TWO DEFINITE INTEGRALS
INVOLVING PRODUCTS OF FOUR LEGENDRE FUNCTIONS
arXiv:1603.03547v2 [math.CA] 29 Apr 2017
YAJUN ZHOU
R1
R1
A BSTRACT. The definite integrals −1 x[Pν ( x)]4 d x and 0 x[Pν ( x)]2 {[Pν ( x)]2 − [Pν (− x)]2 } d x are evaluated in closed form, where Pν stands for the Legendre function of degree ν ∈ C. Special cases of
these integral formulae have appeared in arithmetic studies of automorphic Green’s functions and
Epstein zeta functions.
Keywords: Legendre functions, Bessel functions, asymptotic expansions, automorphic Green’s functions
Subject Classification (AMS 2010): 33C05, 33C15 (Primary), 11F03 (Secondary)
1. I NTRODUCTION
In [15, §6, Eq. 77] and [19, Remark 3.3.4.1, Eq. 3.3.38], we stated the following definite integral
formulae
Z1
2 sin4 (νπ)[ψ(2) (ν + 1) + ψ(2) (−ν) + 28ζ(3)]
4
x[Pν ( x)] d x =
(1.1)
(2ν + 1)2 π4
−1
and
Z1
0
·
¸
4 sin2 (νπ) ψ(0) (ν + 1) + ψ(0) (−ν)
+ γ0 + 2 log 2
x[Pν ( x)] {[Pν ( x)] − [Pν (− x)] } d x =
2
(2ν + 1)2 π2
2
2
2
(1.2)
without elaborating on their proof. Here, the Legendre function of the first kind is defined by the
Mehler–Dirichlet integral [2, p. 22, Eq. 1.6.28]
2
Pν (cos θ ) =
π
Zθ
0
p
cos
(2ν+1)β
2
2(cos β − cos θ )
d β,
θ ∈ (0, π), ν ∈ C;
(1.3)
the polygamma functions ψ(m) ( z) = dm+1 log Γ( z)/ d z m+1 , m ∈ Z≥0 are logarithmic
derivatives
of the
¢
¡
P
Euler gamma function; the Euler–Mascheroni constant γ0 := limn→∞ − log n + nk=1 1k is given by
P
−3
−ψ(0) (1/2) − 2 log2 and Apéry’s constant ζ(3) = ∞
is −ψ(2) (1/2)/14. On the right-hand sides
n=1 n
of (1.1) and (1.2), the expressions are well-defined for ν ∈ C r (Z ∪ {−1/2}), and extend to all ν ∈ C,
by continuity.
Some special cases of (1.1) and (1.2) have shown up in arithmetic studies of automorphic
Green’s functions (see [17, Eqs. 2.3.18 and 2.3.40] as well as [19, Remark 3.3.4.1, Eqs. 3.3.34–
3.3.37]) and Epstein zeta functions [18, Eqs. 4.18 and 4.28], namely,
Z
Z1
Z1
2π4 1
π4
π4
4
4
ζ(3) = −
x[P−1/6 ( x)] d x = −
x[P−1/4 ( x)] d x = −
x[P−1/3 ( x)]4 d x,
(1.4)
189 −1
168 −1
243 −1
Z1
π4
ζ(5) = −
x[P−1/2 ( x)]4 d x,
(1.5)
372 −1
Date: May 2, 2017.
1
2
YAJUN ZHOU
and
Z
8π2 1
x[P−1/6 ( x)]2 {[P−1/6 ( x)]2 − [P−1/6 (− x)]2 } d x = − 3 log 3,
9 0
Z
π2 1
x[P−1/4 ( x)]2 {[P−1/4 ( x)]2 − [P−1/4 (− x)]2 } d x = − log 2,
8 0
Z
3
π2 1
x[P−1/3 ( x)]2 {[P−1/3 ( x)]2 − [P−1/3 (− x)]2 } d x = 2 log2 − log 3,
27 0
2
2 Z1
π
x[P−1/2 ( x)]2 {[P−1/2 ( x)]2 − [P−1/2 (− x)]2 } d x = − ζ(3).
7 0
(1.6)
(1.7)
(1.8)
(1.9)
P
−5
Here, ζ(5) = ∞
n=1 n . These special cases have already been verified by independent methods
in the aforementioned references. For example, (1.6)–(1.8) evaluate, respectively, the following
weight-6 automorphic Green’s functions [cf. 17, Eqs. 2.3.18–2.3.20]:
!
Ã
Ã
p !
p
¶
µ
3
+
i
3
i
−
1
i
i
H/Γ0 (1) 1 + i 3
H
/
Γ (3)
H/Γ (2)
G3
and G 3 0
(1.10)
, i , G3 0
,p
,p ,
2
2
6
3
2
where
H/Γ0 ( N )
G3
2
( z 1 , z 2 ) := −
X
a,b,c,d ∈Z
ad − N bc=1

¯
¯2 
¯
az +b ¯
¯ z1 − N cz22 +d ¯


Q 2 1 +
,
az2 +b
2 Im z1 Im N cz2 +d
(1.11)
+1
with Q 2 ( t) = − 32t + 3 t 4−1 log tt−
1 for t > 1. One can compute these special values of automorphic
Green’s functions by other means (see [4, Chap. IV, Proposition 2] and [19, Remark 3.3.4.1]). In
addition to descending from the symmetry of Epstein zeta functions [18, Eq. 4.18], the evaluation
in (1.5) has also appeared in the studies of lattice sums by Wan and Zucker [13, Eq. 42].
