A Rademacher Type Formula for Partitions and Overpartitions

Abstract

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of 𝑛 and the Zuckerman formula for the Fourier coefficients of 𝜗₄(0∣𝜏)⁻¹ is presented.

1. Background

1.1. Partitions

A partition of an integer 𝑛 is a representation of 𝑛 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 3, namely, 3, 2+1, and 1+1+1. Let 𝑝(𝑛) denote the number of partitions of 𝑛, with the convention that 𝑝(0)=1, and let 𝑓(𝑥) denote the generating function of 𝑝(𝑛), that is, let

𝑓(𝑥)∶=^∞_𝑛=0𝑝(𝑛)𝑥^𝑛. (1.1)

Euler [1] was the first to systematically study partitions. He showed that

𝑓(𝑥)=^∞_𝑚=111−𝑥^𝑚. (1.2) Euler also showed that

1=𝑓(𝑥)^∞_𝑛=−∞(−1)^𝑛𝑥^{𝑛(3𝑛−1)/2}, (1.3) and since the exponents appearing on the right side of (1.3) are 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 above (and indeed all the series and infinite products mentioned in this paper) converge absolutely when |𝑥|<1, which is important for analytic study of these series and products.

Hardy and Ramanujan were the first to study 𝑝(𝑛) analytically and produced an incredibly accurate asymptotic formula [2, page 85, equation (1.74)], namely,

1𝑝(𝑛)=2𝜋√2^{√⌊𝛼𝑛⌋}_𝑘=1√𝑘_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝜔(ℎ,𝑘)𝑒^{−2𝜋𝑖ℎ𝑛/𝑘}𝑑(𝑑𝑛exp((𝜋/𝑘)(2/3)(𝑛−1/24)))𝑛−1/24+𝑂(𝑛^−1/4), (1.4) where

𝜔(ℎ,𝑘)=exp(𝜋𝑖^𝑘−1_𝑟=1𝑟𝑘(ℎ𝑟𝑘−⌊ℎ𝑟𝑘1⌋−2)). (1.5)

𝛼 is an arbitrary constant, and here and throughout (ℎ,𝑘) is an abbreviation for gcd(ℎ,𝑘).

Later Rademacher [3] improved upon (1.4) by finding the following convergent series representation for 𝑝(𝑛):

1𝑝(𝑛)=𝜋√2^∞_𝑘=1√𝑘_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝜔(ℎ,𝑘)𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝑑(𝑑𝑛sinh((𝜋/𝑘)(2/3)(𝑛−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], Livingood [21], Niven [22], and Subramanyasastri [23] to study various restricted partitions functions.

Recently, Bringmann and Ono [24] have given exact formulas for the coefficients of all harmonic Maass forms of weight ≦1/2. The generating functions considered herein are weakly holomorphic modular forms of weight −1/2, and thus they are harmonic Maass forms of weight ≦1/2. Accordingly, the results of this present paper could be derived from the general theorem in [24]. However, here we opt to derive the results via classical method of Rademacher.

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 Zhao [25–43].

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, 2+1, 2+1, 2+1, 2+1, 1+1+1, 1+1+1.

Let 𝑝(𝑛) denote the number of overpartitions of 𝑛 and let 𝑓(𝑥) denote the generating function ∑^∞_𝑛=0𝑝(𝑛)𝑥^𝑛 of 𝑝(𝑛). Elementary techniques are sufficient to show that

𝑓(𝑥)=^∞_𝑚=11+𝑥^𝑚1−𝑥^𝑚=𝑓(𝑥)²𝑓(𝑥²). (1.7)

Note that

1=𝑓(𝑥)^∞_𝑛=−∞(−1)^𝑛𝑥^𝑛2. (1.8) 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 𝑥 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 𝑥 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 𝑝(𝑛) “has no very simple arithmetical interpretation; but the series is none the less, as the direct reciprocal of a simple 𝜗-function, of particular interest." They went on to state that

1𝑝(𝑛)=𝑑4𝜋(𝑒𝑑𝑛√𝑛^𝜋√𝑛√)+322𝜋cos(31𝑛𝜋−6𝑑𝜋)𝑑𝑛(𝑒^{𝜋√𝑛/3})+⋯+𝑂(𝑛^−1/4). (1.9)

In fact, (1.9) was improved to the following Rademacher-type convergent series by Zuckerman [46, page 321, equation (8.53)]:

1𝑝(𝑛)=2𝜋_{𝑘≧120𝑥0008𝑐∤𝑘}√𝑘_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝜔(ℎ,𝑘)²𝑒𝜔(2ℎ,𝑘)^{−2𝜋𝑖𝑛ℎ/𝑘}𝑑(√𝑑𝑛sinh(𝜋𝑛/𝑘)√𝑛). (1.10) A simplified proof of (1.10) was given by Goldberg in his Ph.D. thesis [47].

