Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 976723, 15 pages
doi:10.1155/2011/976723
Research Article
A Rademacher-Type Formula for podn
Vyacheslav Kiria-Kaiserberg
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30458, USA
Correspondence should be addressed to Vyacheslav Kiria-Kaiserberg, vslkrg@yahoo.com
Received 24 May 2011; Accepted 7 September 2011
Academic Editor: Marianna Shubov
Copyright q 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.
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.
1. 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 pn represent the number of partitions of n.
In 1937, Rademacher 1, 2 was able to express pn as a convergent series:
⎛
⎞
∞
sinh
−
1/24
2/3n
π/k
d ⎜
1
⎟
Ak n k ⎝
pn √
⎠,
dn
π 2 k1
n − 1/24
1.1
where
Ak n eπish,k−2nh/k
0≤h<kgcdh,k1
1.2
is a Kloosterman sum and
sh, k k−1 r hr
r1
is a Dedekind sum.
k
hr
1
−
−
k
k
2
1.3
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In 2011, Bruinier and Ono 3 announced a new formula that expresses pn as a finite
sum.
1.1. Formula for pn
Let
∞
∞
pnqn f q :
1
1
−
qm
m1
n0
1.4
be Euler’s generating function for pn. H. Rademacher used the classical circle method to
find the coefficients of qn . 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 qn . Define
2 2
∞
1 − qm
f q
G q :
.
2
2m
f q
m1 1 − q 1.5
Let pj denote the coefficient of qj in the expansion of Gq, that is,
∞
G q p j qj .
1.6
j0
We will find a closed expression for pj. Note that
∞
∞
1 q2m−1 podnqn ,
G −q 2m
1
−
q
n0
m1
1.7
where podn equals the number of partitions of n where no odd part is repeated. Thus
podn −1n pn
−1n
∞
2 ω2 h, k/2 −2πinh/k d
e
k
π k1k,22
ωh, k
dn
0≤h<kh,k1
⎞
√
sinh π 8n − 1/2k
⎟
⎜
×⎝
√
⎠
8n − 1
⎛
1.8
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3
which is simpler than the one given by Sills 4, page 4, Equation 1.13 in 2010:
∞
ωh, kω4h/k, 4, k/k, 4
2
4, k
podn k 1 − −1k π k1
4
ω2h/k, 2, k/k, 2
0≤h<kh,k1
⎛
⎞
sinh
π
48n
−
1/4k
k,
d ⎜
⎟
× e−2πinh/k
√
⎝
⎠,
dn
8n − 1
1.9
where ωh, k is defined as
ωh, k exp πi
hr
1
.
−
−
k
k
2
k−1 r hr
r1
k
1.10
2. Evaluation of the Path Integral
2.1. Convergence and Cauchy Residue Theorem
Considering q as a complex variable in
∞
∞
∞
1 − qm
1
1
m ,
2
m
2m
2m
1−q
1 − q2m
m1 1 − q
m1 1 q
m1 1 − −q
2.1
we see from the right-hand side that infinite product and thus also infinite series are
convergent for |q| < 1 since
∞ n
qk 1
,
1 − qk
n0
2.2
is a geometric series which converges for |q| < 1 for any fixed k ≥ 1.
Next, we note that from
∞
p j qj ,
G q 2.3
j0
we get that
∞ p
G q
j qj
n1
qn1
j0 q
if 0 < q < 1.
2.4
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The series on the right side of 2.4 is a Laurent series of Gq/qn1 . It has a pole of
order n 1 at q 0 with residue pn. Applying Cauchy’s Residue Theorem we get that
1
pn 2πi
C
G q
1
dq 2πi C
qn1
2 2
f q
dq,
f q qn1
2.5
where C is any positively oriented simple closed countour lying inside the unit circle.
2.2. Change of the Variable
The change of the variable q e2πiτ 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 e2πiτ we get log q 2πiτ, so
τ log q/2πi. Choosing the branch cut to be 0, 1, we get
logq Arg q
.