In this article, we establish (1.1) and (1.2) in full generality, drawing on an analytic technique
explored extensively in [15, 16]: the Hansen–Heine scaling limits that relate Legendre functions
Pν of large degrees |ν| → ∞ to Bessel functions (see [14, §5.7 and §5.71] as well as (2.4) and (2.5) in
this article). The article is organized as follows. In §2, we present a collection of integrals, old and
new, over products of four Bessel functions. In §3, we investigate the asymptotic behavior of the
definite integrals in (1.1) and (1.2), based on the analysis in §2, and verify the integral formulae
for all ν ∈ C by contour integrations.
2. S OME DEFINITE INTEGRALS INVOLVING PRODUCTS OF FOUR B ESSEL FUNCTIONS
We use Pν to define the Legendre functions of the second kind, as follows:
π
Q ν ( x ) :=
[cos(νπ)Pν ( x) − Pν (− x)], ν ∈ C r Z; Q n ( x) := lim Q ν ( x),
ν→ n
2 sin(νπ)
n ∈ Z≥0
(2.1)
for −1 < x < 1. For ν ∈ C, −π < arg z ≤ π, the Bessel functions Jν and Yν are defined by
³ z ´2k+ν
∞
X
Jµ ( z) cos(µπ) − J−µ ( z)
(−1)k
,
Yν ( z) := lim
,
Jν ( z) :=
µ→ν
sin(µπ)
k=0 k !Γ( k + ν + 1) 2
(2.2)
in parallel to the modified Bessel functions I ν and K ν :
³ z ´2k+ν
∞
X
I −µ ( z) − I µ ( z)
π
1
,
K ν ( z) := lim
.
I ν ( z ) :=
2 µ→ν
sin(µπ)
k=0 k !Γ( k + ν + 1) 2
(2.3)
As in [15, §3], our arguments in this paper draw heavily on the following asymptotic formulae
that connect Legendre functions Pν and Q ν to Bessel functions J0 and Y0 (see [5, Eqs. 43 and 46]
TWO DEFINITE INTEGRALS INVOLVING PRODUCTS OF FOUR LEGENDRE FUNCTIONS
or [7, Chap. 12, Eqs. 12.18 and 12.25]):
s
¶
µ
¶
1
(2ν + 1)θ
Pν (cos θ ) =
+O
,
J0
sin θ
2
2ν + 1
s
¶
µ
¶
µ
θ
1
(2ν + 1)θ
π
+O
,
Y0
Q ν (cos θ ) = −
2 sin θ
2
2ν + 1
θ
µ
3
(2.4)
(2.5)
where the error bounds are uniform for θ ∈ (0, π/2], so long as |ν| → ∞, −π < arg ν < π [5, 7].
Lemma 2.1. We have the following evaluations:
Z∞
Z∞
1
3
xJ0 ( x)[Y0 ( x)] d x =
x[ J0 ( x)]3 Y0 ( x) d x = − ,
4π
0
0
and a vanishing identity:
Z∞
©
ª
x[ J0 ( x)]2 [ J0 ( x)]2 − 3[Y0 ( x)]2 d x = 0,
where it is understood that
0
R∞
0
f ( x) d x := lim M →+∞
RM
0
(2.6)
(2.7)
f ( x) d x.
Proof. We first point out that the Hankel function H0(1) ( z) = J0 ( z) + iY0 ( z) satisfies
Z+∞
P
z[ H0(1) ( z)]4 d z = 0,
(2.8)
−∞+ i0+
³R +
R+∞
RM ´
i0
where the Cauchy principal value is taken: P −∞+ i0+ f ( z) d z := lim M →+∞ −M + i0+ + i0+ f ( z) d z.
The formula above follows directly from the asymptotic behavior [14, §7.2, Eq. 1]
s
·
µ ¶¸
2 i( z− π )
1
(1)
4
, | z| → +∞, −π < arg z < 2π,
(2.9)
e
1+O
H0 ( z) =
πz
| z|
and an application of Jordan’s lemma to a semi-circle in the upper half-plane. Noting that
H0(1) (− x + i 0+ ) = − J0 ( x) + iY0 ( x) for x > 0, we can convert (2.8) into
½Z∞
¾
Z∞
3
3
8i
x[ J0 ( x)] Y0 ( x) d x −
xJ0 ( x)[Y0 ( x)] d x = 0,
(2.10)
0
0
which implies the first equality in (2.6).
We now use asymptotic analysis and residue calculus to establish a formula
¾
Z+∞ ½
1 − e4 iz
z[ J0 ( z)]2 [ H0(1) ( z)]2 −
P
d z = 0,
π2 z
−∞+ i0+
and read off its imaginary part as
Z
Z∞
1 ∞ sin(4 x) d x
3
4
= 0.
x[ J0 ( x)] Y0 ( x) d x + 2
π −∞
x
0
Thus we arrive at the evaluation
Z∞
1
x[ J0 ( x)]3 Y0 ( x) d x = − ,
4π
0
R∞ sin x
upon invoking the familiar Dirichlet integral −∞ x d x = π.
