A Poisson-Jacobi-type transformation for the sum $\sum_{n=1}^\infty n^{-2m} \exp (-an^2}$ for positive integer $m$

0.14 MB·English

by  R. B. Paris

A Poisson-Jacobi-type transformation for the sum
∞ n−2m exp(−an2) for positive integer m
n=1
P
R. B. Paris
5 School of Engineering, Computing and Applied Mathematics,
1
University of Abertay Dundee, Dundee DD1 1HG, UK
0
2

n
a
J Abstract

4 Weobtain an asymptotic expansion for thesum

] ∞ e−an2
A S(a;w)=
nw
C Xn=1
h. as a→0 in |arg a|< 12π for arbitrary finite w>0. The result when w=2m, where m is
t apositiveinteger,istheanalogueofthewell-knownPoisson-Jacobi transformation forthe
a sum with m=0. Numerical results are given to illustrate theaccuracy of theexpansion.
m
[ Mathematics Subject Classification: 30E15, 33B10, 34E05, 41A30

1
Keywords: Poisson-Jacobi transformation, asymptotic expansion, inverse factorial ex-
v
pansion
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8
6
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. 1. Introduction
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0
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1
The classical Poisson-Jacobitransformation is given by
:
v
∞ ∞
Xi e−an2 = 1 π − 1 + π e−π2n2/a, (1.1)
2ra 2 ra
r nX=1 nX=1
a
where the parameter a satisfies ℜ(a) > 0. This transformation relates a sum of Gaussian
exponentials involving the parameter a to a similar sum with parameter π2/a. In the case
a→0inℜ(a)>0,the convergenceofthe sumonthe left-handsidebecomesslow, whereasthe
sum on the right-hand side converges rapidly in this limit. Various proofs of the well-known
result (1.1) exist in the literature; see, for example, [3, p. 120], [4, p. 60] and [5, p. 124].
In this note we consider the sum
∞ e−an2
S(a;w)= (ℜ(a)>0). (1.2)
nw
nX=1
This sum converges for any finite value of the parameter w provided ℜ(a) > 0; when a = 0
then S(0;w) reduces to the Riemann zeta function ζ(w) when ℜ(w) > 1. Consequently, the
seriesin(1.2)canbeviewedasasmoothedDirichletseriesforζ(w). Theasymptoticexpansion
1
2 R. B. Paris
of S(a;w) as a → 0 in ℜ(a) > 0 is straightforward. The most interesting case arises when
w = 2m, where m is a positive integer, for which we establish a transformation for S(a;2m)
analogous to that in (1.1) valid as a → 0 in ℜ(a) > 0. This similarly involves the series in
(1.2) with a replaced by π2/a, but with each term decorated by an asymptotic series in a. A
recent application of the series with w = 2 and w = 4 has arisen in the geological problem of
thermochronometry in spherical geometry [6].
2. An expansion for S(a;w) as a→0 when w 6=2,4,...
Our starting point is the well-known Cahen-Mellin integral (see, for example, [3, §3.3.1])
1 c+∞i
z−αe−z = Γ(s−α)z−sds (z 6=0, |arg z|< 1π), (2.1)
2πiZ 2
c−∞i
where c > ℜ(α) so that the integration path passes to the right of all the poles of Γ(s−α)
situatedats+α−k(k =0,1,2,...). Forsimplicityinpresentationweshallassumethroughout
real values of w>0. Then, it follows that
∞ e−an2 ∞ n−w c+∞i
S(a;w) = = Γ(s)(an2)−sds
nw 2πi Z
nX=1 nX=1 c−∞i
1 c+∞i
= Γ(s)ζ(2s+w)a−sds,
2πiZ
c−∞i
uponreversaloftheorderofsummationandintegration,whichisjustifiedwhenc>max{0,1−
2
1w}, and evaluation of the inner sum in terms of the Riemann zeta function. The integrand
2
possesses simple poles at s = 1 − 1w and s = −k (k = 0,1,2,...), except if w = 2m+1 is an
2 2
odd positive integer when the pole at s= 1 − 1w is double. The case when w =2m is an even
2 2
positive integer requires a separate investigation which is discussed in Section 3.
Considerthe integraltakenroundthe rectangularcontourwith verticesatc±iT,−c′±iT,
where c′ > 0. The contribution from the upper and lower sides s = σ ±iT, −c′ ≤ σ ≤ c,
vanishes as T →∞ provided |arg a|< 1π, since from the behaviour
2
Γ(σ±it)=O(tσ−21e−12πt), ζ(σ±it)=O(tµ(σ)logAt), (t→∞),
where for σ and t real
µ(σ)=0 (σ >1), 1 − 1σ (0≤σ ≤1), 1 −σ (σ <0),
2 2 2
A=1 (0≤σ ≤1), A=0 otherwise,
themodulusoftheintegrandiscontrolledbyO(Tσ+µ(σ)−12 log Te−∆T),with∆= 1π−|arg a|.