1.3. Partitions Where No Odd Part Is Repeated

Let 𝑝𝑜𝑑(𝑛) denote the number of partitions of 𝑛 where no odd part appears more than once. Let 𝑔(𝑥) denote the generating function of 𝑝𝑜𝑑(𝑛), so we have

𝑔(𝑥)=^∞_𝑛=0𝑝𝑜𝑑(𝑛)𝑥^𝑛=^∞_𝑚=11+𝑥^2𝑚−11−𝑥^2𝑚=𝑓(𝑥)𝑓(𝑥⁴)𝑓(𝑥²). (1.11)

Via another identity of Gauss (see equation (2.2.13), page 23 in [44, 45]), it turns out that

1=𝑔(𝑥)^∞_𝑛=0(−𝑥)^{𝑛(𝑛+1)/2}=^∞_𝑛=−∞(−1)^𝑛𝑥^{2𝑛2−𝑛}; (1.12) 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 𝑝𝑜𝑑(𝑛) is as follows:

2𝑝𝑜𝑑(𝑛)=𝜋_𝑘≧1𝑘(1−(−1)^𝑘+⌊(𝑘,4)4⌋)×_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝜔(ℎ,𝑘)𝜔(4ℎ/(𝑘,4),𝑘/(𝑘,4))𝑒𝜔(2ℎ/(𝑘,2),𝑘/(𝑘,2))^{−2𝜋𝑖𝑛ℎ/𝑘}×𝑑(𝑑𝑛sinh(𝜋(𝑘,4)(8𝑛−1)/4𝑘)√8𝑛−1). (1.13)

Equation (1.13) is the case 𝑟=2 of Theorem 2.1 to be presented in the next section.

2. A Common Generalization

Let us define

𝑓_𝑟(𝑥)∶=𝑓(𝑥)𝑓(𝑥^2𝑟)𝑓(𝑥²), (2.1) where 𝑟 is a nonnegative integer. Thus,

𝑓₀(𝑥)=𝑓(𝑥)=^∞_𝑛=0𝑝(𝑛)𝑥^𝑛,𝑓₁(𝑥)=𝑓(𝑥)=^∞_𝑛=0𝑝(𝑛)𝑥^𝑛,𝑓₂(𝑥)=𝑔(𝑥)=^∞_𝑛=0𝑝𝑜𝑑(𝑛)𝑥^𝑛. (2.2)

Let 𝑝_𝑟(𝑛) denote the coefficient of 𝑥^𝑛 in the expansion of 𝑓_𝑟(𝑛), that is,

𝑓_𝑟(𝑥)=^∞_𝑛=0𝑝_𝑟(𝑛)𝑥^𝑛. (2.3)

Notice that 𝑓_𝑟(𝑥) can be represented by several forms of equivalent infinite products, each of which has a natural combinatorial interpretation: 𝑓_𝑟(𝑥)=^∞_𝑚=11+𝑥^𝑚1−𝑥^2𝑟𝑚=^∞_𝑚=11(1−𝑥^2𝑚−1)(1−𝑥^2𝑟𝑚)=^∞_𝑚=111−𝑥^{2𝑟−1𝑚2𝑟−1−1}_𝜆=1(1+𝑥^{2𝑟−1𝑚+𝜆}). (2.4)(2.5)(2.6)

Thus, 𝑝_𝑟(𝑛) equals each of the following:

(i)the number of overpartitions of 𝑛 where nonoverlined parts are multiples of 2^𝑟 (by (2.4)); (ii)the number of partitions of 𝑛 where all parts are either odd or multiples of 2^𝑟 (by (2.5)), provided 𝑟≧1; (iii)the number of partitions of 𝑛 where nonmultiples of 2^𝑟−1 are distinct (by (2.6)), provided 𝑟≧1.

Theorem 2.1. For 𝑟=0,1,2,3,4, 𝑝_𝑟2(𝑛)=^(𝑟+1)/2√3𝜋_𝑘≧1√𝑘_{(𝑘,2max(𝑟,1))=1}×_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟ℎ,𝑘)×𝑑𝜔(2ℎ,𝑘){𝑑𝑛sinh(𝜋(24𝑛−2^𝑟+1)(1+2^𝑟−1)/(2^𝑟/2⋅6𝑘))√24𝑛−2^𝑟}+√+13𝜋^𝑟_{𝑗=1+⌊𝑟/2⌋}2^{(2−𝑗+𝑟)/2}_𝑘≧1√𝑘_{(𝑘,2𝑟)=2𝑗}×_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)×𝑑𝜔(ℎ,𝑘/2){𝑑𝑛sinh(𝜋(24𝑛−2^𝑟+1)(−1+2^2𝑗−𝑟)/6𝑘)√24𝑛−2^𝑟+1}. (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 of (1.6) are provided in [49], [50, Chapter 14], and [51, Chapter 5].

Of fundamental importance is the path of integration to be used. In [48], Rademacher improved upon his original proof of (1.6) given in [3], 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 ℱ_𝑁 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 given 𝑁, let ℎ_𝑝, ℎ_𝑠, 𝑘_𝑝, and 𝑘_𝑠 be such that ℎ_𝑝/𝑘_𝑝 is the immediate predecessor of ℎ/𝑘 and ℎ_𝑠/𝑘_𝑠 is the immediate successor of ℎ/𝑘 in ℱ_𝑁. It will be convenient to view each ℱ_𝑁 cyclically, that is, to view 0/1 as the immediate successor of (𝑁−1)/𝑁.