To prove (2.7), simply rewrite
Z+∞
P
zJ0 ( z)[ H0(1) ( z)]3 d z = 0
−∞+ i0+
(2.11)
(2.12)
(2.13)
(2.14)
with the knowledge that [ H0(1) ( x)]3 −[ H0(1) (− x+ i 0+ )]3 = [ J0 ( x)+ iY0 ( x)]3 −[− J0 ( x)+ iY0 ( x)]3 = 2[ J0 ( x)]3 −
6 J0 ( x)[Y0 ( x)]2 .
4
YAJUN ZHOU
Remark We note that the following integrals for Re ν > −1/2
Z∞
x[ Jν ( bx)]2 Jν ( cx)Yν ( cx) d x = 0,
0
and
Z∞
0
x[ Jν ( bx)]2 Jν ( cx)Yν ( cx) d x = −
1
,
2πbc
0<b< c
(2.15a)
0< c<b
(2.15b)
have been tabulated in [10, item 2.13.24.2]. Formally, our (2.13) is an average of the two integral
formulae displayed above.
Lemma 2.2. We have the following evaluation:
Z∞
©
ª
14ζ(3)
x [ J0 ( x)]4 − 6[ J0( x)Y0 ( x)]2 + [Y0 ( x)]4 d x = −
.
π4
0
Proof. We recast the stated integral into
Z∞
14ζ(3)
z[ H0(1) ( z)]4 d z = −
Re
.
π4
0
(2.16)
(2.17)
To show the identity above, deform the contour to the positive Im z-axis, use the relation H0(1) ( i y) =
2 iK 0( y)/π, ∀ y > 0, along with the following formula:
Z∞
7ζ(3)
t[K 0 ( t)]4 d t =
,
(2.18)
8
0
which is found in [8, p. 6], [3, Eq. 202] and [1, p. 19, Table 1]. (Thanks to the extensive and
systematic research
on “Bessel moments” in the references just mentioned, it is now possible
R∞
to evaluate 0 tℓ+mn [K m ( t)]n d t for ℓ, m ∈ Z≥0 , n ∈ {1, 2, 3, 4} in closed form, as implemented in
symbolic computation software like Mathematica. In contrast, only sporadic results are known
when five or more Bessel factors are involved [1, §5 and §6].)
Lemma 2.3. We have the following integral identity:
¶
Z∞ µ
1 − cos(4 x)
2
2
2
x[ J0 ( x)] {[Y0 ( x)] − [ J0 ( x)] } +
d x = 0.
π2 x
0
Proof. The claimed formula is equivalent to the statement that
¾
Z∞ ½
1 − e4 iz
(1)
2
2
Re
− z[ J0 ( z)] [ H0 ( z)] d z = 0.
π2 z
0
(2.19)
(2.20)
Deforming the contour of integration to the positive Im z-axis, we see that it suffices to verify
another vanishing identity:
¾
Z∞ ½
1 − e−4 y
2
2
(2.21)
− 4 y[ I 0 ( y)] [K 0 ( y)] d y = 0.
y
0
To put the formula above in broader context, we will demonstrate that
Z∞
©
ª
y I ν−1/2 ( y) I ν+1/2 ( y)K ν−1/2 ( y)K ν+1/2 ( y) − [ I ν ( y)]2 [K ν ( y)]2 d y = 0
(2.22)
0
holds as long as Re ν > −1/2. We start from the following identity [cf. 14, §13.6, Eq. 3]:
Zπ/2
1
J2ν (2 y tan φ) d φ
,
Re ν > − ,
I ν ( y)K ν ( y) =
cos φ
2
0
(2.23)
TWO DEFINITE INTEGRALS INVOLVING PRODUCTS OF FOUR LEGENDRE FUNCTIONS
5
and transform it with the aid of a standard recurrence formula 2∂ Jν ( z)/∂ z = Jν−1 ( z) − Jν+1 ( z) for
Bessel functions [14, §3.2, Eq. 2]. In particular, we have
Zπ/2
J2ν−1 (2 y tan θ ) sin θ d θ
,
Re ν > 0,
(2.24)
I ν−1 ( y)K ν−1 ( y) − I ν ( y)K ν ( y) =
y
0
Zπ/2
J2ν (2 y tan θ ) sin θ d θ
1
I ν−1/2 ( y)K ν−1/2 ( y) − I ν+1/2 ( y)K ν+1/2 ( y) =
,
Re ν > − ,
(2.25)
y
2
0
after integration by parts. If we now denote the left-hand side of (2.22) by f (ν), then
Z∞
Ï
sin θ
×
f (ν) − f (ν − 1/2) =
dy
dθ dφ
cos φ
0
0<θ<π/2,0<φ<π/2
× [ J2ν−1 (2 y tan θ ) J2ν (2 y tan φ) − J2ν (2 y tan θ ) J2ν−1 (2 y tan φ)],
where the integration over y has a closed form [14, §13.42, Eq. 8]:
(
Z∞
bµ−1 /aµ , a > b > 0,
Jµ (at) Jµ−1 ( bt) d t =
0,
0 < a < b,
0
(2.26)
(2.27)
and one may indeed interchange the order of integrations, upon introducing a convergent factor
e−ε y , ε → 0+ . This brings us
µ
µ
¶
¶
Ï
Ï
tan φ 2ν sin θ
tan θ 2ν cos θ
dθ dφ −
d θ d φ. (2.28)
f (ν) − f (ν − 1/2) =
2 cos φ
2 sin φ
0<φ<θ<π/2 tan θ
0<θ<φ<π/2 tan φ
Here, both double integrals are convergent when Re ν > 0, and they exactly cancel each other (as
is evident from variable substitutions θ 7→ π2 − θ and φ 7→ π2 − φ). Thus, we have f (ν) = f (ν − 1/2)
as long as Re ν > 0. Therefore, the relation f (ν) = limn→∞ f (ν + n/2) holds for Re ν > −1/2. When
ν > −1/2, we can compute the limit limn→∞ f (ν + n/2) = 0 through the dominated convergence
theorem. Finally, we claim that f (ν) = 0, Re ν > −1/2 follows from analytic continuation.
R∞ k
Remark It is worth noting that the “Bessel moments” in the form of 0 t [ I 0 ( t)]2 [K 0 ( t)]n−2 d t
(where n ≥ 5, k ≥ 0) appear in the investigations of two-dimensional quantum field theory [1,
Eq. 13]. More generally, in D -dimensional space, the free propagator of a massive particle is
expressible in terms of K (D −2)/2 [3, Eq. 6], and the product of Bessel functions I (D −2)/2 K (D −2)/2
arises from angular average of the aforementioned propagator [3, Eq. 227]. Thus, it might be
appropriate to ask for some physical interpretations or implications of the cancellation formula
in (2.22).
3. P ROOF OF THE MAIN IDENTITIES
In this section, we shall refer to the left- and right-hand sides of (1.1) (resp. (1.2)) as A L (ν) and
A R (ν) (resp. B L (ν) and B R (ν)). It is clear that all these four functions are invariant under the
variable substitution ν 7→ −ν − 1. Furthermore, one can show that the Taylor expansions
8π3
(ν − n)3 + O((ν − n)4 ),
3
2
(2ν + 1) B R (ν) = 2(ν − n) + O((ν − n)2 )
(2ν + 1)2 A R (ν) = 4(ν − n) −
(3.1)
(3.2)
hold around every non-negative integer n. Before achieving our final goal of proving A L (ν) =
A R (ν) and B L (ν) = B R (ν), we need to check that the Taylor expansions of (2ν + 1)2 A L (ν) and
(2ν + 1)2 B L (ν) agree with their respective counterparts in (3.1) and (3.2).
Lemma 3.1. (a) We have an integral formula for ν ∈ C:
Z1
sin(2νπ) cos(νπ)
(2ν + 1)2
x[Pν ( x)]3 Pν (− x) d x =
π
−1
(3.3)
6
YAJUN ZHOU
and a vanishing identity for n ∈ Z≥0 :
¾
½
Z1
4
2
2
xP n ( x)Q n ( x) 2 [Q n ( x)] − [P n ( x)] d x = 0.
π
−1
(3.4)
(b) We have the following Taylor expansions
8π2
(ν − n)3 + O((ν − n)4 ),
3
(2ν + 1)2 B L (ν) = 2(ν − n) + O((ν − n)2 )
(2ν + 1)2 A L (ν) = 4(ν − n) −
(3.5)
(3.6)
as ν approaches n ∈ Z≥0 .
Proof. (a) The proof of (3.3) is found in [15, Corollary 5.3]. It was based on the Tricomi pairing
(see [12, §4.3, Eq. 2] or [6, Eq. 11.237])
¸
· Z1
¸
· Z1
Z1
Z1
f ( ξ) d ξ
g ( ξ) d ξ
dx+
g ( x) P
dx =0
(3.7)
f ( x) P
−1
−1 π( x − ξ)
−1
−1 π( x − ξ)
of the following formulae [15, Proposition 5.2 and Corollary 5.3]
Z1
2 sin(νπ)
P ν ( ξ) P ν ( − ξ) d ξ
P
= [Pν ( x)]2 − [Pν (− x)]2 ,
π
x−ξ
−1
Z1
2 sin(2νπ)
ξ P ν ( ξ) P ν ( − ξ) d ξ
2 sin(νπ)
P
= x{[Pν ( x)]2 − [Pν (− x)]2 } −
,
π
x−ξ
(2ν + 1)π
−1
which are valid for x ∈ (−1, 1), ν ∈ C, along with an integral evaluation [15, Eq. 19(0,ν) ]
Z1
2 cos(νπ)
, ν ∈ C r {−1/2}.
P ν ( ξ) P ν ( − ξ) d ξ =
2ν + 1
−1
To justify (3.4), we need the Tricomi pairing of the following formulae for n ∈ Z≥0 :
Z1
4
P n (ξ)Q n (ξ) d ξ
4
= 2 [Q n ( x)]2 − [P n ( x)]2 ,
P
2
x−ξ
π
π
−1
¾
½
Z1
4
ξP n (ξ)Q n (ξ) d ξ
4
2
2
P
= x 2 [Q n ( x)] − [P n ( x)] .
x−ξ
π2
π
−1
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
To prove (3.11), apply the Hardy–Poincaré–Bertrand formula (see [12, §4.3, Eq. 4] or [6,
Eq. 11.52]) to the Neumann integral representation of Q n [6, Eq. 11.269]:
Z1
P n ( ξ) d ξ
Q n ( x) = P
, ∀ x ∈ (−1, 1), ∀ n ∈ Z≥0.
