EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 620-625
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A quadruple integral involving the Hermite polynomial
Hn(x): derivation and evaluation

Robert Reynolds1,∗, Allan Stauffer1
1 Department of Mathematics and Statistics, Faculty of Science, York University, Toronto,
Ontario, Canada, M3J1P3

Abstract. A closed form expression of a quadruple integral involving the Hermite polynomial
Hn(x) is derived. Special cases are expressed in terms of special functions and fundamental con-
stants. All the results in this work are new.
2020 Mathematics Subject Classifications: 30E20, 33-01, 33-03, 33-04, 33-33B
Key Words and Phrases: Hermite polynomial, quadruple integral, Hurwitz-Lerch zeta function,
Cauchy integral formula

1. Significance Statement

Named for the Frenchman, Charles Hermite (1822-1901) these polynomials are orthog-
onal on the infinite interval −∞ < x < ∞ with a weight function of e−x2 . They arise
in physics, as in the solution of Schrödinger’s differential equation for a simple harmonic
oscillator, which belongs to a broad class of second order differential equations [4]. In
this present work we investigate the quadruple integral involving the Hermite polynomial
Hn(x) and the parameter n dependence on a constant factor raised to a power and its
invariance with respect to the Hurwitz-Lerch zeta function.

2. Introduction

In this paper we derive the quadruple definite integral given by

(1)

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0
t−mxm−1z1−mym−nHn(xα)e

−α2x2−b
(
t2+y2+z2

)

logk
(axy
tz

)
dxdydzdt

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4204

Email addresses: milver@my.yorku.ca (R. Reynolds), stauffer@yorku.ca (A. Stauffer)

https://www.ejpam.com 620 © 2022 EJPAM All rights reserved.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 620-625 621

where the parameters k, a, n,m are general complex numbers and Re(n) < Re(m). This
definite integral will be used to derive special cases in terms of special functions and
fundamental constants. The derivations follow the method used by us in [5]. This method
involves using a form of the generalized Cauchy’s integral formula given by

yk

Γ(k + 1)
=

1

2πi

∫
C

ewy

wk+1
dw. (2)

where C is in general an open contour in the complex plane where the bilinear concomitant
has the same value at the end points of the contour. We then multiply both sides by a
function of x, y, z and t, then take a definite quadruple integral of both sides. This
yields a definite integral in terms of a contour integral. Then we multiply both sides of
Equation (2) by another function of y and take the infinite sums of both sides such that
the contour integral of both equations are the same.

3. Definite Integral of the Contour Integral

We use the method in [5]. The variable of integration in the contour integral is
u = w+m. The cut and contour are in the first quadrant of the complex u-plane. The cut
approaches the origin from the interior of the first quadrant and the contour goes round
the origin with zero radius and is on opposite sides of the cut. Using a generalization of
Cauchy’s integral formula we form the triple integral by replacing y by log

(axy
tz

)
and multi-

plying by t−mxm−1z1−mym−nHn(xα)e
α2

(
−x2

)
−b

(
t2+y2+z2

)
then taking the definite integral

with respect to x ∈ [0,∞), y ∈ [0,∞), z ∈ [0,∞) and t ∈ [0,∞) to obtain

(3)

1

Γ(k + 1)

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0
t−mxm−1z1−mym−nHn(xα)e

−α2x2−b
(
t2+y2+z2

)

logk
(axy
tz

)
dxdydzdt

=
1

2πi

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫
C
aww−k−1t−m−wxm+w−1z−m−w+1Hn(xα)

ym−n+we−α2x2−b
(
t2+y2+z2

)
dwdxdydzdt

=
1

2πi

∫
C

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0
aww−k−1t−m−wxm+w−1z−m−w+1Hn(xα)

ym−n+we−α2x2−b
(
t2+y2+z2

)
dxdydzdtdw

=
1

2πi

∫
C
π22n−3aww−k−1α−m−w csc(π(m+ w))b

1
2
(m+n+w−4)dw

from equation (3.22.2.2) in [1] and equation (3.326.2) in [3] where Re(π(m + w)) >
0, Re(n) < Re(m), |argα|< π/4 and using the reflection formula (8.334.3) in [3] for the
Gamma function. We are able to switch the order of integration over x, y, z and t using



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 620-625 622

Fubini’s theorem since the integrand is of bounded measure over the space C × [0,∞) ×
[0,∞)× [0,∞)× [0,∞).

4. The Hurwitz-Lerch zeta Function and Infinite Sum of the Contour
Integral

In this section we use Equation (2) to derive the contour integral representations for
the Hurwitz-Lerch zeta function.

4.1. The Hurwitz-Lerch zeta Function

The Hurwitz-Lerch zeta function (25.14) in [2] has a series representation given by

Φ(z, s, v) =

∞∑
n=0

(v + n)−szn (4)

where |z|< 1, v 6= 0,−1, .. and is continued analytically by its integral representation
given by

Φ(z, s, v) =
1

Γ(s)

∫ ∞

0

ts−1e−vt

1− ze−t
dt =

1

Γ(s)

∫ ∞

0

ts−1e−(v−1)t

et − z
dt (5)

where Re(v) > 0, and either |z|≤ 1, z 6= 1, Re(s) > 0, or z = 1, Re(s) > 1.

