Proc. R. Soc. A (2012) 468, 2667–2681doi:10.1098/rspa.2011.0664
Published online 18 April 2012
A remarkable identity involving Bessel functionsBY DIEGO E. DOMINICI1,*, PETER M. W. GILL2 AND
TAWEETHAM LIMPANUPARB2
1Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin,Germany
2Research School of Chemistry, Australian National University,Australian Captial Territory 0200, Australia
We consider a new identity involving integrals and sums of Bessel functions. Theidentity provides new ways to evaluate integrals of products of two Bessel functions.The identity is remarkably simple and powerful since the summand and the integrandare of exactly the same form and the sum converges to the integral relatively fast formost cases. A proof and numerical examples of the identity are discussed.
is ubiquitous in mathematical physics and is essential to an understanding of bothgravitation and electrostatics (Kellogg 1967). It is central in classical mechanics(Goldstein 1980), but plays an equally important role in quantum mechanics(Landau & Lifshitz 1965), where it mediates the dominant two-particle interactionin electronic Schrödinger equations of atoms and molecules.
Although the Newtonian kernel has many beautiful mathematical properties,the fact that it is both singular and long-ranged is awkward and expensive froma computational point of view (Hockney & Eastwood 1981) and this has led toa great deal of research into effective methods for its treatment. Of the manyschemes that have been developed, Ewald partitioning (Ewald 1921), multi-polemethods (Greengard 1987) and Fourier transform techniques (Payne et al. 1992)are particularly popular and have enabled the simulation of large-scale particulateand continuous systems, even on relatively inexpensive computers.
A recent alternative (Gilbert 1996; Varganov et al. 2008; Gill & Gilbert 2009;Limpanuparb & Gill 2009, 2011; Limpanuparb 2011; Limpanuparb et al. 2011,
*Author and address for correspondence: Department of Mathematics, State University of NewYork at New Paltz, 1 Hawk Drive, New Paltz, NY 12561-2443, USA ([email protected]).
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2668 D. E. Dominici et al.
2012) to these conventional techniques is to resolve (1.1), a non-separable kernel,into a sum of products of one-body functions
K (r , r !) =$!
l=0
l!
m="l
Ylm(r)Ylm(r !)Kl(r , r !) =$!
n,l=0
l!
m="l
f%nlm(r)fnlm(r !), (1.2)
where Ylm(r) is a spherical harmonic (Olver et al. 2010) of the angular part ofthree-dimensional vector r ,
Kl(r , r !) = 4p
!$
0
Jl+1/2(kr)Jl+1/2(kr !)k&
rr ! dk, (1.3)
Jl(z) is a Bessel function of the first kind (Olver et al. 2010), and r = |r |. Theresolution (1.2) is computationally useful because it decouples the coordinates rand r ! and allows the two-body interaction integral
E[ra , rb] =!!
ra(r)K (r , r !)rb(r !) dr dr !, (1.4)
between densities ra(r) and rb(r) to be recast as
E[ra , rb] =$!
n=0
$!
l=0
l!
m="l
AnlmBnlm , (1.5)
where Anlm is a one-body integral of the product of ra(r) and fnlm(r). If theone-body integrals can be evaluated efficiently and the sum converges rapidly,(1.5) may offer a more efficient route to E[ra , rb] than (1.4).
The key question is how best to obtain the Kl resolution
Kl(r , r !) =$!
n=0
Knl(r)Knl(r !).
Previous attempts (Gilbert 1996; Varganov et al. 2008; Gill & Gilbert 2009;Limpanuparb & Gill 2009) yielded complicated Knl whose practical utility isquestionable but, recently, we have discovered the remarkable identity
!$
0
Jn(at)Jn(bt)t
dt =$!
n=0
3nJn(an)Jn(bn)
n, (1.6)
where 3n is defined by
3n =
"#
$
12, n = 0
1, n ' 1(1.7)
a, b # [0, p], n = 12 ,
32 ,
52 , . . . and we take the appropriate limit for the n = 0 term
in the sum (1.6). This yields (Limpanuparb et al. 2011) the functions
fnlm(r) =%
4p3n
rnJl+1/2(rn) Ylm(r),
and these provide a resolution that is valid, provided that r < p. We note thatfnlm vanish for n = 0 unless l = 0.
