author thanks the anonymous referee for bringing the work of Zuckerman [46] and Goldberg [47] to his attention. This, in turn, led the author to seek the more general result presented here in this final version of the paper.

References

  1. L. Euler, Introductio in Analysin Infinatorum, Marcum-Michaelem Bousquet, Lausanne, Switzerland, 1748.
  2. G. H. Hardy and S. Ramanujan, “Asymptotic formulae in combinatory analysis,” Proceedings of the London Mathematical Society, vol. 17, pp. 75–115, 1918. View at: Google Scholar
  3. H. Rademacher, “On the partition function p(n),” Proceedings of the London Mathematical Society, vol. 43, pp. 241–254, 1937. View at: Google Scholar
  4. E. Grosswald, “Some theorems concerning partitions,” Transactions of the American Mathematical Society, vol. 89, pp. 113–128, 1958. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. E. Grosswald, “Partitions into prime powers,” The Michigan Mathematical Journal, vol. 7, pp. 97–122, 1960. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. M. Haberzetle, “On some partition functions,” American Journal of Mathematics, vol. 63, pp. 589–599, 1941. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. P. Hagis Jr., “A problem on partitions with a prime modulus p3,” Transactions of the American Mathematical Society, vol. 102, pp. 30–62, 1962. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. P. Hagis Jr., “Partitions into odd summands,” American Journal of Mathematics, vol. 85, pp. 213–222, 1963. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. P. Hagis Jr., “On a class of partitions with distinct summands,” Transactions of the American Mathematical Society, vol. 112, pp. 401–415, 1964. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. P. Hagis Jr., “Partitions into odd and unequal parts,” American Journal of Mathematics, vol. 86, pp. 317–324, 1964. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. P. Hagis Jr., “Partitions with odd summands—some comments and corrections,” American Journal of Mathematics, vol. 87, pp. 218–220, 1965. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. P. Hagis Jr., “A correction of some theorems on partitions,” Transactions of the American Mathematical Society, vol. 118, p. 550, 1965. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. P. Hagis Jr., “On the partitions of an integer into distinct odd summands,” American Journal of Mathematics, vol. 87, pp. 867–873, 1965. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. P. Hagis Jr., “Some theorems concerning partitions into odd summands,” American Journal of Mathematics, vol. 88, pp. 664–681, 1966. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. P. Hagis Jr., “Partitions with a restriction on the multiplicity of the summands,” Transactions of the American Mathematical Society, vol. 155, pp. 375–384, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH |