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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 630458, 21 pages
A Rademacher Type Formula for Partitions and Overpartitions
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460-8093, USA
Received 3 July 2009; Revised 21 January 2010; Accepted 28 February 2010
Academic Editor: Stéphane Louboutin
Copyright © 2010 Andrew V. Sills. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  was the first to systematically study partitions. He showed that
Euler also showed that
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,
is an arbitrary constant, and here and throughout is an abbreviation for .
Rademacher's method was used extensively by many practitioners, including Grosswald [4, 5], Haberzetle , Hagis [7–15], Hua , Iseki [17–19], Lehner , Livingood , Niven , and Subramanyasastri  to study various restricted partitions functions.
Recently, Bringmann and Ono  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 . However, here we opt to derive the results via classical method of Rademacher.
Overpartitions were introduced by Corteel and Lovejoy in  and have been studied extensively by them and others including Bringmann, Chen, Fu, Goh, Hirschhorn, Hitczenko, Lascoux, Mahlburg, Robbins, Rødseth, Sellers, Yee, and Zhao [25–43].
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
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
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
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:
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  with the necessary modifications. Additional details of Rademacher's proof of (1.6) are provided in , [50, Chapter 14], and [51, Chapter 5].
Of fundamental importance is the path of integration to be used. In , Rademacher improved upon his original proof of (1.6) given in , 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  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 .
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
Thus (in the -plane) maps to the clockwise-oriented circle (in the -plane) centered at with radius .
So we now have
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 .
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):
Let us now rewrite (3.21) as where
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
Thus, we may revise (3.21) to
3.8. Evaluation of for and
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 . 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 ,
The author thanks the anonymous referee for bringing the work of Zuckerman  and Goldberg  to his attention. This, in turn, led the author to seek the more general result presented here in this final version of the paper.
- L. Euler, Introductio in Analysin Infinatorum, Marcum-Michaelem Bousquet, Lausanne, Switzerland, 1748.
- G. H. Hardy and S. Ramanujan, “Asymptotic formulae in combinatory analysis,” Proceedings of the London Mathematical Society, vol. 17, pp. 75–115, 1918.
- H. Rademacher, “On the partition function ,” Proceedings of the London Mathematical Society, vol. 43, pp. 241–254, 1937.
- E. Grosswald, “Some theorems concerning partitions,” Transactions of the American Mathematical Society, vol. 89, pp. 113–128, 1958.
- E. Grosswald, “Partitions into prime powers,” The Michigan Mathematical Journal, vol. 7, pp. 97–122, 1960.
- M. Haberzetle, “On some partition functions,” American Journal of Mathematics, vol. 63, pp. 589–599, 1941.
- P. Hagis Jr., “A problem on partitions with a prime modulus ,” Transactions of the American Mathematical Society, vol. 102, pp. 30–62, 1962.
- P. Hagis Jr., “Partitions into odd summands,” American Journal of Mathematics, vol. 85, pp. 213–222, 1963.
- P. Hagis Jr., “On a class of partitions with distinct summands,” Transactions of the American Mathematical Society, vol. 112, pp. 401–415, 1964.
- P. Hagis Jr., “Partitions into odd and unequal parts,” American Journal of Mathematics, vol. 86, pp. 317–324, 1964.
- P. Hagis Jr., “Partitions with odd summands—some comments and corrections,” American Journal of Mathematics, vol. 87, pp. 218–220, 1965.
- P. Hagis Jr., “A correction of some theorems on partitions,” Transactions of the American Mathematical Society, vol. 118, p. 550, 1965.
- P. Hagis Jr., “On the partitions of an integer into distinct odd summands,” American Journal of Mathematics, vol. 87, pp. 867–873, 1965.
- P. Hagis Jr., “Some theorems concerning partitions into odd summands,” American Journal of Mathematics, vol. 88, pp. 664–681, 1966.
- 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.
- L.-K. Hua, “On the number of partitions of a number into unequal parts,” Transactions of the American Mathematical Society, vol. 51, pp. 194–201, 1942.
- S. Iseki, “A partition function with some congruence condition,” American Journal of Mathematics, vol. 81, pp. 939–961, 1959.
- S. Iseki, “On some partition functions,” Journal of the Mathematical Society of Japan, vol. 12, pp. 81–88, 1960.
- S. Iseki, “Partitions in certain arithmetic progressions,” American Journal of Mathematics, vol. 83, pp. 243–264, 1961.
- J. Lehner, “A partition function connected with the modulus five,” Duke Mathematical Journal, vol. 8, pp. 631–655, 1941.
- J. Livingood, “A partition function with the prime modulus ,” American Journal of Mathematics, vol. 67, pp. 194–208, 1945.
- I. Niven, “On a certain partition function,” American Journal of Mathematics, vol. 62, pp. 353–364, 1940.