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A remarkable identity 2669
If we write (1.6) as an integral from "$ to $,!$
"$
Jn(at)Jn(bt)t
dt =$!
n="$
Jn(an)Jn(bn)n
, (1.8)
the summand and integrand are of exactly the same form.There have been a number of studies of this kind of sum-integral equality by
various groups, for example, Krishnan & Bhatia in the 1940s (Bhatia & Krishnan1948; Krishnan 1948a,b; Simon 2002) and Boas, Pollard & Shisha in the 1970s(Boas & Stutz 1971; Pollard & Shisha 1972; Boas & Pollard 1973). Their discoverywas inspired by a practice to ‘approximate’ an intractable sum that arises inphysics by an integral. They realized that the ‘approximation’ was in fact exactfor a number of cases.
In this paper, however, our goal is the opposite. We originally aimed to use thesum to approximate the integral but later found that it was exact. Our identity(1.6) is also considerably different from theirs, but we may regard theirs (Boas &Pollard 1973) when c = 0 as a special case of ours when n = 1/2.
The aim of this study was to prove an extended version of the identity (1.6)and demonstrate its viability in approximating the integral of Bessel functions.
2. Preliminaries
The Bessel function of the first kind Jn(z) is defined by Watson (1995)
Jn(z) =$!
n=0
("1)n
G(n + n + 1)n!&z2
'n+2n. (2.1)
It follows from (2.1) that
Jn(z)&z2
'"n
is an entire function of z and we have
limz(0
Jn(z)&z2
'"n
= 1G(n + 1)
.
Gauss’ hypergeometric function is defined by Andrews et al. (1999)
2F1
(a, bc ; z
)=
$!
k=0
(a)k(b)k
(c)k
zk
k! , (2.2)
where (u)k is the Pochhammer symbol (or rising factorial), given by
(u)k = u(u + 1) · · · (u + k " 1).
The series (2.2) converges absolutely for |z | < 1 (Andrews et al. 1999). If Re(c "a " b) > 0, we have (Andrews et al. 1999)
2F1
(a, bc ; 1
)= G(c)G(c " a " b)
G(c " a)G(c " b). (2.3)
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2670 D. E. Dominici et al.
Many special functions can be defined in terms of the hypergeometric function.In particular, the Gegenbauer (or ultraspherical) polynomials C (l)
n (t) are definedby Erdélyi et al. (1981)
C (l)n (x) = (2l)n
n! 2F1
*"n, n + 2l
l + 12
;1 " x
2
+
, (2.4)
with n # N0 andN0 = {0, 1, . . .}.
The even Gegenbauer polynomials can also be expressed in terms of thehypergeometric function by Erdélyi et al. (1981)
C (l)2k (z) = ("1)k (l)k
k! 2F1
*"k, l + k12
; z2
+
, |z | < 1. (2.5)
3. Main result
The discontinuous integral!$
0
Jm(at)Jn(bt)tl
dt,
was investigated by Weber (1873), Sonine (1880) and Schafheitlin (1887). Theyproved that (Watson 1995)
!$
0
Jm(at)Jn(bt)tl
dt = al"n"1bnG((n + m " l + 1)/2)2lG(n + 1)G((l + m " n + 1)/2)
) 2F1
*n + m " l + 1
2,
n " m " l + 12
n + 1;(
ba
)2+
, (3.1)
forRe(m + n + 1) > Re(l) > "1 (3.2)
and 0 < b < a. The corresponding expression for the case when 0 < a < b isobtained from (3.1) by interchanging a, b and also m, n. When a = b, we have(Watson 1995)
!$
0
Jm(at)Jn(at)tl
dt = al"1G(l)2lG((l + m " n + 1)/2)
) G((n + m " l + 1)/2)G((l + n " m + 1)/2)G((n + m + l + 1)/2)
,
provided that Re(m + n + 1) > Re(l) > 0. This result also follows from Gauss’summation formula (2.3) and (3.1).
We will now prove our main theorem.
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A remarkable identity 2671
Theorem 3.1. If 0 < b < a < p, Re(m + n " 2k) > "1, and k # N0, then$!
n=0
3nJm(an)
((1/2)an)m
Jn(bn)((1/2)bn)n
n2k = 4k
a2k+1
G(k + 1/2)G (m " k + 1/2) G(n + 1)
) 2F1
*k + 1
2, k + 1
2" m
n + 1;(
ba
)2+
(3.3)
Proof. From Watson (1995), the Bessel function Jn(z) has the integralrepresentation
Jn+l(z) = ("i)lG(2n)l !((1/2)z)n
G(n + 1/2)G(1/2)G(2n + l)
!p
0eiz cos q sin2n(q)C (n)
l (cos q)dq, (3.4)
valid when Re(n) > " 12 and l # N0. Replacing n by m " 2k, l by 2k, z by an in
(3.4), and changing the integration variable to x = a cos(q), we obtain
Jm(an) = ("1)kG(m " 2k)(2k)!(2an)m"2k
2paG(2m " 2k)
)! a
"aeixn
(1 " x2
a2
)m"2k"1/2
C (m"2k)2k
&xa
'dx , Re(m " 2k) > "1
2.
Since (2.5) implies that C (m"2k)2k (x/a) is an even function, we conclude that
Jm(an)((1/2)an)m
n2k = Amk (a)
! a
0
(1 " x2
a2
)m"2k"1/2
C (m"2k)2k
&xa
'cos(nx) dx , (3.5)
valid when Re(m " 2k) > " 12 , where
Amk (a) = ("1)k(2k)!G(m " 2k)4m"k
pa2k+1G(2m " 2k). (3.6)
Replacing m by n, k by 0 in (3.4), and changing the integration variable from xto y, we have, because C (n)
0 (x) = 1, that
Jn(bn)((1/2)bn)n
= An0(b)
! b
0
(1 " y2
b2
)n"1/2
cos(ny) dy, Re(n) > "12. (3.7)
It follows from (3.5) and (3.7) that$!
n=0
3nJm(an)
((1/2)an)m
Jn(bn)((1/2)bn)n
n2k = Amk (a)An
0(b)
and! a
0
! b
0
(1 " x2
a2
)m"2k"1/2
C (m"2k)2k
&xa
' (1 " y2
b2
)n"1/2
D(x , y) dy dx ,
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2672 D. E. Dominici et al.
where
D(x , y) =$!
n=0
3n cos(nx) cos(ny).
Using the identity (Olver et al. 2010),
12p
$!
n="$einz = d(z), "p < z < p,
where d(z) is the Dirac delta function, we obtain
D(x , y) = 14
$!
n="$[ein(x"y) + ein(x+y)] = p
2[d(x + y) + d(x " y)].
Hence, since 0 < x , y < a < p, we get
$!
n=0
3nJm(an)
((1/2)an)m
Jn(bn)((1/2)bn)n
n2k = p
2Am
k (a)An0(b)
)! b
0
(1 " x2
a2
)m"2k"1/2 (1 " x2
b2
)n"1/2
C (m"2k)2k
&xa
'dx . (3.8)
Using Euler’s transformation (Olver et al. 2010)
2F1
(a, bc ; z
)= (1 " z)c"a"b
2F1
(c " a, c " b
c ; z)
,
in (2.5), we have
(1 " x2
a2
)m"2k"1/2
C (m"2k)2k
&xa
'= ("1)k (m " 2k)k
k!
