A Rademacher Type Formula for Partitions and Overpartitions

Volume 2010, Article ID 630458,21pages doi:10.1155/2010/630458

Research Article

A Rademacher Type Formula for

Partitions and Overpartitions

Andrew V. Sills

Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460-8093, USA

Correspondence should be addressed to Andrew V. Sills,asills@georgiasouthern.edu Received 3 July 2009; Revised 21 January 2010; Accepted 28 February 2010

Academic Editor: St´ephane Louboutin

Copyrightq2010 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 ofnand the Zuckerman formula for the Fourier coefficients ofϑ40|τ−1is presented.

1. Background

1.1. Partitions

A partition of an integernis a representation ofnas a sum of positive integers, where the order of the summands called partsis considered irrelevant. It is customary to write the parts in nonincreasing order. For example, there are three partitions of the integer 3, namely, 3, 21, and 111. Letpndenote the number of partitions ofn, with the convention that p0 1, and letfxdenote the generating function ofpn, that is, let

fx:

∞

n0

pnxn. 1.1

Euler1was the first to systematically study partitions. He showed that

fx

∞

m1 1

Euler also showed that

1 fx

∞

n−∞

−1n_xn3n−1/2_{, 1.3}

and since the exponents appearing on the right side of1.3are 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 aboveand indeed all the series and infinite products mentioned in this paperconverge absolutely when|x|< 1, which is important for analytic study of these series and products.

Hardy and Ramanujan were the first to study pn analytically and produced an incredibly accurate asymptotic formula2, page 85, equation1.74, namely,

pn 1

2π√2

α√n

k1

k

0h<k

h,k1

ωh, ke−2πihn/k d dn

⎛ ⎜ ⎝exp

π/k2/3n−1/24

n−1/24

⎞ ⎟ ⎠

On−1/4 ,

1.4

where

ωh, k exp

πi

k−1

r1 r k

hr

k −

hr k

−1

2

. 1.5

αis an arbitrary constant, and here and throughouth, kis an abbreviation for gcdh, k. Later Rademacher3improved upon1.4by finding the following convergent series representation forpn:

pn 1

π√2

∞

k1

k

0h<k

h,k1

ωh, ke−2πinh/k d dn

⎛ ⎜ ⎝sinh

π/k2/3n−1/24

n−1/24

⎞ ⎟

⎠. _1.6

Rademacher’s method was used extensively by many practitioners, including Grosswald 4, 5, Haberzetle 6, Hagis 7–15, Hua 16, Iseki 17–19, Lehner 20, Livingood21, Niven22, and Subramanyasastri23to study various restricted partitions functions.

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 Zhao25–43.

An overpartition of n is a representation of n 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 3 are 3, 3, 21, 21, 21, 21, 111, 111.

Letpndenote the number of overpartitions ofnand letfxdenote the generating function∞_n0pnxn_{ofpn. Elementary techniques are sufficient to show that}

fx

∞

m1 1xm 1−xm

fx2

fx2. 1.7

Note that

1 fx

∞

n−∞

−1n_xn2

. 1.8

via an identity of Gausssee equation2.2.12, page 23 in44,45, so that the reciprocal of the generating function for overpartitions is a series wherein a coefficient is nonzero if and only if the exponent ofxis 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 ofxis a pentagonal number.

Hardy and Ramanujan, writing more than 80 years before the coining of the term “overpartition,” stated2, page 109-110that the function which we are callingpn“has no very simple arithmetical interpretation; but the series is none the less, as the direct reciprocal of a simpleϑ-function, of particular interest.” They went on to state that

pn 1 4π

d dn

eπ√n

√ n

√

3 2π cos

2 3nπ−

1 6π

d dn

eπ√n/3 · · ·On−1/4 . 1.9

In fact,1.9was improved to the following Rademacher-type convergent series by Zuckerman46, page 321, equation8.53:

pn 1 2π

k1 2k

k

0h<k

h,k1

ωh, k2 ω2h, ke

−2πinh/k d

dn

sinhπ√n/k √

n

.

1.10

1.3. Partitions Where No Odd Part Is Repeated

Letpodndenote the number of partitions ofnwhere no odd part appears more than once. Letgxdenote the generating function ofpodn, so we have

gx

∞

n0

podnxn

∞

m1

1x2m−1 1−x2m

fxfx4

fx2 . 1.11

Via another identity of Gausssee equation2.2.13, page 23 in44,45, it turns out that

1 gx

∞

n0

−xnn1/2

∞

n−∞

−1n_x2n2_−n

; 1.12

so in this case the reciprocal of the generating function under consideration has nonzero coefficients at the exponents which are triangularor equivalently, hexagonalnumbers.

