1_paris.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 3, 2012, 260-281
ISSN 1307-5543 – www.ejpam.com

On the Asymptotics and Zeros of a Class of Fourier Integrals

Richard B. Paris

University of Abertay Dundee, Dundee DD1 1HG, UK

Abstract. We obtain the asymptotic expansion of the Fourier integrals

∫ ∞

0

tν−1 cos
sin
(x t) exp (−tn/n) d t

for large complex values of x and integer n> 2 by means of the asymptotic theory of the Wright func-
tion. Asymptotic approximations for both the real and complex zeros of these integrals are considered.
These results are extended to p-dimensional Fourier integrals of a similar structure.

2010 Mathematics Subject Classifications: 30E15, 33B10, 33C70, 34E05, 41A60

Key Words and Phrases: Fourier integrals, asymptotic expansion, zeros, Wright function

1. Introduction

The asymptotic expansion of the Fourier integrals

∫ ∞

0

cos
sin
(x t) exp (−tn/n) d t (1)

for large complex values of x and integer n ≥ 2 has been considered in [1, 3, 5] and, in the
case of the cosine integral, more recently in [15]. All these authors employed the method of
steepest descents to derive the asymptotics which resulted in long and detailed calculations.
Considerable interest in the real zeros of Fourier integrals originated with the seminal study of
Pólya [14], who showed that the cosine integral in (1) when n = 4,6, . . . has infinitely many
real zeros; generalisations of these results have been obtained in [4] and more recently in
[6, 7, 8]. The first-order asymptotics for the location of the zeros of the cosine integral in (1)
were obtained in [5], with higher-order approximations for these zeros being given in [15].

Our aim in this paper is to show that the large-x asymptotics of the above integrals can be
more readily obtained by making use of the well-established asymptotic theory of the Wright

Email address: r.paris�abertay.a
.uk

http://www.ejpam.com 260 c© 2012 EJPAM All rights reserved.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 261

function defined below. We make a generalisation in (1) to include an algebraic power of t in
the integrand and consider the integrals

Cn,1

Sn,1
(x ;ν) =

∫ ∞

0

tν−1 cos
sin
(x t) exp (−tn/n) d t, Re (ν)>

¨

0
−1

, (2)

where the subscript 1 denotes the dimension of the integral. We then use the asymptotics of
the integrals in (2) to examine their real and complex zeros. An advantage of this approach
is that the same procedure can be applied with little additional effort to the p-dimensional
integrals possessing a similar structure given by

Cn,p

Sn,p
(x ; ~ν) =

∫ ∞

0

. . .

∫ ∞

0

t
ν1−1
1 . . . t

νp−1
p

cos
sin
(x t1 . . . tp) exp [−(tn

1 + · · ·+ tn
p)/n] d t1 . . . d tp, (3)

where ~ν = (ν1,ν2, . . . ,νp), n≥ 2 is an integer and it is supposed for Cn,p(x ; ~ν) that Re(νr)> 0
and for Sn,p(x ; ~ν) that Re(νr)> −1 (1≤ r ≤ p). An integral of this type with p = 2 was given
as the solution of a certain nth-order differential equation by Spitzer [17] well over a century
ago.

The Wright function pΨq(z) (a generalised hypergeometric function) is defined by

pΨq(z) =

∞
∑

k=0

∏p

r=1 Γ(αr k+ ar)
∏q

r=1 Γ(βr k+ br)

zk

k!
, αr k+ ar 6= 0,−1,−2, . . . (4)

where p and q are nonnegative integers, the parameters αr and βr are real and positive and
ar and br are arbitrary complex numbers. In the special case αr = βr = 1, the function

pΨq(z) reduces to a multiple of the ordinary hypergeometric function pFq((a)p; (b)q; z) [16,
p. 40]. The particular function of this class that we shall use has q = 0 and the parameters
αr = 1/n, ar = νr/n. Following the notation used in [12, Chapter 3] for the solution of a
certain nth-order differential equation, we denote this function by Un,p(z; ~ν), where

Un,p(z; ~ν) =
∞
∑

k=0

(np/nz)k

k!

p
∏

r=1

Γ

�

k+ νr

n

�

(n> p ≥ 1; |z| <∞).

The integrals Cn,p(x ; ~ν) and Sn,p(x ; ~ν) in (3) will be shown to be expressed respectively in
terms of the even and odd combinations

Un,p(i x ; ~ν)± Un,p(−i x ; ~ν),

whence the asymptotics for large complex x can be easily constructed from knowledge of that
of Un,p(z; ~ν).

It will be established that, for general values of the parameters νr and p ≥ 1, the integrals
Cn,p(x ; ~ν) and Sn,p(x ; ~ν) possess a dominant algebraic expansion in the sectors
|arg (±x)| < πp/(2n) and an exponentially large expansion in the complementary sectors
|arg (±i x)| < 1

2
π(1− p/n). An infinite sequence of complex zeros of these integrals is found



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 262

in the neighbourhood of the anti-Stokes lines arg x = ±πp/(2n) (together with a symmetrical
distribution in Re(x)< 0), where the exponential and algebraic expansions are of comparable
magnitude. For certain values of νr when n is even, however, the algebraic expansion vanishes
to leave an exponentially small behaviour in the sectors |arg (±x)|< πp/(2n). In these cases,
the integrals Cn,p(x ; ~ν) and Sn,p(x ; ~ν) are found to have an infinite sequence of real zeros.
Asymptotic approximations to the zeros are obtained both in the general case and in the
exponentially small case.

The structure of the paper is as follows. In Section 2 we present the asymptotic expansion
of the function Un,p(z; ~ν) for large complex values of z. From this we construct the asymptotics
of the integrals (3) for large complex x in Section 3. The zeros (both real and complex) in the
one-dimensional case p = 1 are examined in Section 4. Finally, in Section 5 we investigate
the zeros when p = 2 in particular cases and make a conjecture on the real zeros for general
p.

2. The Asymptotic Properties of Un,p(z; ~ν)

We present in this section the asymptotic expansion of the function Un,p(z; ~ν) which is fun-
damental in our discussion of the Fourier integrals Cn,p(x ; ~ν) and Sn,p(x ; ~ν). It was introduced
in [11, 12] as the solution of the nth-order differential equations∗

u(n) ∓
p
∑

r=0

arz
r u(r) = 0 (n> p ≥ 1), (5)

where z is the independent variable and the coefficients ar (1 ≤ r ≤ p − 1) are arbitrary
constants with a0 6= 0 and (without loss of generality) ap = 1. Four classes of solution of (5),
exhibiting different types of asymptotic behaviour for large |z|, have been discussed in detail
in [12, Chapter 3]. The polynomial G(s) of degree p associated with (5) is defined by