(3.13)
−1 2( x − ξ)
R1
To deduce (3.12) from (3.11), use a vanishing identity −1 P n ( x)Q n ( x) d x = 0, ∀ n ∈ Z≥0 [11, item
2.18.13.3].
(b) The symmetry of Legendre polynomials P n (− x) = (−1)n P n ( x), n ∈ Z≥0 gives (2 n + 1)2 A L ( n) =
(2 n + 1)2 B L ( n) = 0. So it remains to check the derivatives of A L (ν) and B L (ν) at ν = n ∈ Z≥0 . In
what follows, we define
¯
∂m ¯¯
{ m}
P n ( x ) :=
Pν ( x), m ∈ Z>0 ,
(3.14)
∂νm ¯ν=n
to simplify notations.
For the linear order Taylor expansions of (2ν + 1)2 A L (ν) and (2ν + 1)2 B L (ν), we compute
Z1
(2 n + 1)2 A ′L ( n) = 4(2 n + 1)2
x[P n ( x)]3 P n{1} ( x) d x
(3.15)
−1
TWO DEFINITE INTEGRALS INVOLVING PRODUCTS OF FOUR LEGENDRE FUNCTIONS
and
(2 n + 1)
2
B′L ( n) = 2(2 n + 1)2
= 2(2 n + 1)
2
Z1
0
Z1
−1
7
x[P n ( x)]2 {P n ( x)P n{1} ( x) − P n (− x)P n{1} (− x)} d x
x[P n ( x)]3 P n{1} ( x) d x.
(3.16)
Differentiating (3.3) at ν = n ∈ Z≥0 , we obtain
Z1
Z1
2
2
{1 }
2
3(2 n + 1)
x[P n ( x)] P n (− x)P n ( x) d x + (2 n + 1)
x[P n ( x)]3 P n{1} (− x) d x
−1
−1
¯
Z1
¯
sin(2
νπ) cos(νπ)
∂
¯
= 2(−1)n ,
= 2(2 n + 1)2(−1)n
x[P n ( x)]3 P n{1} ( x) d x =
¯
∂ν
π
−1
ν= n
(3.17)
(3.18)
which yields (2 n + 1)2 A ′L ( n) = 2(2 n + 1)2 B′L ( n) = 4 for all n ∈ Z≥0 .
For the quadratic order expansion of (2ν + 1)2 A L (ν), we compute
(2 n + 1)2 A ′′L ( n) + 8(2 n + 1) A ′L ( n)
Z1
Z1
2
3 {2 }
2
= 4(2 n + 1)
x[P n ( x)] P n ( x) d x + 12(2 n + 1)
x[P n ( x)]2 [P n{1} ( x)]2 d x +
−1
−1
32
,
2n + 1
(3.19)
and compare it to the second-order derivative of (3.3):
½Z1
¾
Z1
2
{1 }
2
2 {1 }
{1 }
6(2 n + 1)
xP n ( x)P n (− x)[P n ( x)] d x +
x[P n ( x)] P n ( x)P n (− x) d x
−1
−1
½ Z1
¾
Z1
16(−1)n
2
2
{2 }
3 {2 }
+ (2 n + 1) 3
x[P n ( x)] P n (− x)P n ( x) d x +
x[P n ( x)] P n (− x) d x +
2n + 1
−1
−1
½
¾
Z1
Z1
16
2
n
3 {2 }
2
2
{1 }
2
= (−1) 2(2 n + 1)
x[P n ( x)] P n ( x) d x + 6(2 n + 1)
x[P n ( x)] [P n ( x)] d x +
2n + 1
−1
−1
¯
sin(2νπ) cos(νπ)
∂2 ¯¯
= 0.
(3.20)
=
¯
2
π
∂ν ν=n
Here, the integral
Z1
−1
x[P n ( x)]2 P n{1} ( x)P n{1} (− x) d x
(3.21)
vanishes because the integrand is an odd function. This shows that (2ν + 1)2 A L (ν) = 4(ν − n) +
O((ν − n)3 ).
To prepare for the computation of
A ′′′
L ( n)
Z1
Z1
Z1
{1 }
3
2 {1 }
{2 }
= 24
xP n ( x)[P n ( x)] d x + 36
x[P n ( x)] P n ( x)P n ( x) d x + 4
x[P n ( x)]3 P n{3} ( x) d x
−1
−1
(3.22)
−1
occurring in the cubic order expansion, we rewrite (3.4) with 2Q n ( x) = P n{1} ( x) − (−1)n P n{1} (− x)
(a consequence of (2.1)), which leads us to
Z1
Z1
{1 }
n {1 }
3
2
xP n ( x)[P n ( x) − (−1) P n (− x)] d x = π
x[P n ( x)]3 [P n{1} ( x) − (−1)n P n{1} (− x)] d x
−1
−1
2
= 2π
Z1
−1
x[P n ( x)]3 P n{1} ( x) d x =
2π2
.
(2 n + 1)2
It is then clear that
¯
½Z 1
¾
Z1
∂3 ¯¯
12(−1)nπ2
3
3
x
[
P
(
x
)]
P
(
−
x
)
d
x
−
x
[
P
(
−
x
)]
P
(
x
)
d
x
+
ν
ν
ν
ν
∂ν3 ¯
(2 n + 1)2
ν= n
−1
−1
(3.23)
8
YAJUN ZHOU
Z1
xP n ( x){[P n{1}( x)]3 − (−1)n [P n{1} (− x)]3 } d x
−1
Z1
n
+ 18(−1)
x[P n ( x)]2 [P n{1} ( x)P n{2} ( x) − P n{1} (− x)P n{2} (− x)] d x
−1
Z1
− 2(−1)n
x[P n ( x)]3 [P n{3} ( x) − (−1)n P n{3} (− x)] d x = 2(−1)n A ′′′
L ( n ).