τ
2πi
2π
2.6
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 FN , 2.5 becomes
1
pn 2πi
i1 4πiτ 2
2πie2πiτ
f e
i
fe2πiτ e2πiτn1
dτ,
2.7
which simplifies to
i1 4πiτ 2
f e
pn e−2πiτn dτ,
fe2πiτ i
4πiτ 2
f e
e−2πiτn dτ.
2πiτ P N fe
2.8
The above can be written as
P N
4πiτ 2
4πiτ 2
N
f e
f e
−2πiτn
e
e−2πiτn dτ,
dτ fe2πiτ fe2πiτ k1 0≤h<kh,k1 γh,k
where γh, k is the upper arc of the Ford circle Ch, k.
2.9
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5
2.3. Another Change of the Variable
Consider another change of variable
τ
h iz
,
k k
2.10
so that
h
z −ik τ −
k
2.11
dz −ik dτ.
2.12
Under this transformation the Ford circle Ch, k in the τ-plane with center at h/k i1/2k2 and radius 1/2k2 is mapped to a negatively oriented circle Ck 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
Ch, k is given by
τ
1
h
i 2
k
2k
1 iθ
e ,
2k2
0 ≤ θ < 2π.
2.13
Substitution of 2.13 into 2.11 gives
z
1 iθ 1
−ie ,
2k 2k
2.14
which is a circle centered at 1/2k with radius 1/2k. Now we make change of variable in 2.9.
This gives
pn i
N
k−1
k1
e−2πinh/k
0≤h<k h,k1
th,k 4πih/k−4πiz/k 2
f e
2πih/k−2πz/k e2πnz/k dz,
f
e
sh,k
2.15
where
sh,k th,k
k2
kp
k
i,
2
2
kp k kp2
k
ks
−
i
2
2
2
k ks k ks2
are initial and terminal points, respectively.
2.16
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2.4. Modular Transformation
Next, we note that
eπiτ/12
,
f q f e2πiτ ητ
2.17
where ητ is the Dedekind eta function. Rewriting modular functional equation 5, page 96
for ητ in terms of fq fe2πiτ fe2πih/k−2πz/k we get
f e
2πih/k−2πz/k
−1
√
π z −z
iz−1 H
ωh, k exp
,
zf exp 2πi
12k
k
2.18
with hH ≡ −1modk, h, k 1.
To evaluate 2.15 we would like to express
f e4πih/k−4πiz/k 2
2πiτ
2πih/k−2πz/k
G e
G q G e
,
f e2πih/k−2πz/k
2.19
in the same way we did for fq 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 fq2 from fq. Hence, we have
⎧
−1
−1
⎪
ω2 2h, keπ2z −2z/6k 2zf 2 e2πii2z H2 /k
⎪
⎪
⎪
⎪
,
√ ⎨
⎪
ωh, keπz−1 −z/12k zf e2πiiz−1 H2 /k
2πih/k−2πz/k
G e
⎪
2
πz−1 −z/3k
2
4πiiz−1 H1 /k
⎪
e
zf
ω
k/2e
h,
⎪
⎪
⎪
⎪
,
√ ⎩
ωh, keπz−1 −z/12k zf e2πiiz−1 H1 /k
if k, 2 1,
if k, 2 2,
2.20
which simplifies to
⎧
2
2πii2z−1 H2 /k
⎪
e
f
2
⎪
⎪ ω 2h, k −πz/4k √
e
z 2πiiz−1 H /k ,
⎨2
⎪
2
ωh, k
2πih/k−2πz/k
f e
G e
⎪
⎪ ω2 h, k/2
√ ⎪
−1
−1
⎪
⎩
eπz −z/4k zG e2πiiz H1 /k ,
ωh, k
if k, 2 1,
2.21
if k, 2 2,
where hHj ≡ −1modk and j | Hj for j 1, 2.