G(s) =

p
∑

r=0

(−)r ar(−s)r =

p
∏

r=1

(s+ νr), (6)

where (α)r = Γ(α+ r)/Γ(α) is the Pochhammer symbol and −νr (1≤ r ≤ p) are the zeros of
G(s). With Θ ≡ zd/dz, so that the differential operator zr(d/dz)r = Θ(Θ− 1) . . . (Θ− r + 1),
the equation (5) can be written in the alternative form

u(n) ∓
p
∏

r=1

(Θ+ νr)u= 0, (7)

which is a transformation of the generalised hypergeometric differential equation [16, p. 42].
The solution of (5) and (7) with the upper sign that we consider here has the series

expansion

Un,p(z; ~ν) =
∞
∑

k=0

(np/nz)k

k!

p
∏

r=1

Γ

�

k+ νr

n

�

(n> p ≥ 1). (8)

∗We exclude the trivial case p = 0.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 263

Provided we impose the restriction that none of the νr equals a negative integer (νr 6= 0 by
hypothesis, since a0 6= 0), then (8) defines Un,p(z; ~ν) as a uniformly and absolutely convergent
series throughout the finite z-plane. Comparison with (4) shows that Un,p(z; ~ν) is a particular

case of the Wright function with q = 0, αr = 1/n, ar = νr/n and argument np/nz. It is easily
verified by differentiation of the right-hand side of (8) that Un,p(z; ~ν) satisfies the differential
equation (7) with the upper sign. Since (5) is unaltered if z is replaced by Ωz, where Ω
denotes an nth root of unity, a fundamental system of solutions of (5) is given by

Un,p(Ω jz)

Un,p(e
πi/nΩ jz)

«

, Ω j = exp(2πi j/n), j = 0,1, . . . , n− 1 (9)

where the upper and lower sets of solutions correspond to the upper and lower signs, respec-
tively. An integral representation of the solution is given by the Mellin-Barnes integral [12,
p. 61]

Un,p(z; ~ν) =
1

2πi

∫ ∞i

−∞i

Γ(−s)

p
∏

r=1

Γ

�

s+ νr

n

�

(−np/nz)−sds (10)

valid in the sector |arg (−z)| < 1
2
π(1+ p/n), where, with the above-mentioned restrictions on

νr , the path of integration can always be chosen to separate the poles of Γ(−s) from those of
Γ((s+ νr)/n) (1≤ r ≤ p).

The asymptotic expansion of Un,p(z; ~ν) for large |z| follows from that of the Wright func-
tion pΨq(z) in (4) [18, 2]; see also [10]. We define the parameters

κ = 1− p

n
, ϑ =

1

n

p
∑

r=1

νr −
1

2
p (11)

and introduce the formal exponential and algebraic asymptotic expansions defined respec-
tively by

E(z) := (2π)p/2κ−
1
2 (z1/κ/n)ϑ exp (κz1/κ)

∞
∑

j=0

c j(κz1/κ)− j , (12)

H(z) := n

p
∑

r=1

(np/nz)−νr Tn,p(z; ~ν), (13)

where, provided no two of the νr either coincide or differ by an integer multiple of n,

Tn,p(z; ~ν) :=
∞
∑

k=0

(−)k
k!
Γ(nk+ νr)

p
∏

j=1

′
Γ

�

ν j − νr

n
− k

�

(np/nz)−nk

with the prime denoting the omission of the term corresponding to j = r in the product.
The algebraic expansion H(z) results from displacement of the integration path in (10) over
the poles of the product of gamma functions and evaluation of the residues. When these



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 264

restrictions on νr are not satisfied, the algebraic expansion is modified by the presence of
logarithmic terms arising from the formation higher-order poles in the integrand of (10).

The coefficients c j appearing in the exponential expansion E(z) are independent of z with
c0 = 1 and are generated by the n-term recurrence relation [12, §3.4]

c j =
1

nκ j

(

n−1
∑

s=1

c j−sP
(n)
s+1(s− j)−

p−1
∑

s=1

c j−sQs+1(s− j)

)

( j ≥ 1) (14)

with c−1 = c−2 = . . . = c2−n = 0, where

P(n)s (χ) =

s
∑

r=0

r
∑

k=0

(ϑ+χ)r−kκk

�

n− k

r − k

�

S(n−k)
n S/

(n−s)
n−r ,

Qs(χ) =

s
∑

r=0

ap−rκ
r P
(p−r)
s−r (χ),

the ar are the coefficients in the differential equation (5) and S(m)n , S/(m)n are respectively the
Stirling numbers of the first and second kind. Alternatively, these coefficients may be obtained
by means of the algorithm described in [10]; see also [13, §2.2.4]. From [10, Appendix A],
we have the explicit representation of the coefficient c1 in the form

c1 =
1
2
κ

(

p
∑

r=1

νr

�νr

n
− 1
�

− ϑ(1− ϑ)
κ

)

+
p

12n
(n2− np+ 1). (15)

The first few values of the coefficients c j obtained from (14) for different n, p and ~ν are given
in Table 1.

Table 1: The coefficients c j (1≤ j ≤ 5) for different n, p and ~ν .

n= 4, p = 1 n= 6, p = 2 n= 6, p = 3
j ν = 1 ~ν = (1

2
, 2) ~ν = (1

2
, 3

2
, 4)

1 7
48

161
288

7
16

2 385
4608

114625
165888

289
512

3 39655
663552

189038465
143327232

10061
8192

4 665665
127401984

608738148865
165112971264

2011691
524288

5 −1375739365
6115295232

704282046029485
47552535724032

132834185
8388608

Then, we have the asymptotic expansion given by



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 265

Theorem 1. For n > p ≥ 1 and |z| → ∞, the function Un,p(z; ~ν) possesses the asymptotic

expansion†

Un,p(z; ~ν)∼
(

E(z) +H(ze∓πi) |arg z| < π
�

1− p

n

�

H(ze∓πi) |arg (−z)| < 1
2
π
�

1+ p

n

� , (16)

where the upper or lower signs in (16) are chosen according as arg z > 0 or arg z < 0, respec-

tively. The expansion of the fundamental systems in (9) follows immediately by rotation of the

argument z by 2π j/n and (2 j+ 1)π/n.

The function Un,p(z; ~ν) is exponentially large as |z| → ∞ in the sector |arg z| < 1
2
πκ,

whereas in the complementary sector |arg(−z)| < 1
2
π(2 − κ) the dominant asymptotic be-

haviour consists (in general) of p algebraic expansions, each with the controlling behaviour
z−νr , r = 1,2, . . . , p. In the common sectors of validity, 1

2
πκ < |arg z| < πκ, the expansions in

(16) differ only through the presence of the series E(z), which is exponentially small in these
sectors. The rays arg z = ±πκ are Stokes lines on which the expansion E(z) is maximally
subdominant. It was established in [9] that (in the sense of increasing |arg z|) the expansion
E(z) switches off smoothly as these Stokes lines are crossed. The positive real axis is also a
Stokes line where the algebraic expansion is maximally subdominant. The sectorial behaviour
of Un,p(z; ~ν) is illustrated in Fig. 1.

Exponentially large
+ Algebraic

Exponentially small
+ Algebraic

Algebraic

Exponentially small
+ Algebraic

Stokes line

Stokes line

Stokes line

πκ

− πκ

πκ/2

−πκ/2

Figure 1: The sectorial behaviour of Un,p(z; ~ν) for large |z|.

Finally, it is worth remarking that the expansion in (16) remains valid for noninteger
values of n > p; in this case, of course, the function Un,p(z; ~ν) is not a solution of (5) and the
recurrence relation (14) can no longer be employed. The coefficients c j in this case can be
obtained by the algorithm described in [10].

†We remark that the first expansion in (16) was given in [12, 11] only in the narrower sector |arg z| ≤ 1
2
πκ where

the exponential expansion E(z) is dominant.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 266

3. The Asymptotic Expansion of Cn,p(x ; ~ν) and Sn,p(x ; ~ν) for |x | →∞
A Laplace integral representation for Un,p(z; ~ν) has been given in [12, p. 124]. When

p = 1, this takes the form

Un,1(z;ν) = n
1
2
−ϑ
∫ ∞

0

tν−1ezt−tn/n d t (Re (ν)> 0), (17)

which may be easily verified by expanding ezt as a Maclaurin series followed by term-by-term
integration. In a similar manner, we may establish that

Un,p(z; ~ν) = np/2−ϑ
∫ ∞

0

. . .

∫ ∞

0

t
ν1−1
1 . . . t

νp−1
p ezt1 ...tp exp [−(tn

1 + . . .+ tn
p)/n] d t1 . . . d tp (18)

for Re (νr)> 0, 1≤ r ≤ p; see [12, p. 133]. This integral in the case p = 2 was first considered
in [17] as the solution of a certain nth-order differential equation. From (17) and (18), it then
follows from the definition of the integrals Cn,p(x ; ~ν) and Sn,p(x ; ~ν) in (3) that

Cn,p

Sn,p
(x ; ~ν) = ξnϑ−

1
2

p{Un,p(i x ; ~ν)± Un,p(−i x ; ~ν)}, (19)

where ξ = 2−1 for Cn,p(x ; ~ν) and ξ = (2i)−1 for Sn,p(x ; ~ν). From (8), we obtain the series
representations

Cn,p

Sn,p
(x ; ~ν) = nϑ−p/2

∞
∑

k=0

(np/nx)k

k!

p
∏

r=1

Γ

�

k+ νr

n

�

cos
sin
(1

2
πk). (20)

As these integrals are respectively even and odd functions of x , it is sufficient to restrict our
attention to the sector |arg x | ≤ 1

2
π.

We now introduce the formal asymptotic expansions

E± := κ−1/2−ϑ
�

2π

n

�p/2

X ϑ exp (X e±πi/(2κ))

∞
∑

j=0

c jX
− j e±πi(ϑ− j)/(2κ), (21)

Hc,s := n1+ϑ− 1
2

p

p
∑

r=1

(np/n x)−νr T (c,s)
n,p (x ; ~ν), (22)

where the variable X is defined by
X := κx1/κ,

T (c,s)
n,p (x ; ~ν) :=

∞
∑

k=0

(−)k
k!
Γ(nk+ νr)

p
∏

j=1

′
Γ

�

ν j − νr

n
− k

�

(np/n x)−nk cos
sin

1
2
π(nk+ νr) (23)

and the sub- and superscripts c, s refer to the expansion with cosine and sine, respectively. In
addition, we introduce the expansions

Ec :=
1

2
(E+ + E−), Es :=

1

2i
(E+− E−),



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 267

so that

Ec,s = κ
−1/2−ϑ
�

2π

n

�p/2

X ϑ exp
�

X cos
π

2κ

� ∞
∑

j=0

c jX
− j cos

sin

�

X sin
π

2κ
+
π

2κ
(ϑ− j)

�

. (24)

Then from Theorem 1, the asymptotic expansion of Cn,p(x ; ~ν) and Sn,p(x ; ~ν) when p/n ≤ 1
2

(κ ≥ 1
2
) is

Cn,p

Sn,p
(x ; ~ν)∼







Hc,s + Ec,s |arg x | ≤ π(1
2
− p

n
)

Hc,s ±ξE− π(1
2
− p

n
)< arg x ≤ 1

2
π

Hc,s +ξE+ −1
2
π ≤ arg x < π(1

2
− p

n
)

(25)

and when p/n> 1
2

(κ < 1
2
)

Cn,p

Sn,p
(x ; ~ν)∼







Hc,s |arg x | ≤ π( p

n
− 1

2
)

Hc,s ±ξE− π(
p

n
− 1

2
)< arg x ≤ 1

2
π

Hc,s +ξE+ −1
2
π ≤ arg x < π(

p

n
− 1

2
)

(26)

as x → ∞; compare [12, §3.8.2]. The asymptotic structure of Cn,p(x ; ~ν) and Sn,p(x ; ~ν)

as |x | → ∞ is summarised in Fig. 2. When κ ≥ 1
2
, this is seen to consist of an alge-

braic and a subdominant exponentially small expansion in the sectors |arg (±x)| < πp/(2n)

and an exponentially large expansion (with a subdominant algebraic expansion) in the sec-
tors |arg (±i x)| < 1

2
πκ. When κ < 1

2
, there is a purely algebraic expansion in the sectors

|arg (±x)| < π(p/n − 1
2
) and an exponentially large (with a subdominant algebraic expan-

sion) in the sectors |arg (±i x)|< πκ.

Algebraic

+ Exponentially small

Exponentially large

Algebraic

+ Exponentially small

Exponentially large

π

−π

p/(2n)

p/(2n)

(a) κ≥ 1
2

Algebraic

Exponentially large

Algebraic

Exponentially large

1/2

1/2

π

−π

(p/n-     )

(p/n-     )

(b) κ < 1
2

Figure 2: The sectorial behaviour of Cn,p(z; ~ν) and Sn,p(z; ~ν) for large |x |.

We remark that the above expansions take into account the switching on or off of the
exponential expansions due to the Stokes phenomenon as one crosses the rays
arg x = ±π(1

2
− p/n). However, the details of the transition across these rays, together

with those associated with the algebraic expansions on arg x = ±1
2
π, would require further

investigation of the Stokes phenomenon on the lines of that given in [9]. Finally, if some
of the νr are equal, or differ by integer multiples of n, then higher order poles will arise in
the integrand of (10) and the algebraic expansions Hc,s will be modified by the presence of
logarithmic terms; see Section 5 for an example.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 268

4. The Zeros in the Case p = 1

We first examine the case with p = 1, where from (19)

Cn,1

Sn,1
(x ;ν) =

∫ ∞

0

tν−1 cos
sin
(x t) exp (−tn/n) d t, Re (ν)>

¨

0
−1

= ξnϑ−
1
2 {Un,1(i x ;ν)± Un,1(−i x ;ν)}. (27)

In this case the algebraic expansions in (22) simplify to

Hc,s = x−ν
∞
∑

k=0

(−)k
k!
Γ(nk+ ν) (n1/n x)−nk cos

sin
1
2
π(nk+ ν). (28)

Then, from (25), we have the asymptotic expansion‡ for n> 2 (κ > 1
2
) given by

Cn,1

Sn,1
(x ;ν)∼







Hc,s + Ec,s |arg x | ≤ π(1
2
− 1

n
)

Hc,s ± ξE− π(1
2
− 1

n
)< arg x ≤ 1

2
π

Hc,s + ξE+ −1
2
π≤ arg x < π(1

2
− 1

n
)

(29)

as |x | →∞. The coefficients c j in the exponential expansions E± and Ec,s in (21) and (24) can
be computed by the recurrence relation (14). The first few values of these coefficients when
n= 4 and ν = 1 are given in the first column of Table 1.

4.1. Real zeros

When n is even and the parameter ν is an odd (resp. even) integer for Cn,1(x ;ν) (resp.
Sn,1(x ;ν)), the algebraic expansion Hc (resp. Hs) in (28) vanishes to leave an exponentially
small expansion in the sector |arg x |< π(1

2
− 1/n). From (29) and (24), we then have when

κ > 1
2

Cn,1

Sn,1
(x ;ν)∼ κ−ϑ
�

2π

nκ

�1/2

X ϑ exp
�

X cos
π

2κ

� ∞
∑

j=0

c j X
− j cos

sin

�

X sin
π

2κ
+
π

2κ
(ϑ− j)

�

(30)

n even ,ν =

¨

2m+ 1
2m+ 2

(m = 0,1,2, . . .)

as |x | → ∞ in |arg x | < π(1
2
− 1/n). In this case, Cn,1(x ;ν) and Sn,1(x ;ν) possess an infinite

sequence of real zeros; see the appendix.
The leading-order approximation for the real zeros is then given by

cos
sin
Ψ = 0, Ψ = X sin

π

2κ
+
πϑ

2κ

‡We exclude the case n = 2 (κ = 1
2
) since the integrals in (27) can be evaluated in terms of parabolic cylinder

functions.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 269

to produce the zeroth-order approximation

Ψ(0) = (k+ ε)π, ε=

¨

1
2
1

(k = 0,1,2, . . . ).

To next order we have
cos
sin
Ψ+

c1

X

cos
sin

�

Ψ− π
2κ

�

= 0,

which leads to

Ψ(1) = (k+ 1)π∓ arctan

�

Λ

Λ−1

�

, Λ =
X + c1 cos π

2κ

c1 sin π
2κ

.

Thus we obtain for X →∞
Ψ(1) =Ψ(0) +

c1

X
sin

π

2κ
.

This yields the zeroth and first-order approximations X (0), X (1) to the (positive) real zeros of
Cn,1(x ;ν) and Sn,1(x ;ν) given by

X (0) =

�

k+ ε− ϑ
2κ

�

π

sin π
2κ

, X (1) = X (0) +
c1

X (0)
(X = κx1/κ). (31)

This approximation procedure in the case of the real zeros of Cn,1(x ; 1) for even n has been
carried out to fourth order in [15].

We remark that when n = 2 (κ = 1
2
) the functions Cn,1(x ;ν) (with ν odd) and Sn,1(x ;ν)

(with ν even) can be expressed in terms of Hermite polynomials Hn(z) as

C2,1

S2,1
(x ;ν) = (−)m2−

1
2
ν π

1
2 e−x2/2 Hν−1(x/

p
2), ν =

¨

2m+ 1
2m+ 2

(32)

for nonnegative integer m. By a well-known property of the Hermite polynomials, it follows
that the functions on the left-hand side of (32) possess a finite number of real zeros.

4.2. Complex zeros

For simplicity, we shall restrict attention to real positive values of ν . Provided the algebraic
expansions Hc,s do not vanish identically (which can only arise when n is an even integer and
ν is either odd (resp. even)), the complex zeros of Cn,1(x ;ν) and Sn,1(x ;ν) will be situated
near the anti-Stokes lines arg x = ±π/(2n), where the expansions Hc,s and E∓ are comparable
in magnitude. We consider only the neighbourhood of the ray arg x = π/(2n) where

Cn,1

Sn,1
(x ;ν)∼ Hc,s ± ξE−



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 270

for large |x |, since E+ is a subdominant expansion in the sector 0 < arg x < π(1
2
− 1/n). To

leading order the complex zeros of Cn,1(x ;ν) and Sn,1(x ;ν) are then described by§

x−ν
cos
sin
(1

2
πν)Γ(ν)± ξ
�

2π

nκ

�1/2

(−i x)ϑ/κ exp (X e−πi/(2κ)) = 0.

If we put
x = reiφ+πi/(2n), (33)

with r = |x |, then we find

exp {iκr1/κ cos φ/κ} −ΥeiΦ = 0, (34)

where

Υ= λr(ν−
1
2
)/κ exp (κr1/κ sin(φ/κ)), λ=

(π/2nκ)1/2

Γ(ν)

�

�

�

�

cos
sin
(1

2
πν)

�

�

�

�

,

Φ = 1
2
π(1∓ 1

2
)+ (ν − 1

2
)φ/κ+πδ,

with δ = 1 if cos 1
2
πν or sin 1

2
πν > 0, and δ = 0 if cos 1

2
πν or sin 1

2
πν < 0.

The solution of (34) requires κr1/κ cos (φ/κ) = Φ+ 2kπ, Υ = 1 to yield

κr1/κ cos (φ/κ) = (2k+ 1
2
)π+ (δ∓ 1

4
)π+ (ν − 1

2
)
φ

κ
,

κr1/κ sin (φ/κ) = − log (λr(ν−
1
2
)/κ),

where k = 0,1,2, . . . . If the parameter ν is such that |φ| ≪ 1, then we find approximately

κr1/κ ≃ (2k+ 1
2
)π+ (δ∓ 1

4
)π, k = 0,1,2, . . . , (35)

φ ≃ −κarcsin

(

log (λr(ν−
1
2
)/κ)

κr1/κ

)

, (36)

where the upper or lower sign corresponds to Cn,1(x ;ν) or Sn,1(x ;ν), respectively. The asymp-
totic distribution of the complex zeros is then obtained from (33).

§When ν is an odd (resp. even) integer and n is odd the leading term in the algebraic expansion in (28) corre-
sponds to k = 1. The modification required in this case is easily carried out.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 271

4.3. Numerical results

The zeros of Cn,1(x ;ν) and Sn,1(x ;ν) have been calculated by means of the secant method
in Mathematica applied to the combinations Un,1(i x ;ν)± Un,1(−i x ;ν) in (27). The function
Un,1(z;ν) was computed by suitable truncation of its series representation in (8) and asymp-
totic estimates obtained from (33), (35) and (36) for the complex zeros and (31) for the real
zeros were employed to initiate the process. The complex zeros, together with their asymp-
totic approximations, are presented in Tables 2 and 3 for n= 4 and n= 5 and different values
of ν . It will be observed that these zeros arise in conjugate pairs (when ν is real) situated
near the anti-Stokes lines arg x = ±π/(2n). It should also be noted that as ν increases some
real zeros are present; this is discussed more fully at the end of this section.

Table 2: The complex zeros xk of Cn,1(x ;ν) in the right-half plane for different n and ν .

n= 4, ν = 1
2

n= 4, ν = 2
3

k xk Asymptotic xk xk Asymptotic xk

0 3.1041± 1.6890i 3.0410± 1.6533i 3.2753± 1.1203i 3.2777± 1.1125i

1 6.4566± 2.9845i 6.4330± 2.9742i 6.6574± 2.4767i 6.6433± 2.4690i

2 9.2953± 4.1250i 9.2821± 4.1194i 9.4904± 3.6411i 9.4816± 3.6367i

3 11.8692± 5.1697i 11.8604± 5.1660i 12.0592± 4.7025i 12.0528± 4.6995i

4 14.2687± 6.1487i 14.2622± 6.1459i 14.4544± 5.6941i 14.4494± 5.6919i

5 16.5406± 7.0783i 16.5354± 7.0761i 16.7225± 6.6341i 16.7184± 6.6322i

n= 4, ν = 3
2

n= 5, ν = 1
2

k xk Asymptotic xk xk Asymptotic xk

0 1.8582 1.0126± 0.2153i 3.3130± 1.6108i 3.2058± 1.5728i

1 5.3221± 0.6644i 5.3306± 0.7204i 7.1928± 2.7755i 7.1544± 2.7627i

2 8.5087± 1.9237i 8.4553± 1.9032i 10.6035± 3.8433i 10.5819± 3.8362i

3 11.2300± 3.0161i 11.1794± 2.9958i 13.7597± 4.8439i 13.7450± 4.8391i

4 13.7215± 4.0352i 13.6754± 4.0164i 16.7445± 5.7959i 16.7335± 5.7923i

5 16.0595± 4.9977i 16.0161± 4.9817i 19.6017± 6.7107i 19.5931± 6.7078i

When n is even and ν is odd (resp. even), the zeros of Cn,1(x ;ν) (resp. Sn,1(x ;ν)) are all

real; see the appendix. The zeroth-order approximation x
(0)
k

for these zeros is given by the

first equation in (31) with ε = 1
2

(resp. 1). For example, when n = 4, we find κ = 3
4

and

ϑ = 1
4
ν − 1

2
, so that

x
(0)
k
=

�

8π

3
p

3
(k+ ε+ 1

3
− 1

6
ν)

�3/4

(k = 0,1,2, . . . ).

The first-order approximation x
(1)
k

is described by the second equation in (31), with the coef-
ficient c1 obtained from (15). The calculation of the real zeros of Cn,1(x ;ν) and Sn,1(x ;ν) is
presented in Table 4.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 272

Table 3: The complex zeros xk of Sn,1(x ;ν) in the right-half plane for different n and ν .

n= 4, ν = 1
2

n= 4, ν = 1
k xk Asymptotic xk xk Asymptotic xk

0 4.0244± 2.0323i 3.9764± 2.0087i 4.2375± 1.3161i 4.2550± 1.3196i

1 7.1994± 3.2811i 7.1796± 3.2726i 7.4819± 2.5264i 7.4758± 2.5234i

2 9.9589± 4.3936i 9.9471± 4.3886i 10.2567± 3.6293i 10.2497± 3.6261i

3 12.4831± 5.4199i 12.4750± 5.4164i 12.7869± 4.6542i 12.7801± 4.6512i

4 14.8473± 6.3852i 14.8412± 6.3826i 15.1536± 5.6211i 15.1471± 5.6183i

5 17.0930± 7.3047i 17.0872± 7.3022i 17.3992± 6.5430i 17.3930± 6.5404i

n= 4, ν = 3
2

n= 5, ν = 1
2

k xk Asymptotic xk xk Asymptotic xk

0 4.0787, 4.6474 4.4355± 0.4157i 4.3501± 1.9128i 4.2766± 1.8857i

1 7.7747± 1.6364i 7.7229± 1.6162i 8.0767± 3.0502i 8.0444± 3.0395i

2 10.5755± 2.7507i 10.5238± 2.7302i 11.4121± 4.0989i 11.3927± 4.0925i

3 13.1152± 3.7862i 13.0680± 3.7670i 14.5199± 5.0859i 14.5063± 5.0815i

4 15.4858± 4.7625i 15.4430± 4.7448i 17.4695± 6.0278i 17.4592± 6.0244i

5 17.7038± 5.6866i 17.6938± 5.6766i 20.2996± 6.9344i 20.2913± 6.9317i

The manner in which the zeros change as ν increases is shown in Fig. 3(a) for the case
of Cn,1(x ;ν) when n= 4; a similar behaviour applies to Sn,1(x ;ν). This figure shows the first
complex zeros x0 and x1 (and their conjugates) for values of ν increasing from 0.1 to 1 in
steps of 0.1. As ν increases, the zeros approach the real axis and eventually coalesce to form
real zeros. This is found to occur for ν

.
= 0.8216 in the case of x0 and ν

.
= 0.9875 in the case

of x1. The zeros labelled A, B, C , D indicate the zeros when ν = 1; the next real zero in the
sequence when ν = 1 (which results from the coalesence of x2 and its conjugate) is labelled
E. The remaining complex zeros exhibit a cascade effect since they all progressively coalesce
to become real as ν increases in the interval (0.9875,1]. An alternative depiction of the zeros
as ν increases in the interval [1

2
, 1] is shown in Fig. 4.

As ν increases beyond the value ν = 1, the zeros labelled B, C and D, E in Fig. 4(a)
approach one another, coalesce and then move off into the complex plane as new conjugate
pairs. The loci of these complex zeros (in the upper half-plane) are indicated in Fig. 3(b). As ν
continues to increase these loci form loops that return to the real axis, resulting in coalescence
and the formation of real zeros again. The loop formed by B and C exists for ν in the interval
(1.0853,1.8733) and that formed by D and E exists in the interval (1.0025,2.6560). A similar
behaviour is exhibited by the other zeros with the result that when ν = 3, all the zeros are
again real. This pattern then repeats itself for ν in the intervals [3,5], [5,7] and so on. It
then becomes clear from this discussion that the finite number of real zeros of Cn,1(x ;ν) and
Sn,1(x ;ν) for a given value of ν (apart from odd or even integer values) is difficult to predict.