The analogous Rademacher-type formula forpodnis as follows:

podn 2 π

k1

k

1−−1k

k,4 4

×

0h<k

h,k1

ωh, k ω4h/k,4, k/k,4 ω2h/k,2, k/k,2 e

−2πinh/k

× d

dn ⎛ ⎜ ⎝sinh

πk,48n−1/4k √

8n−1

⎞ ⎟ ⎠.

1.13

Equation1.13is the caser2 ofTheorem 2.1to be presented in the next section.

2. A Common Generalization

Let us define

frx:

fxfx2r

whereris a nonnegative integer. Thus,

f0x fx

∞

n0 pnxn,

f1x fx

∞

n0 pnxn,

f2x gx

∞

n0

podnxn_.

2.2

Letprndenote the coefficient ofxnin the expansion offrn, that is,

frx

∞

n0

prnxn. 2.3

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

frx

∞

m1 1xm

1−x2r_m 2.4

∞ m1

1

1−x2m−1_1−x2r_m 2.5

∞ m1

1 1−x2r−1_m

2r−1₋₁

λ1

1x2r−1mλ . 2.6

Thus,prnequals each of the following:

ithe number of overpartitions ofnwhere nonoverlined parts are multiples of 2r_by 2.4;

iithe number of partitions ofnwhere all parts are either odd or multiples of 2r _by 2.5, providedr1;

Theorem 2.1. Forr0,1,2,3,4,

prn 2

r1/2√₃ π

k1

k

k,2maxr,1₁

×

0h<k

h,k1

e−2πinh/kωh, kω2

r_{h, k}

ω2h, k

× d

dn ⎧ ⎪ ⎨ ⎪ ⎩

sinhπ

24n−2r₁₁₂r−1_/2r/2_·6k

√

24n−2r₁

⎫ ⎪ ⎬ ⎪ ⎭

√

3 π

r

j1r/2

22−jr/2 k1

k

k,2r₂j

×

0h<k

h,k1

e−2πinh/kωh, kω

2r−j_h,2−j_k

ωh, k/2

× d

dn ⎧ ⎪ ⎨ ⎪ ⎩

sinhπ

24n−2r₁₋₁₂2j−r_/6k

√

24n−2r₁

⎫ ⎪ ⎬ ⎪ ⎭.

2.7

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 of1.6are provided in49,50, Chapter 14, and51, Chapter 5.

Of fundamental importance is the path of integration to be used. In48, Rademacher improved upon his original proof of1.6given in3, 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 FN of proper Farey fractions of order N is the set of all h/k with h, k 1 and 0 h/k < 1, arranged in increasing order. Thus, for example, F4

{0/1, 1/4,1/3, 1/2, 2/3, 3/4}.

For a givenN, lethp,hs,kp, andksbe such thathp/kpis the immediate predecessor ofh/kandhs/ksis the immediate successor ofh/kinFN. It will be convenient to view each

3.2. Ford Circles and the Rademacher Path

Lethandkbe integers withh, k 1 and 0h < k. The Ford circle52Ch, kis the circle inCof radius 1/2k2_{centered at the point}

h k

1

2k2i. 3.1

The upper arcγh, kof the Ford circleCh, kis those points ofCh, kfrom the initial point

αIh, k: h

k − kp

kk2_k2 p

1

k2_k2 p

i 3.2

to the terminal point

αTh, k:

h k

ks

kk2_k2 s

1

k2_k2 s

i, 3.3

traversed in the clockwise direction. Note that we have

αI0,1 αTN−1, N. 3.4

Every Ford circle is in the upper half plane. For h1/k1, h2/k2 ∈ FN, Ch1, k1and Ch2, k2are either tangent or do not intersect.

The Rademacher pathPNof orderNis the path in the upper half of theτ-plane from itoi1 consisting of

h/k∈FN

γh, k _3.5

traversed left to right and clockwise. In particular, we consider the left half of the Ford circle C0,1and the corresponding upper arcγ0,1to be translated to the right by 1 unit. This is legal given then periodicity of the function which is to be integrated overPN.