) 2F1
,
-.k + 1
2, k + 1
2" m
12
;&xa
'2
/
01, |x | < a. (3.9)
Using (3.9) in (3.8), we get
$!
n=0
3nJm(an)
((1/2)an)m
Jn(bn)((1/2)bn)n
n2k = p
2Am
k (a)An0(b)("1)k (m " 2k)k
k!
)! b
0
(1 " x2
b2
)n"(1/2)
2F1
,
-.k + 1
2, k + 1
2" m
12
;&xa
'2
/
01 dx . (3.10)
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A remarkable identity 2673
Making the change of integration variable x = b&
t in (3.10), and setting u =b/a < 1, we obtain
$!
n=0
3nJm(an)
((1/2)an)m
Jn(bn)((1/2)bn)n
n2k = p
2Am
k (a)An0(b)("1)k (m " 2k)k
k!
) b2
! 1
0t"1/2)(1 " t)n"1/2
2F1
,
-.k + 1
2, k + 1
2" m
12
; u2t
/
01 dt.
Using the formula (Olver et al. 2010),! 1
0xs"1(1 " x)c"s"1
2F1
(a, bs ; zx
)dx = G(s)G(c " s)
G(c) 2F1
(a, bc ; z
),
valid for Re(c) > Re(s) > 0, and | arg(1 " x)| < p, with s = 12 and c = n + 1, we get
$!
n=0
3nJm(an)
((1/2)an)m
Jn(bn)((1/2)bn)n
n2k = p
4bAm
k (a)An0(b)("1)k
) (m " 2k)k
k!G(1/2)G(n + 1/2)
G(n + 1) 2F1
*k + 1
2, k + 1
2" m
n + 1; u2
+
.
Using (3.6), we have
p
4bAm
k (a)An0(b)("1)k (m " 2k)k
k!G(1/2)G(n + 1/2)
G(n + 1)
= 4k
a2k+1
G(k + 1/2)G(m " k + 1/2)G(n + 1)
,
and (3.3) follows. !The special case of theorem 3.1 in which k = 0 was derived by Cooke (1928),
as part of his work on Schlömilch series.
Corollary 3.2. If 0 < b < a < p, Re(m + n " 2k) > "1, k # N0, then!$
0
Jm(at)Jn(bt)tm+n"2k dt =
$!
n=0
3nJm(an)Jn(bn)
nm+n"2k .
Proof. From (3.3), we have$!
n=0
3nJm(an)Jn(bn)
nm+n"2k = 1a2k+12m+n"2k
ambnG(k + 1/2)G(m " k + 1/2)G(n + 1)
) 2F1
*k + 1
2, k + 1
2" m
n + 1;(
ba
)2+
.
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2674 D. E. Dominici et al.
Taking l = m + n " 2k in (3.1), we get!$
0
Jm(at)Jn(bt)tm+n"2k dt = am"2k"1bnG(k + 1/2)
2m+n"2kG(n + 1)G(m " k + 1/2)
) 2F1
*k + 1
2, k " m + 1
2n + 1
;(
ba
)2+
,
and the result follows. Note that since for all k # N0
the conditions (3.2) are satisfied. !Corollary 3.3. If 0 < a, b < p, and n = k + 1
2 with k # N0, then$!
n=0
3nJn(an)Jn(bn)
n=
!$
0
Jn(at)Jn(bt)t
dt
Proof. The result is a consequence of corollary 3.2 (taking m = n = k + 12) and
a special case of the integral (3.1) (Watson 1995)
!$
0
Jn(at)Jn(bt)t
dt =
"22#
22$
12n
&ab
'n
, a * b
12n
(ba
)n
, a ' b.
!
4. Alternative approach
After posting our manuscript on ArXiv,1 we received several comments on thepaper. Tom H. Koornwinder,2 observed that a more conceptual proof of corollary3.2 is obtained by the following proposition. The same method was used byBoas and Pollard in Boas & Pollard (1973) to find integral representations ofsome series.
Proposition 4.1. Suppose that f , g # L2[a, b], with
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A remarkable identity 2675
andl = 1
b " a.