3.2. Ford Circles and the Rademacher Path

Let ℎ and 𝑘 be integers with (ℎ,𝑘)=1 and 0≦ℎ<𝑘. The Ford circle [52] 𝐶(ℎ,𝑘) is the circle in ℂ of radius 1/2𝑘² centered at the point

ℎ𝑘+12𝑘²𝑖. (3.1)

The upper  arc𝛾(ℎ,𝑘) of the Ford circle 𝐶(ℎ,𝑘) is those points of 𝐶(ℎ,𝑘) from the initial point

𝛼_𝐼ℎ(ℎ,𝑘)∶=𝑘−𝑘_𝑝𝑘(𝑘²+𝑘²_𝑝)+1𝑘²+𝑘²_𝑝𝑖 (3.2) to the terminal point

𝛼_𝑇ℎ(ℎ,𝑘)∶=𝑘+𝑘_𝑠𝑘(𝑘²+𝑘²_𝑠)+1𝑘²+𝑘²_𝑠𝑖, (3.3) traversed in the clockwise direction.

Note that we have

𝛼_𝐼(0,1)=𝛼_𝑇(𝑁−1,𝑁). (3.4)

Every Ford circle is in the upper half plane. For ℎ₁/𝑘₁,ℎ₂/𝑘₂∈ℱ_𝑁, 𝐶(ℎ₁,𝑘₁) and 𝐶(ℎ₂,𝑘₂) are either tangent or do not intersect.

The Rademacher path 𝑃(𝑁) of order 𝑁 is the path in the upper half of the 𝜏-plane from 𝑖 to 𝑖+1 consisting of

_{ℎ/𝑘∈ℱ𝑁}𝛾(ℎ,𝑘) (3.5) traversed left to right and clockwise. In particular, we consider the left half of the Ford circle 𝐶(0,1) and the corresponding upper arc 𝛾(0,1) to be translated to the right by 1 unit. This is legal given then periodicity of the function which is to be integrated over 𝑃(𝑁).

3.3. Set Up the Integral

Let 𝑛 and 𝑟 be fixed, with 𝑛>(2^𝑟−1)/24.

Since

𝑓_𝑟(𝑥)=^∞_𝑛=0𝑝_𝑟(𝑛)𝑥^𝑛, (3.6)

Cauchy's residue theorem implies that

𝑝_𝑟1(𝑛)=2𝜋𝑖_𝒞𝑓_𝑟(𝑥)𝑥^𝑛+1𝑑𝑥, (3.7) where 𝒞 is any simply closed contour enclosing the origin and inside the unit circle. We introduce the change of variable

𝑥=𝑒^{2𝜋𝑖𝜏} (3.8) so that the unit disk |𝑥|≦1 in the 𝑥-plane maps to the infinitely tall, unit-wide strip in the 𝜏-plane where 0≦ℜ𝜏≦1 and ℑ𝜏≧0. The contour 𝒞 is then taken to be the preimage of 𝑃(𝑁) under the map 𝑥↦𝑒^{2𝜋𝑖𝜏}.

Better yet, let us replace 𝑥 with 𝑒^{2𝜋𝑖𝜏} in (3.7) to express the integration in the 𝜏-plane:

𝑝_𝑟(𝑛)=_𝑃(𝑁)𝑓_𝑟(𝑒^{2𝜋𝑖𝜏})𝑒^{−2𝜋𝑖𝑛𝜏}𝑑𝜏=_{ℎ/𝑘∈ℱ𝑁𝛾(ℎ,𝑘)}𝑓_𝑟(𝑒^{2𝜋𝑖𝜏})𝑒^{−2𝜋𝑖𝑛𝜏}=𝑑𝜏_{𝑁𝑘=10≦ℎ<𝑘(ℎ,𝑘)=1𝛾(ℎ,𝑘)}𝑓_𝑟(𝑒^{2𝜋𝑖𝜏})𝑒^{−2𝜋𝑖𝑛𝜏}𝑑𝜏. (3.9)

3.4. Another Change of Variable

Next, we change variables again, taking

𝜏=𝑖𝑧+ℎ𝑘, (3.10) so that

ℎ𝑧=−𝑖𝑘(𝜏−𝑘). (3.11) Thus 𝐶(ℎ,𝑘) (in the 𝜏-plane) maps to the clockwise-oriented circle 𝐾_𝑘⁽⁻⁾ (in the 𝑧-plane) centered at 1/2𝑘 with radius 1/2𝑘.

So we now have

𝑝_𝑟(𝑛)=𝑖^𝑁_𝑘=1𝑘⁻¹_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘𝑧𝑇}_𝑧^(ℎ,𝑘)_{𝐼(ℎ,𝑘)arcof𝐾𝑘(−)}𝑒^{2𝑛𝜋𝑧/𝑘}𝑓_𝑟(𝑒^{2𝜋𝑖ℎ/𝑘−2𝜋𝑧/𝑘})𝑑𝑧, (3.12) where 𝑧_𝐼(ℎ,𝑘) (resp., 𝑧_𝑇(ℎ,𝑘)) is the image of 𝛼_𝐼(ℎ,𝑘) (see (3.2)) (resp. 𝛼_𝑇(ℎ,𝑘) [see 3.2]) under the transformation (3.11).