3.3. Set Up the Integral

Letnandrbe fixed, withn >2r_−1/24. Since

frx

∞

n0

Cauchy’s residue theorem implies that

prn 1 2πi _C

frx

xn1 dx, 3.7

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

xe2πiτ 3.8

so that the unit disk|x| 1 in thex-plane maps to the infinitely tall, unit-wide strip in the τ-plane where 0 Rτ 1 andIτ 0. The contourCis then taken to be the preimage of PNunder the mapx→e2πiτ_.

Better yet, let us replacexwithe2πiτ _{in3.7to express the integration in theτ-plane:}

prn

PNfr

e2πiτ e−2πinτdτ

h/k∈FN γh,k fr

e2πiτ e−2πinτdτ

N k1

0h<k

h,k1

γh,kfr

e2πiτ e−2πinτdτ.

3.9

3.4. Another Change of Variable

Next, we change variables again, taking

τ izh

k , 3.10

so that

z−ik

τ− h k

. 3.11

Thus Ch, k in the τ-plane maps to the clockwise-oriented circle K_k− in the z-plane centered at 1/2kwith radius 1/2k.

So we now have

prn i N

k1 k−1

0h<k

h,k1

e−2πinh/k

zTh,k

zIh,k arc ofK_k−

e2nπz/kfr

e2πih/k−2πz/k dz, _3.12

So the transformation3.11maps the upper arcγh, kofCh, kin theτ-plane to the arc onK_k−which initiates at

zIh, k k

k2_k2 p

kp k2_k2

p

i 3.13

and terminates at

zTh, k k

k2_k2 s

− ks

k2_k2 s

i. 3.14

3.5. Exploiting a Modular Transformation

From the theory of modular forms, we have the transformation formula2, page 93, Lemma 4.31:

f

exp

2πih

k −

2πz k

ωh, kexp

πz−1_−z 12k

√ zf

exp

2πiiz−

1_H

k

,

3.15

where√zis the principal branch,h, k 1, andHis a solution to the congruence

hH≡ −1modk. 3.16

From3.15, we deduce the analogous transformation forfrx.

The transformation formula is a piecewise defined function with r 1 cases corresponding toj0,1,2, . . . , r, wherek,2r 2j:

fr

exp

2πih

k −

2πz k

ωh, kω

2r−j_h,2−j_k

ω2h/2−δj0

, k/2−δj0

exp

⎛ ⎜ ⎝π

222j−r1_−2−δ j0

2

24kz

π1−2rz 12k

⎞ ⎟ ⎠

×z 2r−j−1_2−δ j0

×f

exp−2π/kz2Hjπi/k

fexp−22j−r1π/kz22j−r1Hjπi/k

fexp−π2−δj0 2

/kzHjπ

2−δj0 2

i/k

,

whereHjis divisible by 2r−jand is a solution to the congruencehHj≡ −1modk, and

δj0 ⎧ ⎨ ⎩

1 ifj 0,

0 ifj /0 3.18

is the Kroneckerδ-function.

Notice that in particular, forr/2j r,3.17simplifies to

fr

exp

2πih

k −

2πz k

ωh, kω

2r−j_h,2−j_k

ωh, k/2 exp

π 12k

!

22j−r−1 z−1 1−2rz"

×z2r−j_f 2j−r

exp

₋ 2π

kz

2Hjπi

k

.

3.19

Since ther 0 case was established by Zuckerman, and ther1 case by Rademacher, we will proceed with the assumption thatr >1.

Apply3.17to3.12to obtain

prn i r

j0 N

k1

k,2r_j k−1

0h<k

h,k1

e−2πinh/k

× zTh,k

zIh,k arc ofK_k−

ωh, kω2r−j_h,2−j_k

ω2h/2−δj0

, k/2−δj0

×exp

⎛ ⎜ ⎝π

222j−r1_−2−δ j02

24kz

π24n1−2rz 12k

⎞ ⎟ ⎠

×z2r−j−1_2−δ j0

×f

exp−2π/kz2Hjπi/k

fexp−22j−r1π/kz22j−r1Hjπi/k

fexp−π2−δj02/kzHjπ2−δj02i/k

dz.