Proof. Suppose that f , g # L2[a, b]. Then, Parseval’s relation for complexFourier series gives (Zayed 1996)
! b
af (x)g(x) dx =
$!
n="$f (n) ¯g(n).
On the other hand, we know from Parseval’s relation for the Fourier transform(Zayed 1996) that
!$
"$f (w) ¯g(w) dw =
!$
"$f (x)g(x) dx .
Using (4.1), the result follows. !We can use the above proposition to prove the result in corollary 3.2. Using
the formula (Gradshteyn & Ryzhik 2000)! 1
"1(1 " x2)m"1/2 eiwxC (m)
k (x) dx = p21"minG(2m + k)k!G(m)
w"mJm+k(w),
valid for Re(m) > " 12 , k # N0 we obtain, after some substitutions,
1&2p
! a
"aHk(s; m, a) eiws ds = Jm(aw)
wm"k , Re(m) > k " 12, k # N0
where
Hk(s; m, a) = ak"m"1k!G(m " k)2m+k&
2pinG(2m " k)
(1 " s2
a2
)m"k"1/2
C (m"k)k
& sa
'.
If we defineHk(s; m, a) = 0, s /# ["a, a], 0 < a * p,
thenH k(w; m, a) = Jm(aw)
wm"k , Re(m) > k " 12, k # N0.
Using the proposition with a = p, b = "p, we conclude that!$
"$H k(w; m, a) ¯H l(w; n, b) dw =
$!
n="$H k(n; m, a) ¯H l(n; n, b)
or !$
"$
Jm(aw)Jn(bw)wm+n"k"l dw =
$!
n="$
Jm(an)Jn(bn)nm+n"k"l , (4.2)
withRe(m) > k " 1
2, Re(n) > l " 1
2, k, l # N0
and 0 < a, b * p.
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2676 D. E. Dominici et al.
a = p/4
a = 2p/4
a = 3p/4
2pb
0.05
0.10
0.15
0.20
p/4 p/2 3p/4 5p/4 3p/2 7p/4p
Figure 1. The integral!$
0 (J5/2(at)J5/2(bt)/t) dt (solid) and the truncated sum T10(a, b) (dashed),for a = p/4, 2p/4, 3p/4 and 0 * b * 2p. (Online version in colour.)
Corollary 3.2 follows from (4.2), but the disadvantage of this method is thatit does not yield an expression for the series, as we obtained in theorem 3.1.
5. Numerical results
In previous sections, we have proved the equality of integrals and sums of Besselfunctions. However, before the identity is used in practice, we need to considerits convergence behaviour.
Firstly, the Weierstrass M-test shows that the series converges uniformly as longas m + n " 2k > 1 and m, n > 0 because the numerator in corollary 3.2 is bounded|Jm(at)Jn(bt)| * 1 (Watson 1995).
To explore the rate of convergence of the sum in corollary 3.3, we choose m =n = 5/2. The exact value of the integral is
!$
0
J5/2(at)J5/2(bt)t
dt = 15
%|a|a
%|b|b
3min(|a|, |b|)max(|a|, |b|)
45/2
(5.1)
and truncation of the infinite series yields the finite sum
TN (a, b) =N!
n=1
J5/2(an)J5/2(bn)n
(5.2)
and its error
RN (a, b) =!$
0
J5/2(at)J5/2(bt)t
dt " TN (a, b). (5.3)
The integral in (5.1) and sum in (5.2) are illustrated in figure 1 and their difference(5.3) is shown in figure 2 and table 1.
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A remarkable identity 2677
–2p –p 0 p 2pa
–2p
0
p
2p
b
–0.1
0
0.1
–p
Figure 2. The difference between the integral and the truncated sum R20(a, b)/(&|a|/a
5|b|/b) for
a, b # ["2p, 2p]. (Online version in colour.)
Table 1. RN (a, b) for a = p/2 and various b and N .