So the transformation (3.11) maps the upper arc 𝛾(ℎ,𝑘) of 𝐶(ℎ,𝑘) in the 𝜏-plane to the arc on 𝐾_𝑘⁽⁻⁾ which initiates at

𝑧_𝐼𝑘(ℎ,𝑘)=𝑘²+𝑘²_𝑝+𝑘_𝑝𝑘²+𝑘²_𝑝𝑖 (3.13) and terminates at

𝑧_𝑇𝑘(ℎ,𝑘)=𝑘²+𝑘²_𝑠−𝑘_𝑠𝑘²+𝑘²_𝑠𝑖. (3.14)

3.5. Exploiting a Modular Transformation

From the theory of modular forms, we have the transformation formula [2, page 93, Lemma 4.31]:

𝑓(exp(2𝜋𝑖ℎ𝑘−2𝜋𝑧𝑘))=𝜔(ℎ,𝑘)exp(𝜋(𝑧⁻¹−𝑧))√12𝑘𝑧𝑓(exp(2𝜋𝑖𝑖𝑧⁻¹+𝐻𝑘)), (3.15) where √𝑧 is the principal branch, (ℎ,𝑘)=1, and 𝐻 is a solution to the congruence

ℎ𝐻≡−1(mod𝑘). (3.16) From (3.15), we deduce the analogous transformation for 𝑓_𝑟(𝑥).

The transformation formula is a piecewise defined function with 𝑟+1 cases corresponding to 𝑗=0,1,2,…,𝑟, where (𝑘,2^𝑟)=2^𝑗:

𝑓_𝑟(exp(2𝜋𝑖ℎ𝑘−2𝜋𝑧𝑘=))𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)𝜔(2ℎ/(2−𝛿_𝑗0),𝑘/(2−𝛿_𝑗0))exp(𝜋(2+2^{2𝑗−𝑟+1}−(2−𝛿_𝑗0)²)+24𝑘𝑧𝜋(1−2^𝑟)𝑧)×12𝑘𝑧2^{𝑟−𝑗−1}(2−𝛿_𝑗0)×𝑓(exp(−2𝜋/𝑘𝑧+2𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−2^{2𝑗−𝑟+1}𝜋/𝑘𝑧+2^{2𝑗−𝑟+1}𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−𝜋(2−𝛿_𝑗0)²/𝑘𝑧+𝐻_𝑗𝜋(2−𝛿_𝑗0)²,𝑖/𝑘)) (3.17) where 𝐻_𝑗 is divisible by 2^𝑟−𝑗 and is a solution to the congruence ℎ𝐻_𝑗≡−1(mod𝑘), and 𝛿_𝑗0=1if𝑗=0,0if𝑗≠0 (3.18) is the Kronecker 𝛿-function.

Notice that in particular, for ⌊𝑟/2⌋≦𝑗≦𝑟, (3.17) simplifies to 𝑓_𝑟(exp(2𝜋𝑖ℎ𝑘−2𝜋𝑧𝑘=))𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)𝜋𝜔(ℎ,𝑘/2)exp(12𝑘[(2^2𝑗−𝑟−1)𝑧⁻¹+(1−2^𝑟×)𝑧])𝑧2^𝑟−𝑗𝑓_2𝑗−𝑟(exp(−2𝜋+𝑘𝑧2𝐻_𝑗𝜋𝑖𝑘)). (3.19)

Since the 𝑟=0 case was established by Zuckerman, and the 𝑟=1 case by Rademacher, we will proceed with the assumption that 𝑟>1.

Apply (3.17) to (3.12) to obtain 𝑝_𝑟(𝑛)=𝑖^𝑟𝑁_{𝑗=0𝑘=1(𝑘,2𝑟)=𝑗}𝑘⁻¹_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}×^𝑧𝑇_𝑧^(ℎ,𝑘)_{𝐼(ℎ,𝑘)arcof𝐾𝑘(−)}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)𝜔(2ℎ/(2−𝛿_𝑗0),𝑘/(2−𝛿_𝑗0))×exp(𝜋(2+2^{2𝑗−𝑟+1}−(2−𝛿_𝑗0)²)+24𝑘𝑧𝜋(24𝑛+1−2^𝑟)𝑧)×12𝑘𝑧2^{𝑟−𝑗−1}(2−𝛿_𝑗0)×𝑓(exp(−2𝜋/𝑘𝑧+2𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−2^{2𝑗−𝑟+1}𝜋/𝑘𝑧+2^{2𝑗−𝑟+1}𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−𝜋(2−𝛿_𝑗0)²/𝑘𝑧+𝐻_𝑗𝜋(2−𝛿_𝑗0)²𝑖/𝑘))𝑑𝑧. (3.20)

3.6. Normalization

Next, introduce a normalization 𝜁=𝑧𝑘 (this is not strictly necessary, but it will allow us in the sequel to quote various useful results directly from the literature):