3.6. Normalization

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

prn i r

j0 N

k1

k,2r_j

k−5/2 0h<k

h,k1

e−2πinh/k

× ζTh,k

ζIh,k arc ofK−

ωh, kω2r−j_h,2−j_k

ω2h/2−δj0

, k/2−δj0 ×exp ⎛ ⎜ ⎝π

222j−r1_−2−δ j0

2

24ζ

π24n1−2r_ζ 12k2

⎞ ⎟ ⎠

×ζ2r−j−1_2−δ j0

×f

exp−2π/ζ2Hjπi/k

fexp−22j−r1_π/ζ22j−r1_H jπi/k

fexp−π2−δj0 2

/ζHjπ

2−δj0 2

i/k

dζ, 3.21

where

ζIh, k

k2 k2_k2

p

kkp k2_k2

p

i,

ζTh, k

k2 k2_k2

s

− kks

k2_k2 s

i.

3.22

Let us now rewrite3.21as

prn i r

j0 N

k1

k,2maxr,1₂j

k−5/2 0h<k

h,k1

e−2πinh/k ωh, kω

2r−jh,2−jk ω2h/2−δj0

, k/2−δj0

×Ij,1Ij,2

,

3.23

where

Ij,1: ζTh,k

ζIh,k arc exp ⎛ ⎜ ⎝π

222j−r1_−2−δ j02

24ζ

π24n1−2r_ζ 12k2

⎞ ⎟ ⎠

×ζ2r−j−1_2−δ j0

×

⎧ ⎨ ⎩−1

fexp−2π/ζ2Hjπi/k

fexp−22j−r1_π/ζ22j−r1_H jπi/k

fexp−π2−δj02/ζHjπ2−δj02i/k

Ij,2: ζTh,k

ζIh,k arc

exp ⎛ ⎜ ⎝π

222j−r1_−2−δ j02

24ζ

π24n1−2r_ζ 12k2

⎞ ⎟ ⎠

×ζ2r−j−1_2−δ j0

dζ.

3.24

3.7. Estimation

It will turn out that asN → ∞, only Ij,2 forj 0 and r/2 < j r ultimately make a contribution. Note that all the integrations in theζ-plane occur on arcs and chords of the circle Kof radius 1/2 centered at the point 1/2. So, inside and onK, 0<Rζ1 andR1/ζ1.

3.7.1. Estimation of

I

for 1

j

r/

2

The regularity of the integrand allows us to alter the path of integration from the arc connectingζIh, kandζTh, kto the directed segment. By51, page 104, Theorem 5.9, the length of the path of integration does not exceed 2√2k/N, and on the segment connecting ζIh, ktoζTh, k,|ζ|<

√

2k/N. Thus, the absolute value of the integrand is ##

## #exp

π22j−r₋₁ 12ζ

π24n1−2r_ζ 12k2

ζ2r−j####

#

|ζ|1/22r−j/2exp

₂₄

n1−2r_πRζ 12k2

exp

π22j−r₋₁ 12 R

1 ζ

|ζ|1/22r/2exp2πn.

3.25

Thus, for 1jr/2,

##_Ij,2## 2√2k N

√

2k N

1/2

2r/2e2πnC_jk3/2N−3/2 3.26

for a constantC_jrecalling thatnandrare fixed.

3.7.2. Estimation of

I

for 1

j

r/

2

## ## #

ζ2r−j_exp

π22j−r₋₁ 12ζ

π24n1−2r_ζ 12k2

## ## #

×

## ## ## ## #

−1 f

exp−2π/ζ2Hjπi/k

fexp−22j−r1π/ζ22j−r1Hjπi/k

f

exp ₋

4π

ζ

4Hjπi

k

####_#ζ2r−j_exp

π22j−r₋₁ 12ζ

π24n1−2r_ζ 12k2

## ## #

×####−1f2j−r

exp ₋

2π

ζ

2Hjπi

k ##

##

|ζ|1/22r−j/2exp

₂₄

n1−2r_πRζ 12k2

exp

22j−r_−1π 12 R

1 ζ

×####_#∞

m1

p2j−rmexp ₋

2πm

ζ

2Hjπim

k ##

## #

|ζ|1/22r−j/2exp

₂₄

n1−2r_π 12

∞

m1

p2j−rmexp

−π

12

24m−22j−r1

|ζ|1/22r/2e2πn

∞

m1

p0me−2πm

cj|ζ|1/2

3.27

for a constantcj. So, for 1jr/2,

##_Ij,1##2√2k N

√

2k N

1/2

cj< Cjk3/2N−3/2 3.28

for a constantCj.