= 12(−1)
n
(3.24)
−1
This gives the formula
16π2
288
−
.
(2 n + 1)4 (2 n + 1)2
The Taylor expansion in (3.5) can now be verified.
A ′′′
L ( n) =
(3.25)
Proposition 3.2. The integral identity in (1.1) holds for all ν ∈ C, where the evaluations of the
right-hand side at ν ∈ {−1/2} ∪ Z are interpreted as limits of a continuous function in ν.
Proof. Similar to [15, Proposition 3.1], we need to bound the expression
A (ν) :=
(2ν + 1)2
sin4 (νπ)
[ A L (ν) − A R (ν)]
(3.26)
on a square contour C N in the complex ν-plane, with vertices 1−44 N − iN, 1+44 N − iN, 1+44 N + iN and
1−4 N
4 + iN , where N ∈ Z>0 . Observe that both | cot(νπ)| and 1/| sin(νπ)| are bounded on C N , and
the uniform bounds do not depend on N .
First, we consider the case where ν = 1+44 N for a large positive integer N . By the asymptotic
formulae in (2.4) and (2.5), as well as the integral evaluations in (2.6), (2.7) and (2.16), we see
that the predominant contribution to
Z1
©
ª
A L (ν) =
x [Pν ( x)]4 − [Pν (− x)]4 d x
0
=
is
Zπ/2 ½
0
¸4 ¾
·
2 sin(νπ)
Q ν (cos θ )
sin θ cos θ d θ
[Pν (cos θ )] − cos(νπ)Pν (cos θ ) −
π
4
(3.27)
¶¸ ·
µ
¶
µ
¶¸ ¾
Z∞ ½· µ
(2ν + 1)θ 4
(2ν + 1)θ
(2ν + 1)θ 4
θ J0
− cos(νπ) J0
+ sin(νπ)Y0
dθ
2
2
2
0
¶¸
· µ
¶ µ
¶¸ · µ
¶¸ ¾
Z∞ ½· µ
(2ν + 1)θ 4
(2ν + 1)θ
(2ν + 1)θ 2
(2ν + 1)θ 4
4
= − sin (νπ)
θ J0
− 6 J0
Y0
+ Y0
dθ
2
2
2
2
0
¶¸ ½· µ
¶¸
· µ
¶¸ ¾
Z∞ · µ
(2ν + 1)θ 2
(2ν + 1)θ 2
(2ν + 1)θ 2
2
+ 2 sin (νπ)
θ J0
J0
− 3 Y0
dθ
2
2
2
0
µ
¶ µ
¶ ½· µ
¶¸ · µ
¶¸ ¾
Z∞
(2ν + 1)θ
(2ν + 1)θ
(2ν + 1)θ 2
(2ν + 1)θ 2
− 2 sin(νπ) cos(νπ)
θ J0
Y0
J0
+ Y0
dθ
2
2
2
2
0
¶ µ
¶½· µ
¶¸ · µ
¶¸ ¾
µ
Z
(2ν + 1)θ
(2ν + 1)θ 2
(2ν + 1)θ 2
(2ν + 1)θ
sin(4νπ) ∞
Y0
J0
− Y0
dθ
θ J0
−
2
2
2
2
2
0
·
¸
14ζ(3)
4
sin(νπ) cos(νπ)
4
=
sin (νπ) +
,
(3.28)
4
π
π
(2ν + 1)2
which is compatible with the expansion
·
µ
¶¸
cos(νπ)
14ζ(3)
1
4
A R (ν)
=
+
+O
.
4
4
3
2
π
(2ν + 1)
(2ν + 1)2
sin (νπ)
π sin (νπ)
We break down the error bound for A (ν) into three parts.
(3.29)
TWO DEFINITE INTEGRALS INVOLVING PRODUCTS OF FOUR LEGENDRE FUNCTIONS
(i)
9
By the uniform approximations in the Hansen–Heine scaling limits ((2.4) and (2.5)), we have
¶¸ ·
µ
¶
µ
¶¸ ¾
Zπ/2 ½· µ
(2ν + 1)θ
(2ν + 1)θ 4 θ 2 cot θ d θ
(2ν + 1)θ 4
− cos(νπ) J0
+ sin(νπ)Y0
J0
2
2
2
sin4 (νπ)
0
¶
·
¸4 ¾
µ
Zπ/2 ½
2 sin(νπ)
sin θ cos θ d θ
1
4
.
