A Rademacher-Type Formula for pod n

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


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

International Journal of Mathematics and Mathematical Sciences
In 2011, Bruinier and Ono 3 announced a new formula that expresses p n as a finite sum.

Formula for p n
1.4 be Euler's generating function for p n .H. Rademacher used the classical circle method to find the coefficients of q n .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 q n .Define Let p j denote the coefficient of q j in the expansion of G q , that is, G q ∞ j 0 p j q j .1.6 We will find a closed expression for p j .Note that pod n q n , 1.7 where pod n equals the number of partitions of n where no odd part is repeated.Thus International Journal of Mathematics and Mathematical Sciences 3 which is simpler than the one given by Sills 4, page 4, Equation 1.13 in 2010: 1.9 where ω h, k is defined as

Convergence and Cauchy Residue Theorem
Considering q as a complex variable in we see from the right-hand side that infinite product and thus also infinite series are convergent for |q| < 1 since is a geometric series which converges for |q| < 1 for any fixed k ≥ 1.
Next, we note that from we get that G q q n 1 ∞ j 0 p j q j q n 1 if 0 < q < 1.

International Journal of Mathematics and Mathematical Sciences
The series on the right side of 2.4 is a Laurent series of G q /q n 1 .It has a pole of order n 1 at q 0 with residue p n .Applying Cauchy's Residue Theorem we get that where C is any positively oriented simple closed countour lying inside the unit circle.

Change of the Variable
The change of the variable q e 2πiτ maps the unit disk |q| < 1 into an infinite vertical strip of width 1 in the τ-plane.To see this we note that from q e 2πiτ we get log q 2πiτ, so τ log q/2πi.Choosing the branch cut to be 0, 1 , we get As q traverses a circle centered at q 0 of radius e −2π in the positive direction, the point τ varies from i to i 1 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 F N , 2.5 becomes

2.8
The above can be written as f e 2πiτ e −2πiτn dτ, 2.9 where γ h, k is the upper arc of the Ford circle C h, k .

Another Change of the Variable
Consider another change of variable dz −ik dτ.

2.12
Under this transformation the Ford circle C h, k in the τ-plane with center at h/k i1/2k 2 and radius 1/2k 2 is mapped to a negatively oriented circle C k in the z-plane with center at 1/2k and radius 1/2k.This follows from the fact that any point on the Ford circle C h, k is given by Substitution of 2.13 into 2.11 gives which is a circle centered at 1/2k with radius 1/2k.Now we make change of variable in 2.9 .This gives where are initial and terminal points, respectively.

Modular Transformation
Next, we note that f q f e 2πiτ e πiτ/12 η τ , 2.17 where η τ is the Dedekind eta function.Rewriting modular functional equation 5, page 96 for η τ in terms of f q f e 2πiτ f e 2πih/k−2πz/k we get To evaluate 2.15 we would like to express in the same way we did for f q above.Two cases have to be considered: k, 2 1 and k, 2 2. When k, 2 1 we will replace h by 2h and z by 2z, and when k, 2 2, k will be replaced by k/2 in order to obtain f q 2 from f q .Hence, we have

2.21
where hH j ≡ −1 modk and j | H j for j 1, 2. We return to evaluation of 2.15 .To proceed we note that

2.22
Rewriting 2.15 in terms of 2.21 and 2.22 we obtain

Estimation of the First Term
We will estimate the first term in 2.23 and will show that it is small for large N. To do this we change variable again by letting ξ zk.Then the first term in 2.23 becomes e πξ/k 2 2n−1/4 dξ,

2.25
where k and its interior onto a half-plane R w ≥ 1 where R w denotes the real part of complex variable w and I w is the imaginary part .From elementary complex analysis we have that R w x/ x 2 y 2 and I w −y/ x 2 y 2 , where x iy ξ.It is readily seen that the segment 0 < x ≤ 1 in the ξ-plane is mapped to an infinite strip 1, ∞ in the w-plane.So, it follows that inside and on the circle C * k we have that 0 < R ξ ≤ 1 and R 1/ξ ≥ 1.We now show that R 1/ξ 1 on the circle C * k .To see this note that in the polar form ξ 1/2 1/2 e iθ on C * k , 0 ≤ θ ≤ 2π.From this we get that

2.28
So, R 1/ξ 1.Furthermore, we may move path of integration from the arc joining s * h,k and t * h,k to a segment connecting these two points on the circle C * k .By 5, page 104 , Theorem 5.9 the length of the path of integration is bounded by 2 √ 2k/N, and on the segment connecting s * h,k which is a part of the integrand in 2.25 .Then, estimating the integrand in 2.25 we get p * m y m .

2.31
Note that c does not depend on ξ or N. It depends on n, but n remains fixed in the above analysis.So,

2.32
for some constant α, and we have that 2i

2.33
This completes the estimation of the first term in 2.23 .We proceed to the second term.

Estimation of the Second Term
First, we will show that is small for large N. Making change of variable ξ zk as before, we get that

2.36
Then, estimating the integrand, we see that International Journal of Mathematics and Mathematical Sciences 11

2.38
Note that b does not depend on ξ or N. It depends on n, but n is fixed.It follows, therefore, that for some constant β.Then we have that

2.40
Combining the results from 2.33 and 2.40 we have that

2.42
We note that where C * k is a circle in the ξ-plane centered at 1/2 with radius 1/2, as before.It is easily seen that the length of the arc connecting 0 and s * h,k is less then

Combining the Results
We combine the results in 2.44 and 2.45 to get where γ is a constant.We can obtain similar estimate for S 2 and, as before, we get an error term O N −1/2 in the formula for p n .Therefore, we can write

2.47
Letting N → ∞ we have that

2.48
We introduce another change of variable

2.50
Let t πw/4 in 2.50 , then the above becomes 2.51

Bessel Function
In Watson's Treatise on Bessel functions 6, page 181 , we find a formula equivalent to the following: 2.53 and ν 3/2.Then we have

6 International
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∞ m 1 p * m e −2πm |ξ| 1 /2 e 2πn ∞ m 1 p * m y m , where y e −2π c|ξ| 1
International Journal of Mathematics and Mathematical Sciences in ξ-plane centered at 1/2 with radius 1/2.Note also that the mapping w 1/ξ maps the circle C * 27are initial and terminal points obtained from 2.16 , respectively.Under this change of variable circle C k in z-plane with center at 1/2k and radius 1/2k is mapped to a circle C * k