- V. V. Subramanyasastri, “Partitions with congruence conditions,” The Journal of the Indian Mathematical Society, vol. 11, pp. 55–80, 1972.
- K. Bringmann and K. Ono, “Coefficients of harmonic Maass forms,” in Proceedings of the University of Florida Conference on Partitions, q-Series, and Modular Forms, 2008.
- S. Corteel and J. Lovejoy, “Overpartitions,” Transactions of the American Mathematical Society, vol. 356, no. 4, pp. 1623–1635, 2004.
- K. Bringmann and J. Lovejoy, “Dyson's rank, overpartitions, and weak Maass forms,” International Mathematics Research Notices, vol. 2007, no. 19, Article ID rnm063, 34 pages, 2007.
- W. Y. C. Chen and J. J. Y. Zhao, “The Gaussian coefficients and overpartitions,” Discrete Mathematics, vol. 305, no. 1–3, pp. 350–353, 2005.
- S. Corteel, W. M. Y. Goh, and P. Hitczenko, “A local limit theorem in the theory of overpartitions,” Algorithmica, vol. 46, no. 3-4, pp. 329–343, 2006.
- S. Corteel and P. Hitczenko, “Multiplicity and number of parts in overpartitions,” Annals of Combinatorics, vol. 8, no. 3, pp. 287–301, 2004.
- S. Corteel, J. Lovejoy, and A. J. Yee, “Overpartitions and generating functions for generalized Frobenius partitions,” in Mathematics and Computer Science. III, Trends in Mathematics, pp. 15–24, Birkhäuser, Basel, Switzerland, 2004.
- S. Corteel and O. Mallet, “Overpartitions, lattice paths, and Rogers-Ramanujan identities,” Journal of Combinatorial Theory. Series A, vol. 114, no. 8, pp. 1407–1437, 2007.
- A. M. Fu and A. Lascoux, “-identities related to overpartitions and divisor functions,” Electronic Journal of Combinatorics, vol. 12, pp. 1–7, 2005, research paper 38.
- M. D. Hirschhorn and J. A. Sellers, “Arithmetic relations for overpartitions,” Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 53, pp. 65–73, 2005.
- M. D. Hirschhorn and J. A. Sellers, “Arithmetic properties of overpartitions into odd parts,” Annals of Combinatorics, vol. 10, no. 3, pp. 353–367, 2006.
- J. Lovejoy, “Gordon's theorem for overpartitions,” Journal of Combinatorial Theory. Series A, vol. 103, no. 2, pp. 393–401, 2003.
- J. Lovejoy, “Overpartitions and real quadratic fields,” Journal of Number Theory, vol. 106, no. 1, pp. 178–186, 2004.
- J. Lovejoy, “Overpartition theorems of the Rogers-Ramanujan type,” Journal of the London Mathematical Society, vol. 69, no. 3, pp. 562–574, 2004.
- J. Lovejoy, “A theorem on seven-colored overpartitions and its applications,” International Journal of Number Theory, vol. 1, no. 2, pp. 215–224, 2005.
- J. Lovejoy, “Rank and conjugation for the Frobenius representation of an overpartition,” Annals of Combinatorics, vol. 9, no. 3, pp. 321–334, 2005.
- J. Lovejoy, “Partitions and overpartitions with attached parts,” Archiv der Mathematik, vol. 88, no. 4, pp. 316–322, 2007.
- K. Mahlburg, “The overpartition function modulo small powers of 2,” Discrete Mathematics, vol. 286, no. 3, pp. 263–267, 2004.
- N. Robbins, “Some properties of overpartitions,” JP Journal of Algebra, Number Theory and Applications, vol. 3, no. 3, pp. 395–404, 2003.
- Ø. J. Rødseth and J. A. Sellers, “On -ary overpartitions,” Annals of Combinatorics, vol. 9, no. 3, pp. 345–353, 2005.
- G. E. Andrews, The Theory of Partitions, vol. 2 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Mass, USA, 1976.
- G. E. Andrews, The Theory of Partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1998.
- H. S. Zuckerman, “On the coefficients of certain modular forms belonging to subgroups of the modular group,” Transactions of the American Mathematical Society, vol. 45, no. 2, pp. 298–321, 1939.
- L. Goldberg, Transformation of theta-functions and analogues of Dedekind sums, Ph.D. thesis, University of Illinois, Urbana, Ill, USA, 1981.
- H. Rademacher, “On the expansion of the partition function in a series,” Annals of Mathematics, vol. 44, pp. 416–422, 1943.
- H. Rademacher, Lectures on Analytic Number Theory, Tata Institute, Bombay, India, 1954-1955.
- H. Rademacher, Topics in Analytic Number Theory, vol. 169 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 1973.
- T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, vol. 41 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1990.
- L. R. Ford, “Fractions,” The American Mathematical Monthly, vol. 45, no. 9, pp. 586–601, 1938.
- E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, UK, 4th edition, 1927.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 2nd edition, 1944.