[Pν (cos θ )] − cos(νπ)Pν (cos θ ) −
=
Q ν (cos θ )
+O
π
(2ν + 1)5/2
0
sin4 (νπ)
(3.30)
Concretely speaking, the error term is bounded by
¶
¶
µ
Zπ/2 µ
(2ν + 1)θ 2
1
θ cot θ d θ
p
O
2ν + 1 0
2
µ
¶
Z(2ν+1)π/4
1
2 x/(2ν + 1)
=O
xdx ,
p ( x)
tan(2 x/(2ν + 1))
(2ν + 1)3 0
(3.31)
where p( x) is a bivariate cubic polynomial in J0 ( x) and Y0 ( x), whose coefficients involve positive integer powers of cot(νπ) and 1/ sin(νπ). For x ∈ (0, 1), the integrand in (3.31) is bounded;
for x ∈ [1, (2ν + 1)π/4), we have an upper bound for x3/2 | p( x)|. Overall, (3.31) is bounded by
R(2ν+1)π/4 −1/2
O((2ν + 1)−3 0
x
d x) = O((2ν + 1)−5/2 ), as stated in (3.30).
p
(ii) As 1 − θ cot θ = O((2ν + 1)−1 ) for θ ∈ (0, 1/ 2ν + 1), we have an estimate
¶¸ ·
¶
µ
¶¸ ¾
µ
Z1/p2ν+1 ½· µ
(2ν + 1)θ 4
(2ν + 1)θ 4 θ 2 cot θ d θ
(2ν + 1)θ
J0
+ sin(νπ)Y0
− cos(νπ) J0
2
2
2
0
sin4 (νπ)
p
¶¸ ·
µ
¶
µ
¶¸ ¾
Z1/ 2ν+1 ½· µ
(2ν + 1)θ
(2ν + 1)θ 4
dθ
(2ν + 1)θ 4
− cos(νπ) J0
+ sin(νπ)Y0
=
θ J0
2
2
2
sin4 (νπ)
0
µ
¶
1
+O
.
(3.32)
(2ν + 1)11/4
The rationale behind this is similar to (i): one may dissect the error term
!
!
Ã
Ã
¶
Z1/p2ν+1 µ
Zp2ν+1/2
(2ν + 1)θ
1
1
p
p ( x) x d x
θ dθ = O
O
2ν + 1 0
2
(2ν + 1)3 0
p
into contributions from the ranges x ∈ (0, 1) and x ∈ [1, 2ν + 1/2).
(iii) Using the asymptotic behavior of Bessel functions for large positive x (cf. (2.9))
s
s
·
µ ¶¸
·
µ ¶¸
³
³
2
2
1
1
π´
π´
J0 ( x) =
1+O
1+O
, Y0 ( x) =
,
cos x −
sin x −
πx
4
x
πx
4
x
(3.33)
(3.34)
along with integration by parts, we are able to verify that
¶¸ ·
µ
¶
µ
¶¸ ¾
½· µ
Zπ/2
(2ν + 1)θ
(2ν + 1)θ 4 θ 2 cot θ d θ
(2ν + 1)θ 4
− cos(νπ) J0
+ sin(νπ)Y0
J0
p
2
2
2
1/ 2ν+1
sin4 (νπ)
µ
¶
1
=O
(3.35)
(2ν + 1)5/2
and
Z∞
½·
µ
J0
µ
¶
1
=O
.
(2ν + 1)5/2
θ
p
1/ 2ν+1
(2ν + 1)θ
2
¶¸4
·
µ
¶
µ
¶¸ ¾
(2ν + 1)θ
(2ν + 1)θ 4
dθ
− cos(νπ) J0
+ sin(νπ)Y0
2
2
sin4 (νπ)
(3.36)
10
YAJUN ZHOU
In sum, we have the following bound estimate
µ
A (ν) = O p
1
2ν + 1
¶
(3.37)
when ν = 1+44 N for a large positive integer N .
Then, for generic ν ∈ C N satisfying Re ν ≥ −1/2, we argue that (3.37) remains valid, upon modifying steps (i)–(iii) by contour deformations. For example, a variation on the derivations in step
(i) brings us
µ
¶
¶
µ
Z|2ν+1|π/4
1
1
2 x/(2ν + 1)
O
,
(3.38)
xdx = O
p ( x)
tan(2 x/(2ν + 1))
(2ν + 1)3 0
(2ν + 1)5/2
while we can bound the following integral on a circular arc in the complex z-plane
Z|2ν+1|π/4
2 z/(2ν + 1)
zdz
(3.39)
p( z)
tan(2 z/(2ν + 1))
(2ν+1)π/4
in the spirit of Jordan’s lemma.
By virtue of the reflection symmetry A (ν) = A (−ν − 1), we have thus confirmed the error bound
in (3.37) for all ν ∈ C N . Furthermore, in view of (3.1) and (3.5), we are sure that A (ν) is analytic
in the region bounded by the contour C N .
Finally, by Cauchy’s integral formula, we see that
I
1
A ( z) d z
A (ν) =
lim
(3.40)
2π i N →∞ C N z − ν
vanishes identically, hence A L (ν) = A R (ν), ∀ν ∈ C.
Proposition 3.3. The integral identity in (1.2) holds for all ν ∈ C, where the evaluations of the
right-hand side at ν ∈ {−1/2} ∪ Z are interpreted as limits of a continuous function in ν.
Proof. We will only show that
B (ν) :=
(2ν + 1)2
sin2 (νπ)
µ
1
[B L (ν) − B R (ν)] = O p
2ν + 1
¶
(3.41)
for ν = 1+44 N , where N is a large positive integer. The rest of the procedures are essentially similar
to those in Proposition 3.2.
Through direct expansions of the digamma functions, we have
µ ¶
(2ν + 1)2
2 cot(νπ) 4(γ0 + 2 log2 + log ν)
1
.