In the case of odd integer n we find a similar behaviour of the zeros. Fig. 5 shows the
distribution of the zeros of Cn,1(x ;ν) when n= 3 for ν in the interval [1

2
, 1]. When ν = 1

2
, the

zeros lie close to the anti-Stokes lines arg x = ±π/6. As ν increases, the first complex zero



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 273

Table 4: The real zeros xk of Cn,1(x ;ν) and Sn,1(x ;ν) on the positive axis, together with their
zeroth and first-order approximations, for different even n and integer ν . The corresponding
value of the coefficient c1 is given.

n= 4, ν = 1, c1 =
7

48
n= 6, ν = 1, c1 =

11
36

Cn,1(x ;ν) Cn,1(x ;ν)

k xk x
(0)
k

x
(1)
k

xk x
(0)
k

x
(1)
k

0 2.441968 2.4063 2.4512 2.500814 2.3407 2.4517
1 4.797244 4.7842 4.7985 4.932583 4.9032 4.9427
2 6.813581 6.8060 6.8140 7.232399 7.2095 7.2326
3 8.647288 8.6422 8.6475 9.389764 9.3743 9.3903
4 10.359390 10.3556 10.3595 11.454280 11.4425 11.4545
5 11.981848 11.9788 11.9819 13.447433 13.4380 13.4476

n= 4, ν = 2, c1 = − 5
48

n= 6, ν = 2, c1 = − 1
36

Sn,1(x ;ν) Sn,1(x ;ν)

k xk x
(0)
k

x
(1)
k

xk x
(0)
k

x
(1)
k

0 3.246903 3.2615 3.2421 3.446131 3.4114 3.4054
1 5.478116 5.4852 5.4770 5.843645 5.8473 5.8445
2 7.430167 7.4346 7.4297 8.088659 8.0892 8.0874
3 9.221748 9.2249 9.2215 10.210708 10.2115 10.2102
4 10.903062 10.9055 10.9029 12.247705 12.2484 12.2474
5 12.501541 12.5035 12.5015 14.218715 14.2193 14.2185

and its conjugate coalesce to form a pair of real zeros, followed by the second complex zero
and its conjugate, with the other zeros remaining complex when ν = 1. The real zero with
the greatest real part moves off to infinity as ν → 1, with the result that when ν = 1 there are
just 3 real zeros¶ together with an infinite string of complex zeros and their conjugates. As ν
increases further, more real zeros can form but their number always remains finite.

5. The Case p ≥ 2

For general real ν , an infinite string of complex zeros of Cn,p(x ; ~ν) and Sn,p(x ; ~ν) will be
situated in the right-half plane near the anti-Stokes lines arg x = ±πp/(2n). In the neigh-
bourhood of arg x = πp/(2n) we have from (25) and (26)

Cn,p

Sn,p
(x ; ~ν)∼ Hc,s ± ξE− (37)

for large |x |, where the exponential expansion E− is defined in (21) and, in the simplest
situation where the parameters νr do not coincide or differ by integer multiples of n, the

¶The statement made in [5, p. 69] that Cn,1(x; 1) has no real zeros when n is odd is seen to be incorrect.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 274

2
 4
 6
 8
 10

-4

-2

0

2

4

A B C D E

x x
0

1

(a) The zeros x0 and x1 and their conjugates
of Cn,1(x;ν) when n = 4 and ν = 0.1(0.1)1.
The zeros labelled A, B, C , D and E indicate
the real zeros when ν = 1.

6
 7
 8
 9
 10
0

0.5

1

1.5

2

2.5

BC

DE

(b) The loci in the upper-half plane of the ze-
ros B, C and D, E after coalescence. The ar-
rows indicate the sense of increasing ν .

Figure 3

algebraic expansions Hc,s are defined by (22) and (23). We recall that ξ = 2−1 for Cn,p(x ; ~ν)
and ξ = (2i)−1 for Sn,p(x ; ~ν). If one of these parameters, say ν1, is much smaller than the
others, then the term containing (np/nx)−ν1 will be the dominant term in Hc,s as |x | →∞ and
a similar procedure to that described in Section 4.2 for the case p = 1 can be followed. If,
on the other hand, the νr are comparable then the complex zeros can be estimated by direct
solution of the leading-order form of (37). To illustrate, we consider the case p = 2 with
ν1 =

1
2

and ν2 =
3
2
. Then (37) to leading order yields

nϑ

¨

(n2/n x)−
1
2Γ(1

2
)Γ (1/n)

cos
sin
(1

4
π) + (n2/n x)−

3
2Γ(3

2
)Γ (−1/n)

cos
sin
(3

4
π)

«

± 2πξ

nκ
1
2

(−i x)ϑ/κ exp (X e−πi/(2κ)) = 0. (38)

Solution of this equation can be carried out using the secant method in Mathematica.
When p ≥ 2, it becomes possible to encounter a more complicated structure for the alge-

braic expansion. When some of the νr either coincide or differ by integer multiples of n, some
of the poles in the integrand of (10) are of higher order and logarithmic terms can appear.
This is discussed fully in [12, §3.5]. As an example, we let n = 6, p = 2 and consider the two
cases (ν1,ν2) = (1,1) and (ν1,ν2) = (1,7). For the first case all the poles in (10) in Re(s) < 0
are double, whereas for the second case the pole at s = −1 is simple with those at s = −7−6k

(k = 0,1,2, . . . ) being double. If we write ν1,2 ≡ a±3m, where m = 0,1,2, . . . , we have a = 1,
m = 0 for the first case and a = 4, m = 1 for the second case‖. From [12, p. 81], the algebraic
expansion of U6,2(z; ~ν) in (22) when ν1,2 = a± 3m becomes

H(z) =6(6
1
3 z)−a+3m

m−1
∑

k=0

(−)k
k!
Γ(a− 3m+ 6k)Γ(m− k)(6

1
3 z)−6k

‖In terms of the differential equation (5) we have the coefficients a0 = 1, a1 = 3 for the first case and a0 = 7,
a1 = 9 for the second case.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 275

2.5
 5
 7.5
 10
 12.5
 15
 17.5
 20

-10

-5

0

5

10

(a) ν = 0.50

5
 10
 15
 20

-10

-5

0

5

10

(b) ν = 0.85

5
 10
 15
 20

-10

-5

0

5

10

(c) ν = 0.99

5
 10
 15
 20

-10

-5

0

5

10

(d) ν = 1

Figure 4: The distribution of the zeros of Cn,1(x ;ν) in the right-half plane when n = 4 and
ν = [0.50,0.85,0.99,1]. The rays arg x = ±π/8 are the anti-Stokes lines.

+ (−)m6(6
1
3 z)−a−3m

∞
∑

k=0

Γ(a+ 3m+ 6k)

k!(k+m)!
(6

1
3 z)−6k

× {6 log (6
1
3 z) +ψ(k+ 1) +ψ(k+m+ 1)− 6ψ(a+ 3m+ 6k)},

where ψ denotes the psi function and the first sum is interpreted as zero when m = 0.
Then, some routine algebra shows that the algebraic expansions associated with C6,2(x ; ~ν)
and S6,2(x ; ~ν) are

Hc =
π

2x

∞
∑

k=0

(−)k(6k)!

(k!)2
(6

1
3 x)−6k, (39)

Hs =
1

x

∞
∑

k=0

(−)k(6k)!