2
The residue at the double pole s=−m when w =2m+1 (m=0,1,2,...) is given by
(−a)m
{γ− 1log a+ 1ψ(m+1)},
m! 2 2
where γ is Euler’s constant and ψ(x) is the logarithmic derivative of the gamma function.
Displacement of the integration path to the left over the poles then yields (provided w 6=2m)
S(a;w)=J(a;w)+N−1′(−)k ζ(w−2k)ak+R , (2.2)
N
k!
kX=0
Poisson-Jacobi-type transformation 3
where
1Γ(1 − 1w)a(w−1)/2 (w 6=2m+1)
2 2 2
J(a;w)= ,
 (−a)m{γ− 1log a+ 1ψ(m+1)} (w =2m+1),
m! 2 2
N is a positive integer such that N > 1w+ 1 and the prime on the sum over k denotes the
2 2
omission of the term corresponding to k =m when w =2m+1.
The remainder R is
N
1 −c+∞i
R = Γ(s)ζ(w+2s)a−sds, c=N − 1. (2.3)
N 2πiZ 2
−c−∞i
Itis showninthe appendix, whenw 6=2,4,..., thatRN =O(aN−12)asa→0in|arg a|< 21π,
with the constantimplied inthe O-symbol growingatleastlike Γ(N+1−1w). This establishes
2
that the above series over k diverges as N → ∞ and that (2.2) is therefore an asymptotic
expansion.
We remark that the algebraic expansion (2.2) also contains a subdominant exponentially
small component as a → 0; compare [3, §8.1.5] for the particular case w = 0. We do not
consider this further in the present paper.
3. An expansion for S(a;2m) when m=1,2,...
The case w = 2m, where m is a positive integer, is more interesting as this leads to the
analogue of the Poisson-Jacobitransformation (1.1). There is now only a finite set of poles of
the integrand in (2.1) at s = 1 − 1w and s = 0,−1,−2,...,−m, since the poles of Γ(s) at
2 2
s = −m−k (k = 1,2,...) are cancelled by the trivial zeros of the zeta function ζ(2m+2s)
at s = −m−1,−m−2,.... This has the consequence that the integrand is holomorphic in
ℜ(s) < −m, so that further displacement of the contour can produce no additional algebraic
terms in the expansion of S(a;2m). Thus, we find when w =2m
m (−)k
S(a;2m)= 12Γ(21 −m)am−21 + k! ζ(2m−2k)ak+IL, (3.1)
Xk=0
where, upon making the change of variable s→−s,
1
I = Γ(−s)ζ(2m−2s)asds (3.2)
L
2πiZ
L
and L denotes a path parallel to the imaginary axis with ℜ(s)>m.
We now employ the functional relation for ζ(s) given by [5, p. 269]
ζ(s)=2sπs−1ζ(1−s)Γ(1−s)sin1πs (3.3)
2
to convertthe argumentofthe zeta functionin(3.2)into onewithrealpartgreaterthanunity.
The integral in (3.2) can then be written in the form
(−)m(2π)2m Γ(2s−2m+1) a s
ζ(2s−2m+1) ds.
2πi ZL Γ(s+1) (cid:16)4π2(cid:17)
Since onthe integrationpathℜ(2s−2m+1)>1, wecanexpandthe zeta functionandreverse
the order of summation and integration to obtain
∞
IL =(−)mπ2m−21 n2m−1Kn(a;m), (3.4)
nX=1
4 R. B. Paris
where
1 Γ(s−m+ 1)Γ(s−m+1) a s
K (a;m):= 2 ds,
n 2πiZL Γ(s+1) (cid:16)π2n2(cid:17)
and we have employed the duplication formula for the gamma function
Γ(2z)=22z−1π−21 Γ(z)Γ(z+ 1).
2
The integrals K (a;m) have no poles in the half-plane ℜ(s) > m, so that we can displace
n
the path L as far to the right as we please. On such a displaced path |s| is everywhere large.
The quotient of gamma functions may then be expanded by making use of the result given in
[3, p. 53]
Γ(s−m+ 1)Γ(s−m+1) M−1
2 = (−)jc Γ(s+ϑ−j)+ρ (s)Γ(s+ϑ−M) (3.5)
j M
Γ(s+1)
Xj=0
for positive integer M, where ϑ= 1 −2m,
2
(m) (m+ 1) 2−2j(2m)
c = j 2 j = 2j
j
j! j!
and ρ (s)=O(1) as |s|→∞ in |arg s|<π. Substitution of this expansion into the integrals
M
K (a;m) then produces
n
M−1
1 a s
K (a;m) = (−)jc Γ(s+ϑ−j) ds+R
n Xj=0 j 2πiZL (cid:16)π2n2(cid:17) M
= M−1(−)jc a 2m+j−21 e−π2n2/a+R (3.6)
j π2n2 M
Xj=0 (cid:16) (cid:17)
by (2.1), where
1 a s
R = ρ (s)Γ(s+ϑ−M) ds.
M 2πiZL M (cid:16)π2n2(cid:17)
Boundsforthe remainderR havebeenconsideredin[3, p.71,Lemma2.7],whereitisshown
M
that
R =O a M−ϑe−π2n2/a (3.7)
M (cid:18)(cid:16)π2n2(cid:17) (cid:19)
as a→0 in the sector |arg a|< 1π.
2
Collecting together the results in (3.2), (3.4), (3.6) and (3.7), we obtain
a 2m−1 ∞ e−π2n2/a M−1 −a j a M
I =(−)m 2  c +O .
L (cid:16)π(cid:17) nX=1 n2m Xj=0 j(cid:18)π2n2(cid:19) (cid:18)(cid:16)π2n2(cid:17) (cid:19)
 