3.7.3. Estimation of

I

∞

n0

p∗_rnxn f

x2r fx

fx2r−1 . 3.29

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

## #exp

π121−r 24ζ

π24n1−2r_ζ 12k2

ζ2r−1####

#

×####_#−1f

exp−2π/ζ2H0πi/k

fexp−21−rπ/ζ21−rH0πi/k

fexp−π/ζH0πi/k

####_#

####_#exp

π121−r 24ζ

π24n1−2r_ζ 12k2

ζ2r−1####

#

×####_#∞

m1

p∗mexp

−21−r_πm

ζ

21−r_H 0πim k

## ## #

exp

π121−r 24 R

1 ζ

exp

π24n1−2r_Rζ 12k2

|ζ|1/22r−1/2

×####_#∞

m1

p∗mexp

−21−r_πm

ζ

exp

21−r_H 0πim k

## ## #

e2πn|ζ|1/22r−1/2

×∞

m1

##_p∗_m##_exp

π121−r 24 R

1 ζ−2

1−r_πmR1

ζ

e2πn|ζ|1/22r−1/2

∞

m1

##_p∗_m##_exp− π

24·2r−1R 1 ζ

24m−1−2r−1

e2πn|ζ|1/22r−1/2

∞

m1

##p∗m##exp−π 24

24m−1−2r−1

< e2πn|ζ|1/22r−1/2

∞

m1 ##

#p∗24m−1−2r−1 ###y24m−1−2r−1 whereye−π/24

c0|ζ|1/2

3.30

for a constantc0. So,

|I0,1| 2√2k

N

√

2k N

1/2

c0< C0k3/2N−3/2 3.31

3.7.4. Estimation of

I

for 1

r/

2

j

r

4

Again, the regularity of the integrand allows us to alter the path of integration from the arc connectingζIh, kandζTh, kto the directed segment.

With this in mind,

## ## #exp

π22j−r₋₁ 12ζ

π24n1−2r_ζ 12k2

ζ2r−j####

#

×####−1f2j−r

exp ₋

2π

ζ

2Hjπi

k ##

##

####_#exp

π22j−r₋₁ 12ζ

exp

π24n1−2r_ζ 12k2

ζ2r−j####

#

×####_#∞

m1

p2j−rmexp ₋

2πm ζ

exp

_2H jπim

k ##

## #

exp

π22j−r−1 12 R

1 ζ

exp

π24n1−2r_Rζ 12k2

|ζ|1/22r−j/2

×####_#∞

m1

p2j−rmexp

−2πmR1

ζ

exp _2H

jπim

k ##

## #

e2πn|ζ|1/22r/2

∞

m1

p2j−rmexp

−π

12R 1 ζ

24m−22j−r 1

e2πn|ζ|1/22r/2

∞

m1

p0mexp

−π

12R 1

ζ24m−2

r₁

cj|ζ|1/2

3.32

for a constantcj. So,

##_Ij,1##2√2k N

√

2k N

1/2

cj< Cjk3/2N−3/2 3.33

3.7.5. Combining the Estimates

k1

k,2r₁

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω2

r_{h, k}

ω2h, k I0,1

i

r/2

j1 N

k1

k,2r₂j

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω

2r−j_h,2−j_k

ωh, k/2

Ij,1Ij,2

i

r

j1r/2 N

k1

k,2r₂j

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω

2r−j_h,2−j_k

ωh, k/2 Ij,1 ## ## ## ## # < r

j0 N

k1 k−1

h0

Cjk−1N−3/2

r/2

j1 N

k1 k−1

h0

Cjk−1N−3/2

CN− 3 2

N

k1 1,

⎛

⎝_whereCr

j0 Cj

r/2

j1 C_j

⎞ ⎠

ON−1/2 .

3.34

Thus, we may revise3.21to

prn i N

k1

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω2

r_{h, k}

ω2h, k I0,2

i

r

j1r/2

N

k1

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω

2r−jh,2−jk

ωh, k/2 Ij,2O

N−1/2 .

3.35

3.8. Evaluation of

I

for

j

0

and

1

r/

2

j

r

Ij,2

K−exp

π22j−r ₋₁ 12ζ

π24n1−2r_ζ 12k2

where

Ij,3: ζIh,k

0

, Ij,4: 0

ζTh,k

, 3.37

andIj,3andIj,4have the same integrand as3.36 and analogously forI0,2.