(k!)2
(6

1
3 x)−6k{log (6

1
3 x)+ 1

3
ψ(k+ 1)−ψ(6k+ 1)} (40)

when (ν1,ν2) = (1,1), and

Hc =
π

12x7

∞
∑

k=0

(−)k(6k+ 6)!

k! (k+ 1)!
(6

1
3 x)−6k, (41)

Hs =
1

x
+

1

6x7

∞
∑

k=0

(−)k(6k+ 6)!

k! (k+ 1)!
(6

1
3 x)−6k {log (6

1
3 x)+ 1

6
ψ(k+ 1) + 1

6
ψ(k+ 2)−ψ(6k+ 7)}

(42)



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 276

2
 4
 6
 8
 10
 12
 14

-10

-5

0

5

10

(a) ν = 0.50

2
 4
 6
 8
 10
 12
 14

-10

-5

0

5

10

(b) ν = 0.85

2
 4
 6
 8
 10
 12
 14

-10

-5

0

5

10

A

(c) ν = 0.999

2
 4
 6
 8
 10
 12
 14

-10

-5

0

5

10

(d) ν = 1

Figure 5: The distribution of the zeros of Cn,1(x ;ν) in the right-half plane when n = 3 and
ν = [0.50,0.85,0.999,1]. In (c) the zero labelled A moves off to infinity as ν → 1. The rays
arg x = ±π/6 are the anti-Stokes lines.

when (ν1,ν2) = (1,7). The leading terms in these expansions can then be used in (38) to
estimate the corresponding zeros. We show some results for the complex zeros when n = 4
and n= 6 with p = 2 in Table 5.

We now consider the conditions for Cn,p(x ; ~ν) and Sn,p(x ; ~ν) to have all real zeros∗∗. From

(25), such zeros can only occur when 1
2
< κ < 1 (that is, when p/n < 1

2
), since when

κ < 1
2

the expansions in (26) on the real axis are purely algebraic with no exponentially small

contribution. The special case κ= 1
2
, where there can be finitely many real zeros, is discussed

below. When κ > 1
2
, an infinite sequence of real zeros will arise when the expansions Hc,s

vanish. From (22) and (23), this will only occur when n is even and the νr are distinct odd

(resp. even) integers which do not differ by integer multiples of n (Condition A). From (24) and
(25), we then find for n even and the parameters νr satisfying the above condition that

Cn,p

Sn,p
(x ; ~ν)∼ κ− 1

2
−ϑ
�

2π

n

�p/2

X ϑ exp
�

X cos
π

2κ

� ∞
∑

j=0

c jX
− j cos

sin

�

X sin
π

2κ
+
π

2κ
(ϑ− j)

�

as |x | → ∞ in the sector |arg x | < π(1
2
− p/n). If some of the integer νr either coincide or

differ by an integer multiple of n, then the algebraic expansion will not vanish — compare
the expansions in (39) – (42) — and complex zeros will arise. In this latter case, it is still

∗∗This is a conjecture as we have no proof that integrals with p ≥ 2 can have all real zeros.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 277

Table 5: The complex zeros xk in the right-half plane when p = 2 for different n and ~ν .

S4,2(x ; ~ν), (ν1,ν2) = (
1
2
, 3

2
) C6,2(x ; ~ν), (ν1,ν2) = (

1
2
, 3

2
)

k xk Asymptotic xk xk Asymptotic xk

0 2.2338± 2.6142i 2.2188± 2.6008i 2.7381± 2.5479i 2.6624± 2.5189i

1 3.3260± 3.6474i 3.3209± 3.6428i 5.1859± 3.8429i 5.1620± 3.8322i

2 4.1507± 4.4378i 4.1480± 4.4354i 7.1864± 4.9344i 7.1735± 4.9282i

3 4.8405± 5.1042i 4.8388± 5.1026i 8.9491± 5.9093i 8.9406± 5.9051i

4 5.4454± 5.6916i 5.4442± 5.6905i 10.5570± 6.8058i 10.5509± 6.8027i

5 5.9905± 6.2230i 5.9896± 6.2221i 12.0532± 7.6444i 12.0484± 7.6419i

C6,2(x ; ~ν), (ν1,ν2) = (1,1) S6,2(x ; ~ν), (ν1,ν2) = (1,1)
k xk Asymptotic xk xk Asymptotic xk

0 3.0327± 2.2880i 2.9696± 2.2521i 3.4932± 2.7751i 3.4448± 2.7529i

1 5.4610± 3.5587i 5.4396± 3.5468i 5.8058± 4.0221i 5.7869± 4.0128i

2 7.4473± 4.6490i 7.4355± 4.6424i 7.7366± 5.0838i 7.7257± 5.0782i

3 10.7994± 6.5263i 10.7937± 6.5231i 9.4550± 6.0386i 9.4477± 6.0348i

4 12.2888± 7.3683i 12.2844± 7.3658i 11.0314± 6.9204i 11.0260± 6.9175i

5 13.6931± 8.1648i 13.6896± 8.1628i 12.5033± 7.7475i 12.4991± 7.7452i

possible†† to have some real zeros in addition to the complex zeros situated near the anti-
Stokes lines arg x = ±πp/(2n).

The procedure for the calculation of the real zeros follows that described in Section 4.1
for the case p = 1. For example, when n = 6, p = 2 (κ = 2

3
), we have the leading-order

approximation from (31)

X (0) =
p

2

�

k+ ε+ 3
4
− 1

8
(ν1 + ν2)

�

π, X = 2
3

x3/2,

where ε= 1
2

for Cn,p(x ; ~ν) and ε= 1 for Sn,p(x ; ~ν). The first-order approximation X (1) can be
similarly computed according to (31); typical results are shown in Table 6.

Finally, we briefly discuss the case of even n and odd (resp. even) integer values of νr

when κ = 1
2

(that is, when p = 1
2
n). Although the functions Cn, 1

2
n(x ; ~ν) and Sn, 1

2
n(x ; ~ν)

are also exponentially small as x → ±∞ when the νr satisfy Condition A, it transpires that
they can be evaluated as polynomials multiplied by exp (−x2/2) and so possess finitely many
real zeros. This situation may be compared with the case n = 2, p = 1 in (32), where
C2,1(x ;ν) (resp. S2,1(x ;ν)) for odd (resp. even) integer ν is expressible in terms of Hermite
polynomials.

To show this, we consider only the case of Cn, 1
2

n(x ; ~ν); the treatment of Sn, 1
2

n(x ; ~ν) is

††For example, the function C6,2(x; ~ν) with ~ν = (1, 7) has 4 positive real zeros.



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 278

Table 6: The real zeros xk and their approximations on the positive axis when p = 2 for dif-
ferent even n and integer ~ν satisfying Condition A. The corresponding value of the coefficient
c1 is given.

(ν1,ν2) = (1,3), c1 =
5

36
(ν1,ν2) = (3,5) c1 = − 7

36
C6,2(x ; ~ν) C6,2(x ; ~ν)

k xk x
(0)
k

x
(1)
k

xk x
(0)
k

x
(1)
k

0 2.945831 2.9233 2.9477 1.283599 1.4054 1.2535
1 5.150494 5.1428 5.1506 4.092473 4.1094 4.0921
2 6.955470 6.9512 6.9555 6.072944 6.0808 6.0729
3 8.550716 8.5479 8.5507 7.765281 7.7701 7.7653
4 10.008981 10.0070 10.0090 9.288354 9.2917 9.2884
5 11.367831 11.3662 11.3678 10.694775 10.6974 10.6948

(ν1,ν2) = (2,4) c1 = − 7
36

(ν1,ν2) = (4,6), c1 =
5

36
S6,2(x ; ~ν) S6,2(x ; ~ν)

k xk x
(0)
k

x
(1)
k

xk x
(0)
k

x
(1)
k

0 3.524152 3.5414 3.5181 2.254113 2.2309 2.2726
1 5.613666 5.6216 5.6123 4.648327 4.6405 4.6502
2 7.361496 7.3663 7.3610 6.527559 6.5233 6.5281
3 8.920297 8.9237 8.9200 8.166473 8.1636 8.1667
4 10.352462 10.3550 10.3523 9.654715 9.6526 9.6549
5 11.691308 11.6933 11.6912 11.035942 11.0343 11.0360

similar. From (20) when p = 1
2
n, we find

Cn, 1
2

n(x ; ~ν) = π
1
2 nϑ−p/2

∞
∑

k=0

(−1
4
nx2)k

k!Γ(k+ 1
2
)

p
∏

r=1

Γ

�

2k+ νr

n

�

.