𝑝_𝑟(𝑛)=𝑖^𝑟𝑁_{𝑗=0𝑘=1(𝑘,2𝑟)=𝑗}𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}×^𝜁𝑇_𝜁^(ℎ,𝑘)_{𝐼(ℎ,𝑘)arcof𝐾(−)}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)𝜔(2ℎ/(2−𝛿_𝑗0),𝑘/(2−𝛿_𝑗0))×exp(𝜋(2+2^{2𝑗−𝑟+1}−(2−𝛿_𝑗0)²)+24𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)×𝜁2^{𝑟−𝑗−1}(2−𝛿_𝑗0)×𝑓(exp(−2𝜋/𝜁+2𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−2^{2𝑗−𝑟+1}𝜋/𝜁+2^{2𝑗−𝑟+1}𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−𝜋(2−𝛿_𝑗0)²/𝜁+𝐻_𝑗𝜋(2−𝛿_𝑗0)²𝑖/𝑘))𝑑𝜁, (3.21) where 𝜁_𝐼𝑘(ℎ,𝑘)=²𝑘²+𝑘²_𝑝+𝑘𝑘_𝑝𝑘²+𝑘²_𝑝𝜁𝑖,_𝑇𝑘(ℎ,𝑘)=²𝑘²+𝑘²_𝑠−𝑘𝑘_𝑠𝑘²+𝑘²_𝑠𝑖. (3.22)

Let us now rewrite (3.21) as 𝑝_𝑟(𝑛)=𝑖^𝑟𝑁_{𝑗=0𝑘=1(𝑘,2max(𝑟,1))=2𝑗}𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)𝜔(2ℎ/(2−𝛿_𝑗0),𝑘/(2−𝛿_𝑗0))×(ℐ_𝑗,1+ℐ_𝑗,2), (3.23) where ℐ_𝑗,1∶=^𝜁𝑇_𝜁^(ℎ,𝑘)_{𝐼(ℎ,𝑘)arc}exp(𝜋(2+2^{2𝑗−𝑟+1}−(2−𝛿_𝑗0)²)+24𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)×𝜁2^{𝑟−𝑗−1}(2−𝛿_𝑗0)×{−1+𝑓(exp(−2𝜋/𝜁+2𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−2^{2𝑗−𝑟+1}𝜋/𝜁+2^{2𝑗−𝑟+1}𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−𝜋(2−𝛿_𝑗0)²/𝜁+𝐻_𝑗𝜋(2−𝛿_𝑗0)²ℐ𝑖/𝑘))}𝑑𝜁,_𝑗,2∶=^𝜁𝑇_𝜁^(ℎ,𝑘)_{𝐼(ℎ,𝑘)arc}exp(𝜋(2+2^{2𝑗−𝑟+1}−(2−𝛿_𝑗0)²)+24𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)×𝜁2^{𝑟−𝑗−1}(2−𝛿_𝑗0)𝑑𝜁. (3.24)

3.7. Estimation

It will turn out that as 𝑁→∞, only ℐ_𝑗,2 for 𝑗=0 and ⌊𝑟/2⌋<𝑗≦𝑟 ultimately make a contribution. Note that all the integrations in the 𝜁-plane occur on arcs and chords of the circle 𝐾 of radius 1/2 centered at the point 1/2. So, inside and on 𝐾, 0<ℜ𝜁≦1 and ℜ(1/𝜁)≧1.

3.7.1. Estimation of ℐ_𝑗,2 for 1≦𝑗≦⌊𝑟/2⌋

The regularity of the integrand allows us to alter the path of integration from the arc connecting 𝜁_𝐼(ℎ,𝑘) and 𝜁_𝑇(ℎ,𝑘) to the directed segment. By [51, page 104, Theorem 5.9], the length of the path of integration does not exceed 2√2𝑘/𝑁, and on the segment connecting 𝜁_𝐼(ℎ,𝑘) to 𝜁_𝑇(ℎ,𝑘), |𝜁|<√2𝑘/𝑁. Thus, the absolute value of the integrand is

|exp(𝜋(2^2𝑗−𝑟−1)+12𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)𝜁2^𝑟−𝑗|=|𝜁|^1/22^{(𝑟−𝑗)/2}exp((24𝑛+1−2^𝑟)𝜋ℜ𝜁12𝑘²)exp(𝜋(2^2𝑗−𝑟−1)ℜ112𝜁)≦|𝜁|^1/22^𝑟/2exp(2𝜋𝑛). (3.25)

Thus, for 1≦𝑗≦⌊𝑟/2⌋,

|ℐ_𝑗,22√|≦2𝑘𝑁(√2𝑘𝑁)^1/22^𝑟/2𝑒^2𝜋𝑛≦𝐶^′_𝑗𝑘^3/2𝑁^−3/2 (3.26)

for a constant 𝐶^′_𝑗 (recalling that 𝑛 and 𝑟 are fixed).

3.7.2. Estimation of ℐ_𝑗,1 for 1≦𝑗≦⌊𝑟/2⌋

We have the absolute value of the integrand:

|𝜁2^𝑟−𝑗exp(𝜋(2^2𝑗−𝑟−1)+12𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)|×|−1+𝑓(exp(−2𝜋/𝜁+2𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−2^{2𝑗−𝑟+1}𝜋/𝜁+2^{2𝑗−𝑟+1}𝐻_𝑗𝜋𝑖/𝑘))𝑓(exp(−4𝜋𝜁+4𝐻_𝑗𝜋𝑖𝑘|))=|𝜁2^𝑟−𝑗exp(𝜋(2^2𝑗−𝑟−1)+12𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)|×|−1+𝑓_2𝑗−𝑟(exp(−2𝜋𝜁+2𝐻_𝑗𝜋𝑖𝑘))|=|𝜁|^1/22^{(𝑟−𝑗)/2}exp((24𝑛+1−2^𝑟)𝜋ℜ𝜁12𝑘²)exp((2^2𝑗−𝑟−1)𝜋ℜ112𝜁)×|^∞_𝑚=1𝑝_2𝑗−𝑟(𝑚)exp(−2𝜋𝑚𝜁+2𝐻_𝑗𝜋𝑖𝑚𝑘)|≦|𝜁|^1/22^{(𝑟−𝑗)/2}exp((24𝑛+1−2^𝑟)𝜋)12^∞_𝑚=1𝑝_2𝑗−𝑟𝜋(𝑚)exp(−12(24𝑚−2^2𝑗−𝑟+1))≦|𝜁|^1/22^𝑟/2𝑒^∞2𝜋𝑛_𝑚=1𝑝₀(𝑚)𝑒^−2𝜋𝑚=𝑐_𝑗|𝜁|^1/2 (3.27)

for a constant 𝑐_𝑗. So, for 1≦𝑗≦⌊𝑟/2⌋,

|ℐ_𝑗,12√|≦2𝑘𝑁(√2𝑘𝑁)^1/2𝑐_𝑗<𝐶_𝑗𝑘^3/2𝑁^−3/2 (3.28)

for a constant 𝐶_𝑗.

3.7.3. Estimation of 𝐼_0,1

Let 𝑝^∗_𝑟(𝑥) be defined by

^∞_𝑛=0𝑝^∗_𝑟(𝑛)𝑥^𝑛=𝑓(𝑥^2𝑟)𝑓(𝑥)𝑓(𝑥^2𝑟−1). (3.29)

Again, the regularity of the integrand allows us to alter the path of integration from the arc connecting 𝜁_𝐼(ℎ,𝑘) and 𝜁_𝑇(ℎ,𝑘) to the directed segment.

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

|exp(𝜋(1+2^1−𝑟)+24𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)𝜁2^𝑟−1|×|−1+𝑓(exp(−2𝜋/𝜁+2𝐻₀𝜋𝑖/𝑘))𝑓(exp(−2^1−𝑟𝜋/𝜁+2^1−𝑟𝐻₀𝜋𝑖/𝑘))𝑓(exp(−𝜋/𝜁+𝐻₀|𝜋𝑖/𝑘))=|exp(𝜋(1+2^1−𝑟)+24𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)𝜁2^𝑟−1|×|^∞_𝑚=1𝑝^∗(𝑚)exp(−2^1−𝑟𝜋𝑚𝜁+2^1−𝑟𝐻₀𝜋𝑖𝑚𝑘)|=exp(𝜋(1+2^1−𝑟)ℜ124𝜁)exp(𝜋(24𝑛+1−2^𝑟)ℜ𝜁12𝑘²)|𝜁|^1/22^(𝑟−1)/2×|^∞_𝑚=1𝑝^∗(𝑚)exp(−2^1−𝑟𝜋𝑚𝜁2)exp(^1−𝑟𝐻₀𝜋𝑖𝑚𝑘)|≦𝑒^2𝜋𝑛|𝜁|^1/22^(𝑟−1)/2×^∞_𝑚=1|𝑝^∗(𝑚)|exp(𝜋(1+2^1−𝑟)ℜ124𝜁−2^1−𝑟1𝜋𝑚ℜ𝜁)=𝑒^2𝜋𝑛|𝜁|^1/22^{∞(𝑟−1)/2}_𝑚=1|𝑝^∗𝜋(𝑚)|exp(−24⋅2^𝑟−1ℜ1𝜁(24𝑚−1−2^𝑟−1))≦𝑒^2𝜋𝑛|𝜁|^1/22^{∞(𝑟−1)/2}_𝑚=1|𝑝^∗𝜋(𝑚)|exp(−24(24𝑚−1−2^𝑟−1))<𝑒^2𝜋𝑛|𝜁|^1/22^{∞(𝑟−1)/2}_𝑚=1|𝑝^∗(24𝑚−1−2^𝑟−1)|𝑦^{24𝑚−1−2𝑟−1}(where𝑦=𝑒^−𝜋/24)=𝑐₀|𝜁|^1/2 (3.30)

for a constant 𝑐₀. So,

|ℐ_0,12√|≦2𝑘𝑁(√2k𝑁)^1/2𝑐₀<𝐶₀𝑘^3/2𝑁^−3/2 (3.31)

for a constant 𝐶₀.

3.7.4. Estimation of ℐ_𝑗,1 for 1+⌊𝑟/2⌋≦𝑗≦𝑟≦4

Again, the regularity of the integrand allows us to alter the path of integration from the arc connecting 𝜁_𝐼(ℎ,𝑘) and 𝜁_𝑇(ℎ,𝑘) to the directed segment.