We return to evaluation of 2.15. To proceed we note that
−1
−1
G e2πiiz H1 /k 1 G e2πiiz H1 /k − 1 .
2.22
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7
Rewriting 2.15 in terms of 2.21 and 2.22 we obtain
N
pn i
k−1
k1k,21
N
i
2
0≤h<kh,k1
k−1
k1k,22
ω 2h, k −2πinh/k
e
ωh, k
2
2
2πii2z−1 H2 /k
√ f e
z 2πiiz−1 H /k eπz/k2n−1/4 dz
2
f e
sh,k
th,k
ω2 h, k/2 −2πinh/k
e
ωh, k
0≤h<kh,k1
th,k
√ −1
z 1 G e2πiiz H1 /k − 1
sh,k
× eπz/k2n−1/4π/4zk dz
N
2i
k−1
k1k,21
N
i
ω 2h, k −2πinh/k
e
ωh, k
0≤h<kh,k1
k−1
k1k,22
2
2
2πii2z−1 H2 /k
e
f
√
z 2πiiz−1 H /k eπz/k2n−1/4 dz
2
f e
sh,k
th,k
ω2 h, k/2 −2πinh/k
e
J1 h, k J2 h, k,
ωh, k
0≤h<kh,k1
2.23
where
J1 h, k J2 h, k th,k √
sh,k
th,k
zeπz/k2n−1/4π/4zk dz ,
√ 2πiiz−1 H1 /k − 1 eπz/k2n−1/4π/4zk dz.
z G e
2.24
sh,k
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 N. To do this
we change variable again by letting ξ zk. Then the first term in 2.23 becomes
2i
N
k−5/2
k1k,21
ω 2h, k −2πinh/k
e
ωh, k
0≤h<kh,k1
2
t∗
h,k
s∗h,k
−1
f 2 e2πii2ξ/k H2 /k
2
ξ eπξ/k 2n−1/4 dξ,
−1
2πiiξ/k
H
/k
2
f e
2.25
where
s∗h,k kkp
k2
i,
2
2
2
k kp k kp2
2.26
t∗h,k k2
kks
−
i,
2
2
2
k ks k ks2
2.27
are initial and terminal points obtained from 2.16, respectively. Under this change of
variable circle Ck in z-plane with center at 1/2k and radius 1/2k is mapped to a circle Ck∗
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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 Ck∗ and its interior onto a half-plane Rw ≥ 1 where Rw denotes the real part of
complex variable w and Iw is the imaginary part. From elementary complex analysis we
have that Rw x/x2 y2 and Iw −y/x2 y2 , 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 Ck∗ we have that 0 < Rξ ≤ 1 and R1/ξ ≥ 1. We now
show that R1/ξ 1 on the circle Ck∗ . To see this note that in the polar form ξ 1/2 1/2eiθ
on Ck∗ , 0 ≤ θ ≤ 2π. From this we get that
1
2
2
ξ 1 eiθ 1 cos θ i sin θ
21 cos θ − i sin θ
1 cos θ2 sin2 θ
21 cos θ
2 sin θ
−i
2 2 cos θ
2 2 cos θ
1−i
2.28
sin θ
.
1 cos θ
So, R1/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 Ck∗ . 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
√
and t∗h,k , |ξ| < 2k/N.