Application of the multiplication formula for the gamma function

Γ(mz) = (2π)
1
2
(1−m)mmz− 1

2

m−1
∏

r=0

Γ(z +
r

m
), (m= 2,3, . . . )

with m = 1
2
n to the factor Γ(k+ 1

2
), then leads to the representation

Ĉn, 1
2

n(x ; ~ν)≡
2

1
2 Cn, 1

2
n(x ; ~ν)

(2π)p/2nϑ−p/2
=

∞
∑

k=0

Ξ(k)

k!
(−1

2
x2)k, (43)

where

Ξ(k) =

p
∏

r=1

Γ
�

2k

n
+
νr

n

�

Γ
�

2k

n
+ 2r−1

n

� .



R. Paris / Eur. J. Pure Appl. Math, 5 (2012), 260-281 279

Whenever the νr are distinct odd integers such that Ξ(k) reduces to a polynomial in k, the
sum in (43) may be evaluated in closed form in terms of derivatives of exp (−x2/2). For
example, in the particular case n= 4, p = 2, where

Ξ(k) =
Γ(1

2
k+ 1

4
ν1)Γ(

1
2
k+ 1

4
ν2)

Γ(1
2
k+ 1

4
)Γ(1

2
k+ 3

4
)

,

we see that when ν1, ν2 are distinct odd integers whose difference is not a multiple of 4,
Ξ(k) reduces to a polynomial in k. The degree of this polynomial depends on ~ν : when
(ν1,ν2) = (1,3) we have Ξ(k) = 1, when (ν1,ν2) = (1,7) we have Ξ(k) = 1

2
k + 3

4
, when

(ν1,ν2) = (1,11) we have Ξ(k) = (1
2
k+ 3

4
)(1

2
k+ 7

4
), and so on. Thus we find‡‡

Ĉ4,2(x ; (1,3)) = e−x2/2

Ĉ4,2(x ; (1,7)) =
∞
∑

k=0

(−1
2

x2)k

k!
(1

2
k+ 3

4
) = 1

4
(3− x2) e−x2/2

Ĉ4,2(x ; (1,11)) =
∞
∑

k=0

(−1
2

x2)k

k!
(1

2
k+ 3

4
)(1

2
k+ 7

4
) = 1

16
(x4− 12x2+ 21) e−x2/2.

When ν1, ν2 are odd integers that either coincide or differ by a multiple of 4, Ξ(k) contains
a gamma function in the numerator. It follows from the asymptotic theory of the Wright
function [18, 2, 10] that the large-x behaviour of C4,2(x ; ~ν)must then contain a non-vanishing
algebraic component, with the result that there will be infinite strings of complex zeros in this
case.

6. Concluding Remarks

The asymptotic expansion of the integrals Cn,p(x ; ~ν) and Sn,p(x ; ~ν) defined in (3) for
large complex x has been obtained by application of the asymptotic theory of a particular
case of the Wright function. The case corresponding to p = 1, where the integrals are one-
dimensional Fourier integrals, extends the results of previous authors. The zeros of Cn,p(x ; ~ν)
and Sn,p(x ; ~ν) have been considered which, in general, are found to lie in infinite strings in the
complex plane situated near the anti-Stokes lines arg x = ±πp/(2n) in the right-half plane,
with a symmetrical distribution in the left-half plane.

An infinite sequence of real zeros of Cn,p(x ; ~ν) (resp. Sn,p(x ; ~ν)) is found to occur only

when n is even, p/n < 1
2

and the parameters νr (1 ≤ r ≤ p) are distinct odd (resp. even)
integers which do not differ by integer multiples of n. In this case, the integrals in (3) may
also be written over doubly infinite intervals in the form

Cn,p(x ; ~ν) = 2−p

∫ ∞

−∞
. . .

∫ ∞

−∞
t
2m1
1 . . . t

2mp

p ei x t1...tp exp [−(tn
1 + · · ·+ tn

p)/n] d t1 . . . d tp,

‡‡We employ the result
∑∞

k=0 kr(−z)k/k!= (−)r(zd/dz)re−z for r = 0, 1, 2, . . . .



REFERENCES 280

with νr = 2mr + 1, and

iSn,p(x ; ~ν) = 2−p

∫ ∞

−∞
. . .

∫ ∞

−∞
t
2m1+1
1 . . . t

2mp+1
p ei x t1 ...tp exp [−(tn

1 + · · ·+ tn
p)/n] d t1 . . . d tp,

with νr = 2mr+2, where mr (1≤ r ≤ p) are distinct nonnegative integers which do not differ
by a multiple of 1

2
n.

Finally, from (9), we remark that the integrals Cn,p(x ; ~ν) and Sn,p(x ; ~ν) are the even and
odd solutions of the nth-order differential equation (5), with the upper or lower sign chosen
according as 1

2
n is even or odd,respectively, and the coefficients ar given in terms of the νr by

(6).

Appendix: The Zeros of Cn,1(x ;ν) and Sn,1(x ;ν) for Even n and Integer ν

Let n= 1,2, . . ., m= 0,1,2, . . . and define

ψn(z) :=

∫ ∞

−∞
exp (−t2n/2n)eiz t d t

for complex z. Then ψ1(z) =
p

2πexp (−z2/2) has no zeros. In [14], Pólya proved that for
n ≥ 2, ψn(z) has infinitely many zeros all of which are real. These results were extended in
[7], where the following theorem was established:

Theorem 2. For k = 0,1,2, . . . and n= 1,2, . . . all the zeros of ψ(k)n (z) are real and simple.

Then, from (2), some straightforward rearrangement shows that

C2n,1(x ; 2m+ 1) =
1

2

∫ ∞

−∞
t2m exp (−t2n/2n)ei x t d t =

(−)m
2
ψ(2m)

n (x)

and

S2n,1(x ; 2m+ 2) =
1

2i

∫ ∞

−∞
t2m+1 exp (−t2n/2n)ei x t d t =

(−)m
2
ψ(2m+1)

n (x).

It then follows from the above theorem that when n ≥ 2 and m = 0,1,2, . . . the zeros of
C2n,1(x ; 2m+ 1) and S2n,1(x ; 2m+ 2) are all real and, moreover, simple.

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