4.2. Infinite sum of the Contour Integral

Using equation (2) and replacing y by log(a) + iπ(2y + 1) − log(2) then multiplying
both sides by
−iπ221−meiπm(2y+1) taking the infinite sum over y ∈ [0,∞) and simplifying in terms of
the Hurwitz-Lerch zeta function we obtain

(6)

− 1

Γ(k + 1)
ik+1πk+2eiπm2k+n−2α−mb

1
2
(m+n−4)

Φ

(
e2imπ,−k, −2i log(a)− i log(b) + 2i log(α) + 2π

4π

)
= − 1

2πi

∞∑
y=0

∫
C
iπ22n−2aww−k−1α−m−weiπ(2y+1)(m+w)b

1
2
(m+n+w−4)dw

= − 1

2πi

∫
C

∞∑
y=0

iπ22n−2aww−k−1α−m−weiπ(2y+1)(m+w)b
1
2
(m+n+w−4)dw

=
1

2πi

∫
C
π22n−3aww−k−1α−m−w csc(π(m+ w))b

1
2
(m+n+w−4)dw

from equation (1.232.3) in [3] where Im(π(m+ w)) > 0 in order for the sum to converge.



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 620-625 623

5. Definite Integral in terms of the Lerch Function

Theorem 1. For all k, a, b, α, n,m ∈ C, Re(n) < Re(m), |argα|< π/4,

(7 )

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0
t−mxm−1z1−mym−nHn(xα)e

−α2x2−b
(
t2+y2+z2

)

logk
(axy
tz

)
dxdydzdt

= −iikπk+2eiπm2k+n−2α−mb
1
2
(m+n−4)

Φ

(
e2imπ,−k, −2i log(a)− i log(b) + 2i log(α) + 2π

4π

)
Proof. The right-hand sides of relations (3) and (6) are identical; hence, the left-hand

sides of the same are identical too. Simplifying with the Gamma function yields the desired
conclusion.

Example 1. The degenerate case.

(8)

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0
t−mxm−1z1−mym−nHn(xα)e

−α2x2−b
(
t2+y2+z2

)
dxdydzdt

= π22n−3α−m csc(πm)b
1
2
(m+n−4)

Proof. Use equation (7) and set k = 0 and simplify using entry (2) in Table below
(64:12:7) in [4].

Example 2.

(9)

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0

√
zy

1
2
−nHn(xα)e

−α2x2−b
(
t2+y2+z2

)
√
t
√
x log

(axy
tz

) dxdydzdt

=
1√
α
iπ2n−4b

n
2
− 7

4

(
ψ(0)

(
−2i log(a)− i log(b) + 2i log(α) + 2π

8π

)
− ψ(0)

(
−2i log(a)− i log(b) + 2i log(α) + 6π

8π

))
Proof. Use equation (7) set m = 1/2 and simplify in terms of the Hurwitz zeta function

ζ(s, v) then apply l’Hopital’s rule as k → −1 and simplify in terms of the digamma function
ψ(0)(x) using equation (64:4:1) in [4].



R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 620-625 624

Example 3.

(10)

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0

√
zy

1
2
−nHn(xα)e

−t2−α2x2−y2−z2

√
t
√
x log

(
−xy

tz

) dxdydzdt

= −
iπ2n−4

(
H i log(α)

4π

−H i log(α)
4π

− 1
2

)
√
α

Proof. Use equation (9) and set a = −1, b = 1 and simplify in terms of the Harmonic
number function Hn.

Example 4.∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0

zy−nHn(x)t
−m−pe−t2−x2−y2−z2

x log
(xy
tz

)(
tmxpypz−p − xmymz−mtp

)
dxdtdzdt

= π2n−2
(
tanh−1

(
eiπm

)
− tanh−1

(
eiπp

))
(11)

Proof. Use equation (7) and form a second equation by replacing m → p and taking
their difference and setting k = −1, a = 1, b = 1, α = 1 and simplify.

Example 5.

(12)
∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0

3
√
zy

1
2
−nHn(x)e

−t2−x2−y2−z2
(

6
√
x 6
√
y − 6

√
t 6
√
z
)

t2/3
√
x log

(xy
tz

) dxdydzdt

= π2n−4 log(3)

Proof. Use equation (11) and set m = 1/2, p = 2/3 and simplify.

6. Discussion

In this paper, we have presented a novel method for deriving a new integral involving
the Hermite polynomial Hn(x) along with some interesting definite integrals using contour
integration. The results presented were numerically verified for both real and imaginary
and complex values of the parameters in the integrals using Mathematica by Wolfram.

Acknowledgements

This research is supported by NSERC Canada under grant 504070.



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