(3.42)
+
B R (ν) =
+O
2
2
π
ν
π
sin (νπ)
In the meantime, we approximate B L (ν) by
Z(2ν+1)π/4
ª
©
e
B(ν) =
[ J0 ( x)]2 [ J0 ( x)]2 − [cos(νπ) J0 ( x) + sin(νπ)Y0 ( x)]2
4x d x
2 x/(2ν + 1)
tan(2 x/(2ν + 1)) (2ν + 1)2
0
Z(2ν+1)π/4
ª 2 x/(2ν + 1) 4 x sin2 (νπ) d x
©
=
[ J0 ( x)]2 [ J0 ( x)]2 − [Y0 ( x)]2 − 2 cot(νπ) J0 ( x)Y0 ( x)
tan(2 x/(2ν + 1)) (2ν + 1)2
0
(3.43)
and estimate the error bounds. By analogy to the proof of Proposition 3.2(i), we assert that
¶
µ
1
(2ν + 1)2
e
.
(3.44)
[B L (ν) − B(ν)] = O p
sin2 (νπ)
2ν + 1
As in the proof of Proposition 3.2(ii)–(iii), we have
¶
µ
Z(2ν+1)π/4
2 x/(2ν + 1)
2 cot(νπ)
1
3
−8
,
(3.45)
xdx =
+O p
[ J0 ( x)] Y0 ( x)
tan(2 x/(2ν + 1))
π
0
2ν + 1
TWO DEFINITE INTEGRALS INVOLVING PRODUCTS OF FOUR LEGENDRE FUNCTIONS
11
according to (2.13), while
¶
Zp2ν+1 µ
ª 1 − cos(4 x)
©
2 x/(2ν + 1)
2
2
2
dx
x [ J0 ( x)] [Y0 ( x)] − [ J0 ( x)] +
2
π x
tan(2 x/(2ν + 1))
0
p
¶
¶
µ
Z 2ν+1 µ
ª 1 − cos(4 x)
©
xdx
1
2
2
2
=
x [ J0 ( x)] [Y0 ( x)] − [ J0 ( x)] +
+O
π2 x
(2ν + 1)2
(2ν + 1)3/4
0
µ
¶
¶
Z∞ µ
ª 1 − cos(4 x)
©
1
xdx
2
2
2
+O
x [ J0 ( x)] [Y0 ( x)] − [ J0 ( x)] +
=− p
π2 x
(2ν + 1)2
(2ν + 1)3/4
2ν+1
¶
µ
1
,
(3.46)
=O p
2ν + 1
p
follows from (2.19) and integration by parts on x ∈ ( 2ν + 1, +∞). This subsequently entails
¶
µ
¶
Z(2ν+1)π/4 µ
ª 1 − cos(4 x)
©
2 x/(2ν + 1)
1
2
2
2
dx =O p
x [ J0 ( x)] [Y0 ( x)] − [ J0 ( x)] +
. (3.47)
π2 x
tan(2 x/(2ν + 1))
0
2ν + 1
Next, we compute a definite integral with the help of integration by parts and a Fourier series
expansion [9, item 5.4.2.1]:
Z(2ν+1)π/4
Z
2 π/2
1 − cos(4 x) 2 x/(2ν + 1)
dx = 2
sin2 ((2ν + 1)θ ) cot θ d θ
2
tan(2 x/(2ν + 1))
π x
π 0
0
#
"
Zπ/2
Z
∞ cos(2 k θ )
X
2(2ν + 1)
2(2ν + 1) π/2
dθ
=−
sin(2(2ν + 1)θ ) logsin θ d θ =
sin(2(2ν + 1)θ ) log 2 +
k
π2
π2
0
0
k=1
¶
∞ µ2
2 cos2 (νπ)
1
1
1X
=
[1 + (−1)k cos(2νπ)]
−
−
log 2 +
4 k=1 k k + 2ν + 1 k − 2ν − 1
π2
µ
µ
¶
¶¸
·
3
1
3ν + 1
ψ(0) (2ν) + γ0 + log 2 cos(2νπ)
(0)
(0)
(0)
ν+
ν+
= 2
− 2ψ (ν + 1) + ψ
ψ
+
+
2
2
2π ν (2ν + 1)
π2
4π2
µ ¶
1
γ0 + 2 log2 + log ν
.
(3.48)
+O
=
2
ν
π
This eventually demonstrates that
µ
¶
2 cot(νπ) 4(γ0 + 2 log 2 + log ν)
1
B L (ν) =
+O p
,
+
π
π2
sin2 (νπ)
2ν + 1
(2ν + 1)2
so (3.41) holds.
(3.49)
While our foregoing treatments of (1.1) and (1.2) were motivated by their significance in certain
arithmetic problems, the methods just outlined can be equally applied to other integrals with
four Legendre factors. In a sense, our previous work [15, 16] and this article have provided a
framework for evaluating “Legendre moments” for (up to) four Legendre factors. This forms an
analog for the researches on “Bessel moments” [8, 3, 1] for (up to) four Bessel factors, which were
motivated by Feynman diagrams in quantum field theory.
Acknowledgements. The author is grateful to an anonymous referee for thoughtful comments
on polishing the presentation of this paper.
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P ROGRAM IN A PPLIED AND C OMPUTATIONAL M ATHEMATICS, P RINCETON U NIVERSITY, P RINCETON, NJ 08544,
USA
E-mail address: [email protected]