Next, let us define p∗ m by
−1
f 2 e2πii2ξ/k H2 /k
p∗ mqm ,
−1
f e2πiiξ/k H2 /k
m0
∞
2.29
which is a part of the integrand in 2.25. Then, estimating the integrand in 2.25 we get
2
2πii2ξ/k−1 H2 /k f
e
ξeπξ/k2 2n−1/4 × −1 −1
2πiiξ/k
H
/k
2
f e
⎛
⎞
−1
∞
H
2πim
iξ/k
2
2
⎟
⎜
|ξ|1/2 eπξ/k 2n−1/4 × p∗ m exp⎝
⎠
m1
k
∞
2πm
2πimH2 1/2 πξ/k2 2n−1/4 ∗
|ξ| e
exp
× p m exp −
m1
ξ
k
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∞
2πm
2πimH
2
2
≤ |ξ|1/2 eπ/k 2n−1/4Rξ p∗ m exp −
exp
m1
ξ
k
≤ |ξ|
1/2 2πn
e
≤ |ξ|1/2 e2πn
9
∞ p∗ m exp −2πmR 1
ξ
m1
∞ p∗ me−2πm
m1
|ξ|1/2 e2πn
∞ p∗ mym ,
where y e−2π
m1
c|ξ|1/2 ,
2.30
where
c e2πn
∞ p∗ mym .
2.31
m1
Note that c does not depend on ξ or N. It depends on n, but n remains fixed in the above
analysis. So,
∗
t h,k
s∗ h,k
−1
√ 1/2 √
f 2 e2πii2ξ/k H2 /k
2k
2 2N
1/2
πξ/k2 2n−1/4
ξ dξ ≤ c|ξ| ≤ c
e
−1
N
N
2πiiξ/k
H
/k
2
f e
2.32
< αk3/2 N −3/2 ,
for some constant α, and we have that
∗
N
ω2 2h, k −2πinh/k t h,k
−5/2
2i
e
k
ωh, k
k1k,21
s∗ h,k
0≤h<kh,k1
−1
f 2 e2πii2ξ/k H2 /k
πξ/k2 2n−1/4
ξ e
dξ
−1
2πiiξ/k
H
/k
2
f e
≤2
N
2.33
αk−1 N −3/2
k1k,210≤h<kh,k1
≤ 2αN −3/2
N
1 2αN −1/2 .
k1k,21
This completes the estimation of the first term in 2.23. We proceed to the second term.
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International Journal of Mathematics and Mathematical Sciences
2.6. Estimation of the Second Term
First, we will show that
J2 h, k th,k
√ 2πiiz−1 H1 /k − 1 eπz/k2n−1/4π/4zk dz
z G e
2.34
sh,k
is small for large N. Making change of variable ξ zk as before, we get that
J2 h, k k
−3/2
t∗
h,k
s∗h,k
−1
2
ξ G e2πiiξ/k H1 /k − 1 eπξ/k 2n−1/4π/4ξ dξ,
2.35
where s∗h,k and t∗h,k are as in 2.26, respectively. As before, we define p∗∗ m by
∞
−1
p∗∗ mqm G e2πiiξ/k H1 /k − 1.
m0
Then, estimating the integrand, we see that
ξeπξ/k2 2n−1/4π/4ξ × G e2πiiξ/k−1 H1 /k − 1
∞
2πimik/ξ H1 1/2 πξ/k2 2n−1/4 π/4ξ ∗∗
|ξ| e
e
− 1
× p m exp
m0
k
∞
−2πm
2πimH1 1/2 π/k2 2n−1/4Rξ π/4R1/ξ ∗∗
≤ |ξ| e
e
exp
p m exp
m1
ξ
k
≤ |ξ|1/2 e2πn eπ/4R1/ξ
∞ p∗∗ m exp −2πmR 1
ξ
m1
|ξ|1/2 e2πn
∞ p∗∗ m exp −2πm π R 1
4
ξ
m1
≤ |ξ|1/2 e2πn
∞ p∗∗ m exp −2πm π
4
m1
|ξ|1/2 e2πn
∞ p∗∗ me−π/48m−1
m1
≤ |ξ|1/2 e2πn
∞ p∗∗ 8m − 1e−π/48m−1
m1
2.36
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|ξ|1/2 e2πn
∞ p∗∗ 8m − 1x8m−1 ,
11
where x e−π/4
m1
b|ξ|1/2 ,
2.37
where
∞ p∗∗ 8m − 1x8m−1 .