With this in mind, |exp(𝜋(2^2𝑗−𝑟−1)+12𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)𝜁2^𝑟−𝑗|×|−1+𝑓_2𝑗−𝑟(exp(−2𝜋𝜁+2𝐻_𝑗𝜋𝑖𝑘))|=|exp(𝜋(2^2𝑗−𝑟−1)12𝜁)exp(𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)𝜁2^𝑟−𝑗|×|^∞_𝑚=1𝑝_2𝑗−𝑟(𝑚)exp(−2𝜋𝑚𝜁)exp(2𝐻_𝑗𝜋𝑖𝑚𝑘)|=exp(𝜋(2^2𝑗−𝑟−1)ℜ112𝜁)exp(𝜋(24𝑛+1−2^𝑟)ℜ𝜁12𝑘²)|𝜁|^1/22^{(𝑟−𝑗)/2}×|^∞_𝑚=1𝑝_2𝑗−𝑟1(𝑚)exp(−2𝜋𝑚ℜ𝜁)exp(2𝐻_𝑗𝜋𝑖𝑚𝑘)|≦𝑒^2𝜋𝑛|𝜁|^1/22^∞𝑟/2_𝑚=1𝑝_2𝑗−𝑟𝜋(𝑚)exp(−ℜ112𝜁(24𝑚−2^2𝑗−𝑟+1))≦𝑒^2𝜋𝑛|𝜁|^1/22^∞𝑟/2_𝑚=1𝑝₀𝜋(𝑚)exp(−ℜ112𝜁(24𝑚−2^𝑟+1))=𝑐_𝑗|𝜁|^1/2 (3.32) for a constant 𝑐_𝑗. So,

|ℐ_𝑗,12√|≦2𝑘𝑁(√2𝑘𝑁)^1/2𝑐_𝑗<𝐶_𝑗𝑘^3/2𝑁^−3/2 (3.33)

for a constant 𝐶_𝑗, when 1+⌊𝑟/2⌋≦𝑗≦𝑟.

3.7.5. Combining the Estimates

One has

|𝑖^𝑁_{𝑘=1(𝑘,2𝑟)=1}𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟ℎ,𝑘)ℐ𝜔(2ℎ,𝑘)^0,1+𝑖^{⌊𝑟/2⌋𝑁}_{𝑗=1𝑘=1(𝑘,2𝑟)=2𝑗}𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)𝜔(ℎ,𝑘/2)(ℐ_𝑗,1+ℐ_𝑗,2)+𝑖^𝑟𝑁_{𝑗=1+⌊𝑟/2⌋𝑘=1(𝑘,2𝑟)=2𝑗}𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)ℐ𝜔(ℎ,𝑘/2)^𝑗,1|<^𝑟𝑁_𝑗=0𝑘=1^𝑘−1_ℎ=0𝐶_𝑗𝑘⁻¹𝑁^−3/2+^{⌊𝑟/2⌋𝑁}_𝑗=1𝑘=1^𝑘−1_ℎ=0𝐶^′_𝑗𝑘⁻¹𝑁^−3/2≦𝐶^′′𝑁⁻32^𝑁_𝑘=11,(where𝐶^′′=^r_𝑗=0C_𝑗+^⌊r/2⌋_𝑗=1C^′_𝑗)=𝑂(𝑁^−1/2). (3.34) Thus, we may revise (3.21) to

𝑝_𝑟(𝑛)=𝑖^𝑁_𝑘=1𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟ℎ,𝑘)ℐ𝜔(2ℎ,𝑘)^0,2+𝑖^𝑟𝑁_{𝑗=1+⌊𝑟/2⌋𝑘=1}𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)ℐ𝜔(ℎ,𝑘/2)^𝑗,2+𝑂(𝑁^−1/2). (3.35)

3.8. Evaluation of ℐ_𝑗,2 for 𝑗=0 and 1+⌊𝑟/2⌋≦𝑗≦𝑟

Write ℐ_𝑗,2 as

ℐ_𝑗,2=_𝐾(−)exp(𝜋(2^2𝑗−𝑟−1)+12𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²)𝜁2^𝑟−𝑗𝑑𝜁−ℐ_𝑗,3−ℐ_𝑗,4, (3.36) where

ℐ_𝑗,3∶=^𝜁𝐼₀^(ℎ,𝑘),ℐ_𝑗,4∶=⁰_{𝜁𝑇(ℎ,𝑘)}, (3.37) and ℐ_j,3 and ℐ_𝑗,4 have the same integrand as (3.36) (and analogously for 𝐼_0,2).

3.8.1. Estimation of ℐ_𝑗,3 and ℐ_𝑗,4

We note that the length of the arc of integration in ℐ_𝑗,3 is less than 𝜋𝑘/√2𝑁, and on this arc|𝜁|<√2𝑘/𝑁. [50, page 272]. Also, ℜ1/𝜁=1 on 𝐾 [50, page 271, equation (120.2)]. Further, 0<ℜ𝜁<2𝑘²/𝑁² [50, page 271, equation (119.6)]. The absolute value of the integrand is thus |2^𝑟−𝑗𝜁|^1/2exp((24𝑛+1−2^𝑟)𝜋ℜ𝜁12𝑘²+(2^2𝑗−𝑟−1)𝜋ℜ112𝜁)<2^{(𝑟−𝑗)/2}2^1/4𝑘^1/2𝑁^−1/2exp((24𝑛+1−2^𝑟)𝜋6𝑁²+(2^2𝑗−𝑟−1)𝜋)12 (3.38) so that

|ℐ_𝑗,2|<𝜋𝑘2^−1/2𝑁⁻¹2^{(𝑟−𝑗)/2}2^1/4𝑘^1/2𝑁^−1/2exp((24𝑛+1−2^𝑟)𝜋6𝑁²+(2^2𝑗−𝑟−1)𝜋)12=𝜋𝑘^3/2𝑁^−3/22^{(2𝑟−2𝑗−1)/4}exp((24𝑛+1−2^𝑟)𝜋6𝑁²+(2^2𝑗−𝑟−1)𝜋)12=𝑂(𝑘^3/2𝑁^−3/2exp((24𝑛+1−2^𝑟)𝜋6𝑁²)). (3.39)

By the same reasoning, |ℐ_𝑗,3|=𝑂(𝑘^3/2𝑁^−3/2exp((24𝑛+1−2^𝑟)𝜋/6𝑁²)).