From (3.1) we now have the following theorem:
Theorem 1. Let m and M be positive integers. Then, when w = 2m, we have the expansion
valid as a→0 in |arg a|< 1π
2
m (−)k
S(a;2m)= 1Γ(1 −m)am−21 + ζ(2m−2k)ak
2 2 k!
kX=0
Poisson-Jacobi-type transformation 5
+(−)m a 2m−21 ∞ Υn(a;m)e−π2n2/a, (3.8)
π n2m
(cid:16) (cid:17) nX=1
where Υ (a;m) has the asymptotic expansion
n
M−1(m) (m+ 1) −a j a M
Υ (a;m)= j 2 j +O .
n Xj=0 j! (cid:18)π2n2(cid:19) (cid:18)(cid:16)π2n2(cid:17) (cid:19)
This is the analogue of the Poisson-Jacobi transformation in (1.1). In the case m = 0, the
quotient of gamma functions in (3.5) is replaced by the single gamma function Γ(s+ 1), with
2
the result that c = 1, c = 0 (j ≥ 1) and Υ (a;m) = 1 for all n ≥ 1. Then (3.8) reduces to
0 j n
(1.1) and is valid for all values of the parameter a (not just a→0) satisfying |arg a|< 1π.
2
Remark 1. We note that the values of the zeta function appearing in (3.8) can be expressed
alternatively in terms of Bernoulli numbers by the result [2, p. 605]
(2π)2n
ζ(2n)= |B |.
2n
2(2n)!
4. Numerical results and concluding remarks
From the well-known values [2, p. 605]
π2 π4
ζ(2)= , ζ(4)= ,
6 90
we obtain from Theorem 1 the expansions in the cases m=1 and m=2 given by
π2 a a 3 ∞ e−π2n2/a M−1 −a j
S(a;2)= 6 + 2 −(πa)12 −(cid:16)π(cid:17)2 nX=1 n2 (cid:26)Xj=0(32)j(cid:18)π2n2(cid:19) +O(aM)(cid:27) (4.1)
and
π4 π2a a2 2
1 3
S(a;4)= − − + π2a2
90 6 4 3
a 7 ∞ e−π2n2/a M−1(5) (2) −a j
+ 2 2 j j +O(aM) (4.2)
(cid:16)π(cid:17) nX=1 n4 (cid:26)Xj=0 j! (cid:18)π2n2(cid:19) (cid:27)
valid as a→0 in |arg a|< 1π.
2
In Table 1 we show the results of numerical calculations for the case m = 2. For different
values of the parameter a we present the value of the absolute error in the computation of
S(a;4) from (4.2). In the computations, we have used only the n= 1 term (since the order of
1exp(−4π2/a) was found to be less than the error), with the expansion for Υ (a;2) optimally
4 1
truncated (corresponding to truncation at, or near, the least term in modulus) at index j ≃
0
(π2/a)−5. Itisseenthatthe errorwhena=0.1isextremely smallandthat, onlywhena≃2
2
does the relative error start to become significant.
To conclude, we mention that a similar treatment can be carried out for the sum
∞ e−anp
S (a;w)≡ (a→0, ℜ(a)>0)
p nw
nX=1
6 R. B. Paris
Table 1: Values of the absolute error in the computation of S(a;4) from (4.2). The value of the index j0
correspondstooptimaltruncationoftheexpansionΥ1(a;2).
a S(a;4) Error j
0
0.10 0.952696 9.662×10−86 96
0.20 0.849025 9.768×10−43 46
0.25 0.803169 4.045×10−34 36
0.50 0.615128 7.769×10−17 17
0.75 0.475493 4.656×10−11 10
1.00 0.369026 3.642×10−8 6