b e2πn
2.38
m1
Note that b does not depend on ξ or N. It depends on n, but n is fixed. It follows, therefore,
that
√
2k
N
|J2 h, k| ≤ b
1/2 √
2 2N
< βk3/2 N −3/2 ,
N
2.39
for some constant β. Then we have that
2πinh
N
N
2
−
ω h, k/2
−5/2
k J2 h, k <
e
k
βk−1 N −3/2
i
k1k,22
ωh,
k
0≤h<kh,k1
k1k,220≤h<kh,k1
N
≤ βN −3/2
2.40
1 βN −1/2 .
k1k,22
Combining the results from 2.33 and 2.40 we have that
N
pn i
k−5/2
k1k,22
N
i
k
k1k,22
−5/2
ω2 h, k/2 −2πinh/k
e
J1 h, k O βN −1/2 2αN −1/2
ωh, k
0≤h<kh,k1
ω2 h, k/2 −2πinh/k
e
J1 h, k O N −1/2 .
ωh, k
0≤h<kh,k1
2.41
Finally, we turn our attention to
J1 h, k k−3/2
t∗
h,k
ξeπξ/k
s∗h,k
2
2n−1/4π/4ξ
dξ.
2.42
We note that
J1 h, k Ck∗
−
0
s∗h,k
−
t∗
h,k
0
Ck∗
−S1 − S2 ,
2.43
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International Journal of Mathematics and Mathematical Sciences
where Ck∗ 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
2π
∗ sh,k 2
√ k
≤ π s∗h,k ≤ π 2 .
N
2.44
From the discussion above we know that R1/ξ 1 and 0 < Rξ ≤ 1 on Ck∗ . So, the integrand
in S1 could be estimated as
ξeπξ/k2 2n−1/4π/4ξ |ξ|1/2 eπξ/k2 2n−1/4 eπ/4ξ |ξ|1/2 eπ/k
≤ 21/4
2
2n−1/4Rξ π/4R1/ξ
e
2.45
k1/2 2πn π/4
e e .
N 1/2
2.7. Combining the Results
We combine the results in 2.44 and 2.45 to get
|S1 | < γk3/2 N −3/2 ,
2.46
where γ is a constant. We can obtain similar estimate for S2 and, as before, we get an error
term ON −1/2 in the formula for pn. Therefore, we can write
N
pn i
k−5/2
k1k,22
×
Ck∗
ξe
ω2 h, k/2 −2πinh/k
e
ωh, k
0≤h<kh,k1
πξ/k2 2n−1/4π/4ξ
dξ O N
−1/2
2.47
.
Letting N → ∞ we have that
∞
pn i
k−5/2
k1k,22
×
Ck∗
ξe
ω2 h, k/2 −2πinh/k
e
ωh, k
0≤h<kh,k1
πξ/k2 2n−1/4π/4ξ
2.48
dξ.
We introduce another change of variable
ξ
1
,
w
dξ −
1
.
w2
2.49
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13
Then 2.48 becomes
∞
1 ω2 h, k/2 −2πinh/k
e
k−5/2
i k1k,22
ωh, k
0≤h<kh,k1
pn ×
1∞i
w−5/2 eπ/k
2
2n−1/41/wπw/4
2.50
dw.
1−∞i
Let t πw/4 in 2.50, then the above becomes
pn 2π
×
π 3/2
8
∞
k−5/2
k1k,22
π/4∞i
t−5/2 etπ
2
ω2 h, k/2 −2πinh/k 1
e
ωh, k
2πi
0≤h<kh,k1
/4k 2n−1/41/t
2
2.51
dt.
π/4−∞i
2.8. Bessel Function
In Watson’s Treatise on Bessel functions 6, page 181, we find a formula equivalent to the
following:
c∞i
1/2zν
2πi
Iν z t−ν−1 etz /4t dt,
2
if c > 0,
Rν > 0.
2.52
c−∞i
Let
z
2
!