We may therefore revise (3.35) to 𝑝_𝑟(𝑛)=𝑖^𝑁_𝑘=1𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟ℎ,𝑘)𝜔(2ℎ,𝑘)×_𝐾(−)𝜁2^𝑟−1exp{𝜋(2^1−𝑟+1)+24𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²}𝑑𝜁+𝑖^𝑟𝑁_{𝑗=1+⌊𝑟/2⌋𝑘=1}𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)×𝜔(ℎ,𝑘/2)_𝐾(−)𝜁2^𝑟−𝑗exp{𝜋(2^2𝑗−𝑟−1)+12𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²}𝑑𝜁+𝑂(𝑁^−1/2) (3.40) and upon letting 𝑁 tend to infinity, obtain

𝑝_𝑟(𝑛)=𝑖^∞_𝑘=1𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟ℎ,𝑘)×𝜔(2ℎ,𝑘)_𝐾(−)𝜁2^𝑟−1exp{𝜋(2^1−𝑟+1)+24𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²}𝑑𝜁+𝑖^𝑟∞_{𝑗=1+⌊𝑟/2⌋𝑘=1}k^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)×𝜔(ℎ,𝑘/2)_𝐾(−)𝜁2^𝑟−𝑗exp{𝜋(2^2𝑗−𝑟−1)+12𝜁𝜋(24𝑛+1−2^𝑟)𝜁12𝑘²}𝑑𝜁. (3.41)

3.9. The Final Form

We may now introduce the change of variable 𝜁=𝜋(2+2^{2𝑗−𝑟+1}−(2−𝛿_𝑗0)²)24𝑡 (3.42) (where the first summation in (3.41) is the 𝑗=0 term separated out for clarity), which allows the integral to be evaluated in terms of 𝐼_3/2, the Bessel function of the first kind of order 3/2 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 (8) on page 177 of [54]. 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 𝑐=𝜋(2+2^{2𝑗−𝑟+1}−(𝛿_𝑗0)²)/24,

𝑝_𝑟(𝑛)=𝑖^∞_𝑘=1𝑘^−5/2_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟ℎ,𝑘)2𝜔(2ℎ,𝑘)^(𝑟−1)/22(𝜋^1−𝑟+1)24^3/2×^𝑐+𝑖∞_{𝑐−𝑖∞}𝑡^−5/2exp{𝑡+(1+2^1−𝑟)(24𝑛+1−2^𝑟)𝜋²288𝑘²𝑡}𝑑𝑡+𝑖^𝑟∞_{𝑗=1+⌊𝑟/2⌋𝑘=1}𝑘^−5/2×_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)𝜔(ℎ,𝑘/2)×2^{(𝑟−𝑗)/2}2(𝜋^2𝑗−𝑟−1)12^3/2×^𝑐+𝑖∞_{𝑐−𝑖∞}𝑡^−5/2exp{𝑡+(2^2𝑗−𝑟−1)(24𝑛−2^𝑟+1)𝜋²144𝑘²𝑡=𝜋}𝑑𝑡(24𝑛−2^𝑟+1)^3/4×{(1+2^𝑟−1)^3/42^(𝑟−2)/4_{𝑘≧1(𝑘,2max(𝑟,1))=1)}𝑘⁻¹×_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟ℎ,𝑘)𝐼𝜔(2ℎ,𝑘)^3/2(𝜋(24𝑛−2^𝑟+1)(1+2^𝑟−1)2^𝑟/2)+⋅6𝑘_{𝑟𝑗=1+⌊𝑟/2⌋}(2^2𝑗−𝑟−1)^3/42^{(2−𝑗+𝑟)/2}_{𝑘≧1(𝑘,2𝑟)=2𝑗}𝑘⁻¹×_{0≦ℎ<𝑘(ℎ,𝑘)=1}𝑒^{−2𝜋𝑖𝑛ℎ/𝑘}𝜔(ℎ,𝑘)𝜔(2^𝑟−𝑗ℎ,2^−𝑗𝑘)𝐼𝜔(ℎ,𝑘/2)^3/2(𝜋(24𝑛−2^𝑟+1)(1+2^2𝑗−𝑟)6𝑘)}, (3.43) which, after application of the formula [51, page 110] 𝐼_3/2(𝑧)=2𝑧𝜋𝑑(𝑑𝑧sinh𝑧𝑧), (3.44) is equivalent to Theorem 2.1.

Acknowledgments

The author thanks the anonymous referee for bringing the work of Zuckerman [46] and Goldberg [47] to his attention. This, in turn, led the author to seek the more general result presented here in this final version of the paper.