The excellent agreement region in figure 1, an apparently flat plateau infigure 2, and the decaying error from the 2nd to the 7th column of table 1 stronglysuggest that corollary 3.3 is true over the larger domain |a| + |b| < 2p for n = 5/2.Additional numerical experiments not shown here suggest that it is true for alln. Furthermore, we believe that this larger domain conjecture also applies tocorollary 3.2. The extension of the 0 < a, b * p proof to other three quadrantsis trivial but we have not yet managed to find a proof for the diamond domain|a| + |b| < 2p.
Table 1 reveals that the rate of convergence of the Bessel sum is stronglydependent on the value of a and b. If a and b are in the domain, where theintegral equals the infinite sum, the difference (5.3) for general m, n, k is only due
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2678 D. E. Dominici et al.
to truncation and can be recast as
RN (a, b) =$!
n=N+1
Jm(an)Jn(bn)nm+n"2k .
The dependence of the truncation error follows directly from the asymptoticexpansion for the Bessel function. From Watson (1995), the first two terms inthat expansion are given by
Jn(z) +(
2pz
)1/2 3cos
(z " 1
2np " 1
4p
)" (n " 1/2)(n + 1/2)
2zsin
(z " 1
2np " 1
4p
)4.
(5.4)
For the parameters considered here, namely, m = n = 52 , we then have, for the
moment keeping only the first term in the expansion,
Jm(an)Jn(bn) + 2np
&ab
cos(
an " 3p
2
)cos
(bn " 3p
2
)
= 2np
&ab
sin(an) sin(bn).
With a = p/2, we then have
Jm
&p
2n'
Jn(bn) + 1n
(2p
)3/2
b"1/2 sin&p
2n'
sin(bn),
in which
sin&p
2n'
=60 n even,("1)(n"1)/2 n odd.
Choosing N + 1 to be odd, the truncation error is then$!
again resulting in terms of order N "4 after cancellation and a sum of order N "3.On the other hand, for b = a = p/2,
sin&p
2n'
sin&p
2n'
= sin2&p
2n'
=60 n even1 n odd
.
We then have$!
n=N+1
Jm((p/2)n)Jn((p/2)n)n
+(
2p
)2 31
(N + 1)2 + 1(N + 3)2 + 1
(N + 3)2 + · · ·4,
and thus the sum is of order N "1.For b = p, the first term in the asymptotic expansion of Jn(pn) is zero, since
sin(pn) = 0 for all n. The second term in the expansion then gives
Jn(pn) + (n " 1/2)(n + 1/2)&2p2n3/2
cos(pn),
from which, with m = n " 52 ,
$!
n=N+1
Jm((p/2)n)Jn(pn)n
+ ("1)N /2 6&
2p3
)3
1(N + 1)3 " 1
(N + 3)3 + 1(N + 5)3 " · · ·
4.
Again, successive terms cancel, leaving terms of order N "4 and the sum is of orderN "3. These explain the N "1 and N "3 orders found in table 1. In this case (m =n = 5/2), we have found that the decay rate is also of order N "p, with 1 * p * 3,for other values of a, b not shown here. For general, parameters a, b, m, n and k,the truncation error may be obtained directly using (5.4).
6. Concluding remark
We provide a proof of a new identity involving integral and sum of Bessel functionsand discuss the rate of convergence of the sum in the identity. Generalization ofthe identity and detailed investigation of its convergence rate should be exploredin the future work.
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2680 D. E. Dominici et al.
The work of D. Dominici was supported by a Humboldt Research Fellowship for ExperiencedResearchers from the Alexander von Humboldt Foundation. P.M.W. Gill thanks the AustralianResearch Council (grants DP0984806 and DP1094170) for funding and the NCI National Facility fora generous grant of supercomputer time. T. Limpanuparb acknowledges the financial support fromthe Royal Thai Government (DPST programme). We wish to express our gratitude to LawrenceGlasser, Tom H. Koornwinder, Christophe Vignat and two anonymous referees who provided uswith invaluable suggestions and comments that greatly improved this paper.
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