"1/2
π2
1
2n −
4
4k2
2.53
and ν 3/2. Then we have
pn 2π
π 3/2
8
∞
k−5/2
k1k,22
ω2 h, k/2 −2πinh/k
e
ωh, k
0≤h<kh,k1
⎛ π
π −3/2 2n − 1/4−3/4
×
I3/2 ⎝
2n −
−3/4
−3/2
k
4
k
⎞
1⎠
4
⎛ ∞
2
2π2n − 1/4−3/4 π
ω
k/2
h,
e−2πinh/k I3/2 ⎝
2n −
k−1
√
ωh, k
k
8
k1k,22
0≤h<kh,k1
⎞
1⎠
.
4
2.54
14
International Journal of Mathematics and Mathematical Sciences
Note that Bessel functions of this order can be expressed as
I3/2 z 2z d sinh z
.
π dz
z
2.55
2z cosh z sinh z
−
.
π
z
z2
2.56
Expanding 2.55 we have that
I3/2 z Substituting 2.53 into 2.56, we get
√
π
π 8n − 1
1 1/2
I3/2
2n −
I3/2 z I3/2
k
4
2k
#
⎛
⎞
√
√
$ √
$ 2 π 8n − 1/2k
8n
−
1/2k
8n
−
1/2k
cosh
π
sinh
π
%
⎜
⎟
− √
⎝ √
2 ⎠
π
π 8n − 1/2k
π 8n − 1/2k
⎛
√
⎞
√
8n
−
1/2k
2
cosh
π
8n
−
1/2k
4k/π
sinh
π
8n − 1 ⎜
⎟
−
√
⎝
⎠
π√
π
k
8n − 1
8n − 1
k
k
⎛
⎞
√
√
8n
−
1/2k
4k
sinh
π
π 8n − 1
1
⎟
⎜
−
√
⎠.
⎝2 cosh
√
2k
π 8n − 1
π
8n − 1/k
2.57
1/4
Multiplying 2.57 by
2π2n − 1/4−3/4
2π
,
√
8
8n − 13/4
2.58
we get
√
√
√
2 2 cosh π 8n − 1/2k − 4k sinh π 8n − 1/2k /π 8n − 1
8n − 13/4
√
8n − 1/k
.
2.59
International Journal of Mathematics and Mathematical Sciences
15
2.9. Final Form
Finally, we rewrite 2.54 in terms of 2.59 to get
∞
pn k−1
k1k,22
×
ω2 h, k/2 −2πinh/k
e
ωh, k
0≤h<kh,k1
√
√
√
2 2 cosh π 8n − 1/2k − 4k sinh π 8n − 1/2k /π 8n − 1
8n − 13/4
√
8n − 1/k
2.60
.
Thus,
podn −1n
∞
k−1
k1k,22
×
ω2 h, k/2 −2πinh/k
e
ωh, k
0≤h<kh,k1
√
√
√
2 2 cosh π 8n − 1/2k − 4k sinh π 8n − 1/2k /π 8n − 1
8n − 13/4
√
8n − 1/k
2.61
,
or equivalently
podn −1n
∞
2
π k1k,22
⎛
⎞
√
8n
−
1/2k
sinh
π
ω h, k/2 −2πinh/k d ⎜
⎟
e
k
√
⎝
⎠.
ωh,
k
dn
8n
−
1
0≤h<kh,k1
2
2.62
Acknowledgment
The authors wishes to express gratitude and appreciation to Dr. A. Sills for his suggestions
and criticism.
References
1 H. Rademacher, “On the expansion of the partition function in a series,” Annals of Mathematics, vol. 44,
no. 2, pp. 416–422, 1943.
2 H. Rademacher, “On the partition function pn,” Proceedings of the London Mathematical Society, vol. 43,
no. 2, pp. 241–254, 1937.
3 J. Bruinier and K. Ono, “An algebraic formula for the partition function,” http://www.aimath.org/
news/partition/brunier-ono.
4 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.
5 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.
6 G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK,
2nd edition, 1944.
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