International Journal of Mathematics and Mathematical SciencesVolume 2011 (2011), Article ID 976723, 15 pagesdoi:10.1155/2011/976723

Research Article

A Rademacher-Type Formula for

Vyacheslav Kiria-Kaiserberg

Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30458, USA

Received 24 May 2011; Accepted 7 September 2011

Academic Editor: Marianna Shubov

Copyright © 2011 Vyacheslav Kiria-Kaiserberg. 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.

Abstract.

A Rademacher-type formula for the Fourier coefficients of the generating function for the partitions of where no odd part is repeated is presented.

1. Partitions

A partition of a positive integer is a representation of as a sum of positive integers where order of summands (parts) does not matter. Let represent the number of partitions of . In 1937, Rademacher [1, 2] was able to express as a convergent series: where is a Kloosterman sum and is a Dedekind sum.

In 2011, Bruinier and Ono [3] announced a new formula that expresses as a finite sum.

1.1. Formula for

Let be Euler's generating function for . H. Rademacher used the classical circle method to find the coefficients of . There are many other infinite products to which this method could be applied. We introduce one of these infinite products here and derive the formula for the coefficients of . Define Let denote the coefficient of in the expansion of , that is,

We will find a closed expression for . Note that where equals the number of partitions of where no odd part is repeated. Thus which is simpler than the one given by Sills [4, page 4, Equation (1.13)] in 2010: where is defined as

2. Evaluation of the Path Integral

2.1. Convergence and Cauchy Residue Theorem

Considering as a complex variable in we see from the right-hand side that infinite product and thus also infinite series are convergent for since is a geometric series which converges for for any fixed .

Next, we note that from we get that

The series on the right side of (2.4) is a Laurent series of . It has a pole of order at with residue . Applying Cauchy's Residue Theorem we get that where is any positively oriented simple closed countour lying inside the unit circle.

2.2. Change of the Variable

The change of the variable maps the unit disk into an infinite vertical strip of width 1 in the -plane. To see this we note that from we get , so . Choosing the branch cut to be , we get

As traverses a circle centered at of radius in the positive direction, the point varies from to along a horizontal segment as could be easly deduced from (2.6).

Replacing the segment by the Rademacher path composed of upper arcs of the Ford circles formed by the Farey series , (2.5) becomes which simplifies to The above can be written as where is the upper arc of the Ford circle .

2.3. Another Change of the Variable

Consider another change of variable

so that

Under this transformation the Ford circle in the -plane with center at and radius is mapped to a negatively oriented circle in the -plane with center at and radius . This follows from the fact that any point on the Ford circle is given by

Substitution of (2.13) into (2.11) gives which is a circle centered at with radius . Now we make change of variable in (2.9). This gives where

are initial and terminal points, respectively.

2.4. Modular Transformation

Next, we note that where is the Dedekind eta function. Rewriting modular functional equation [5, page 96] for in terms of we get with , .

To evaluate (2.15) we would like to express in the same way we did for above. Two cases have to be considered: and . When we will replace by and by , and when , will be replaced by in order to obtain from . Hence, we have which simplifies to where and for .

We return to evaluation of (2.15). To proceed we note that Rewriting (2.15) in terms of (2.21) and (2.22) we obtain where

2.5. Estimation of the First Term

We will estimate the first term in (2.23) and will show that it is small for large . To do this we change variable again by letting . Then the first term in (2.23) becomes where are initial and terminal points obtained from (2.16), respectively. Under this change of variable circle in -plane with center at and radius is mapped to a circle in -plane centered at with radius . Note also that the mapping maps the circle and its interior onto a half-plane (where denotes the real part of complex variable and is the imaginary part). From elementary complex analysis we have that and , where . It is readily seen that the segment in the -plane is mapped to an infinite strip in the -plane. So, it follows that inside and on the circle we have that and . We now show that on the circle . To see this note that in the polar form on , . From this we get that So, .

Furthermore, we may move path of integration from the arc joining and to a segment connecting these two points on the circle . By [5, page 104], Theorem  5.9 the length of the path of integration is bounded by , and on the segment connecting and , .

Next, let us define by which is a part of the integrand in (2.25). Then, estimating the integrand in (2.25) we get where Note that does not depend on or . It depends on , but remains fixed in the above analysis. So, for some constant , and we have that

This completes the estimation of the first term in (2.23). We proceed to the second term.

2.6. Estimation of the Second Term

First, we will show that is small for large . Making change of variable as before, we get that where and are as in (2.26), respectively. As before, we define by

Then, estimating the integrand, we see that where Note that does not depend on or . It depends on , but is fixed. It follows, therefore, that for some constant . Then we have that

Combining the results from (2.33) and (2.40) we have that

Finally, we turn our attention to

We note that where is a circle in the -plane centered at with radius , as before. It is easily seen that the length of the arc connecting 0 and is less then From the discussion above we know that and on . So, the integrand in could be estimated as

2.7. Combining the Results

We combine the results in (2.44) and (2.45) to get where is a constant. We can obtain similar estimate for and, as before, we get an error term in the formula for . Therefore, we can write Letting we have that

We introduce another change of variable

Then (2.48) becomes Let in (2.50), then the above becomes

2.8. Bessel Function

In Watson's Treatise on Bessel functions [6, page 181], we find a formula equivalent to the following: Let

and . Then we have

Note that Bessel functions of this order can be expressed as Expanding (2.55) we have that

Substituting (2.53) into (2.56), we get Multiplying (2.57) by we get

2.9. Final Form

Finally, we rewrite (2.54) in terms of (2.59) to get Thus, or equivalently

Acknowledgment

The authors wishes to express gratitude and appreciation to Dr. A. Sills for his suggestions and criticism.

References

H. Rademacher, “On the expansion of the partition function in a series,” Annals of Mathematics, vol. 44, no. 2, pp. 416–422, 1943.

H. Rademacher, “On the partition function ,” Proceedings of the London Mathematical Society, vol. 43, no. 2, pp. 241–254, 1937.

J. Bruinier and K. Ono, “An algebraic formula for the partition function,” http://www.aimath.org/news/partition/brunier-ono

A. V. Sills, “A Rademacher type formula for partitions and overpartitions,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 630458, 21 pages, 2010.

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.

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 2nd edition, 1944.