1.50 0.223285 2.856×10−5 3
2.00 0.135356 7.500×10−4 1
forpositiveevenintegerw andp. Thecasew =0andp>0,correspondingtothe Euler-Jacobi
series, has been considered in [3, §8.1]; see also [1] for a hypergeometric approach when p is a
rational fraction. The details of the small-a expansion of S (a;w) will be presented elsewhere.
p
Acknowledgement. The author wishes to acknowledge B. Guralnik for having brought this
problem to his attention
Appendix: A bound for the remainder R
N
Let ψ =arg a and integer N > 1w+ 1. Upon replacement of s by −s followedby use of (3.3),
2 2
the remainder R in (2.3) becomes
N
(2π)w N−12+∞i Γ(1−w+2s) sinπ(s− 1w) a s
R = ζ(1−w+2s) 2 ds.
N 2πi ZN−1−∞i Γ(1+s) sinπs (cid:16)4π2(cid:17)
2
With s=N − 1 +it, t∈(−∞,∞) we have
2
a N−1 ∞ Γ(2N −w+2it)
|R |≤(2π)w−1 2 ζ(2N −w) e−ψt dt,
N (cid:16)4π2(cid:17) Z−∞ (cid:12)(cid:12) Γ(N + 12 +it) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
since |ζ(x+it)|≤ζ(x) (x>1) and
sinπ(N − 1 − 1w+it) |cosπ(1w−it)| (cos2 1πw+sinh2πt)12
2 2 = 2 = 2 ≤1.
(cid:12) sinπ(N + 1 +it) (cid:12) coshπt coshπt
(cid:12) 2 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
It then follows that
a N−1 ∞ Γ(N − 1w+ 1 +it)
|R |=O 2 e−ψt|Γ(N − 1w+it)| 2 2 dt . (A.1)
N (cid:18)(cid:16)π2(cid:17) Z−∞ 2 (cid:12)(cid:12) Γ(N + 21 +it) (cid:12)(cid:12) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
Using the argumentpresentedin [3, p. 126],we set N− 1w− 1 =M+δ, with −1 <δ ≤ 1
2 2 2 2
so that M ≤N −1, to find
Γ(N − 1w+ 1 +it) Γ(1+δ+it)
2 2 =P(t)g(t), g(t):= ,
(cid:12) Γ(N + 1 +it) (cid:12) (cid:12) Γ(1 +it) (cid:12)
(cid:12) 2 (cid:12) (cid:12) 2 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Poisson-Jacobi-type transformation 7
where
P(t)=(1 +t2)−12 Mr=1{(r+δ)2+t2}21 ≤(1 +t2)−12 ≤2.
4 QNr=−11{(r+ 12)2+t2}21 4
Q
From the upper bound for the gamma function Γ(z) with z =x+it, x>0 [3, p. 35]
|Γ(z)| < Γ(x)(1+t2/x2)12x−14e−|t|φ(t)e1/(6|z|), φ(t)=arctan(|t|/x)
< Γ(x)(1+τ2)21x−41 exp[x{ω(τ)− 1π|τ|}e1/(6x)
2
< exΓ(x)(1+τ2)12x−14e−21πx|τ|e1/(6x),
wherewehaveputτ =t/x,definedω(τ)=|τ|arctan(1/|τ|)andusedthefactthat0≤ω(τ)<1
for τ ∈ [0,∞), with the limit 1 being approached as τ → ∞. Substituting the above bounds
into (A.1), we see on setting x=N − 1w that
2
a N−1 ∞
|R |=eNΓ(N−1w+1)O 2 (1+τ2)N/2g(τ){e−∆+τ +e−∆−τ}dτ ,
N 2 (cid:18)(cid:16)π2(cid:17) Z0 (cid:19)
where ∆± = (N − 21w)(12π±ψ). Since g(τ) = O(τδ+21) as τ → ∞, the integral is convergent
provided |ψ|< 1π and is manifestly an increasing function of N.
2
Hence
RN =O(aN−12) (a→0, |arg a|< 12π), (A.2)
with the constant implied in the O-symbol growing at least like Γ(N+1−1w) as N increases.
2
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