3.8.1. Estimation of

I

and

I

We note that the length of the arc of integration inIj,3 is less thanπk/

√

2N, and on this arc|ζ|<√2k/N.50, page 272. Also,R1/ζ1 onK50, page 271, equation120.2. Further, 0<Rζ <2k2_/N2_{50, page 271, equation119.6. The absolute value of the integrand is thus}

##

#2r−j_ζ##_#1/2_exp

24n1−2r_πRζ 12k2

22j−r _−1π 12 R

1 ζ

<2r−j/221/4k1/2N−1/2exp

24n1−2r_π 6N2

22j−r_−1π 12

3.38

so that

##_Ij,2##< πk2−1/2N−12r−j/221/4k1/2N−1/2exp

24n1−2r_π 6N2

22j−r_−1π 12

πk3/2N−3/222r−2j−1/4exp

24n1−2r_π 6N2

22j−r_−1π 12

O

k3/2N−3/2exp

₂₄

n1−2r_π 6N2

.

3.39

By the same reasoning,|Ij,3|Ok3/2N−3/2exp24n1−2rπ/6N2. We may therefore revise3.35to

prn i N

k1

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω2

r_{h, k}

ω2h, k

×

K−

ζ2r−1_exp $

π21−r₁ 24ζ

π24n1−2r_ζ 12k2

% dζ

i

r

j1r/2

N

k1

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω

2r−j_h,2−j_k

ωh, k/2

×

K−

ζ2r−j_exp $

π22j−r₋₁ 12ζ

π24n1−2r_ζ 12k2

%

dζON−1/2

and upon lettingNtend to infinity, obtain

prn i

∞

k1

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω2

r_{h, k}

ω2h, k

×

K−

ζ2r−1_exp $

π21−r₁ 24ζ

π24n1−2r_ζ 12k2

% dζ

i

r

j1r/2 ∞

k1

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω

2r−j_h,2−j_k

ωh, k/2

×

K−

ζ2r−j_exp $

π22j−r₋₁ 12ζ

π24n1−2r_ζ 12k2

% dζ.

3.41

3.9. The Final Form

We may now introduce the change of variable

ζ

π222j−r1_−2−δ j0

2

24t

3.42

where the first summation in3.41is thej0 term separated out for clarity, which allows the integral to be evaluated in terms ofI3/2, the Bessel function of the first kind of order 3/2 with purely imaginary argument53, page 372,§17.7when we bear in mind that a “bent” path of integration is allowable according to the remark preceding Equation8on page 177 of54. See also51, 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, forcπ222j−r1_−δ

j02/24,

prn i

∞

k1

k−5/2 0h<k

h,k1

e−2πinh/kωh, kω2

r_{h, k}

ω2h, k 2

r−1/2

π2 1−r₁

24 3/2

× ci∞

c−i∞t −5/2_exp

$ t

121−r_24n1−2r_π2 288k2_t

% dt

i

r

j1r/2 ∞

k1 k−5/2

×

0h<k

h,k1

e−2πinh/kωh, kω

2r−j_h,2−j_k

ωh, k/2 ×2

r−j/2

π2 2j−r₋₁

12 3/2

× ci∞

c−i∞t −5/2_exp

$ t

22j−r_−124n−2r_1π2 144k2_t

π 24n−2r₁3/4

×

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

12r−13/4 2r−2/4

k1

k,2maxr,1₁ k−1

×

0h<k

h,k1

e−2πinh/kωh, kω2

r_{h, k}

ω2h, k I3/2 ⎛ ⎜ ⎝π

24n−2r₁₁₂r−1 2r/2_·6k

⎞ ⎟ ⎠

r j1r/2

22j−r−1 3/422−jr/2 k1

k,2r₂j k−1

×

0h<k

h,k1

e−2πinh/kωh, kω

2r−j_h,2−j_k

ωh, k/2 I3/2 ⎛ ⎜ ⎝π

24n−2r₁₁₂2j−r 6k

⎞ ⎟ ⎠

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ ,

3.43

which, after application of the formula51, page 110

I3/2z

2z π

d dz

sinhz

z

, 3.44

is equivalent toTheorem 2.1.

Acknowledgments

The author thanks the anonymous referee for bringing the work of Zuckerman 46 and Goldberg47to his attention. This, in turn, led the author to seek the more general result presented here in this final version of the paper.

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