EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 810-820
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Double Integrals Stemming from the Boltzmann
Equation in the Kinetic Theory of Gasses

H. M. Srivastava1,2,3,4

1 Department of Mathematics and Statistics, University of Victoria,
Victoria, British Columbia V 8W3R4, Canada
2 Department of Medical Research, China Medical University Hospital,
China Medical University, Taichung 40402, Taiwan, Republic of China
3 Department of Mathematics and Informatics, Azerbaijan University,
71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
4 Section of Mathematics, International Telematic University Uninettuno,
I-00186 Rome, Italy

Abstract. The main object of this article is to revisit a certain double integral involving Kummer’s
confluent hypergeometric function 1F1, which arose in the study of the collision terms of the
celebrated Boltzmann equation in the kinetic theory of gases. Here, in this article, we propose to
investigate some novel extensions and generalizations of this family of double integrals. We also
point out some relevant connections of the results, which are presented here, with other related
recent developments in the theory and applications of hypergeometric functions.

2020 Mathematics Subject Classifications: 33C15, 33C20, 33E12, 11M35, 33C60, 76P05,
82B40, 82C40
Key Words and Phrases: Kinetic theory of gase, Boltzmann equation, Kummer’s confluent
hypergeometric function, Generalized hypergeometric functions, Fox-Wright function, General
Mittag-Leffler type, Hurwitz-Lerch type functions

1. Introduction and Motivation

Introduced in the year 1872 by the Austrian physicist and philosopher, Ludwig Boltz-
man (1844–1906), the celebrated Boltzmann equation is known to describe the statistical
behaviour of a thermodynamic system which is not in a state of equilibrium. In a recent
study of the collision terms of the Boltzmann equation occurring in the kinetic theory of
gases, the problem of evaluation of the following double integral arose (see, for detail, [3]):

∆ :=

∫ π

0

∫ π

0
1F1

 α;

γ;
λ1 + λ2 cosψ + λ cos θ cosψ

 dψ dθ, (1)

DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4429

Email addresses: harimsri@math.uvic.ca (H. M. Srivastava)

https://www.ejpam.com 810 © 2022 EJPAM All rights reserved.



H. M. Srivastava / Eur. J. Pure Appl. Math, 15 (3) (2022), 810-820 811

where λ, λ1 and λ2 are constants. Also, the (Kummer’s) confluent hypergeometric function

1F1, which is involved in the integral in (1) above (see, for details, [2]), corresponds to the
special case of the generalized hypergeometric function pFq (p, q ∈ N0) when p = q = 1.
Indeed, in terms of the general Pochhammer symbol or the shifted factorial (κ)ν , since

(1)n = n! (n ∈ N0 := N ∪ {0} = {0, 1, 2, · · · }),

which is defined (for κ, ν ∈ C), in terms of the (Euler’s) Gamma function, by

(κ)ν :=
Γ(κ+ ν)

Γ(κ)
=


1 (ν = 0; κ ∈ C \ {0})

κ(κ+ 1) · · · (κ+ n− 1) (ν = n ∈ N; κ ∈ C),
(2)

it being understood conventionally that (0)0 := 1 and assumed tacitly that the Γ-quotient
exists, a generalized hypergeometric function, with p numerator parameters αj ∈ C (j =
1, · · · , p) and q denominator parameters γj ∈ C \ Z−

0 (j = 1, · · · , q), is given by

pFq

 α1, · · · , αp;

γ1, · · · , γq;
z

 :=

∞∑
n=0

(α1)n · · · (αp)n
(γ1)n · · · (γq)n

zn

n!

=
Γ(γ1) · · ·Γ(γq)
Γ(α1) · · ·Γ(αp)

·
∞∑
n=0

Γ(α1 + n) · · ·Γ(αp + n)

Γ(γ1 + n) · · ·Γ(γq + n)

zn

n!

=: pFq (α1, · · · , αp; γ1, · · · , γq; z) , (3)

under appropriate conditions for convergence of the infinite series (see, for details, [21, p.
3 et seq.]), given by (see also [1], [10], [11] [12], [14], [15] and [24])

(i) converges absolutely for |z| <∞ if p ≦ q,
(ii) converges absolutely for |z| < 1 if p = q + 1, and
(iii) diverges for all z (z ̸= 0) if p > q + 1.

Under the constraint min{ℜ(α),ℜ(γ)} > 1, it was shown for the double integral in (1)
that (see [3, p. 13])

∆ =
π

R

(
γ − 1

α− 1

) 1F1

 α− 1;

γ − 1;
λ1 +R

− 1F1

 α− 1;

γ − 1;
λ1 −R

 , (4)

where, for convenience,
R2 = λ2 + λ22. (5)

The long and involved derivation of the integral formula (4) by Deshpande [3, pp. 11–
13] made use of such konen results as (for example) a contour integral representation of



H. M. Srivastava / Eur. J. Pure Appl. Math, 15 (3) (2022), 810-820 812

Kummer’s confluent hypergeometric function 1F1 [4, p. 272], a certain Neumann expan-
sion involving the modified Bessel function Iν(z) and the Gegenbauer (or ultraspherical)
polynomials Cν

n(z) (see [5, p. 98]), and the addition theorem for the Legendre (or spheri-
cal) polynomials Pn(z) in terms of the associated Legendre polynomials Pm

n (z) (see [7, p.
35] and [5, p. 244]). In a sequel to [3], a direct and much shorter evaluation of the double
integral in (4) was given by Srivastava [16] who did actually extend the integral formula
(4) to the following general form (see [16, p. 8, Eq. (22)]):

∆∗ :=

∫ π

0

∫ π

0
pFq

 α1, · · · , αp;

γ1, · · · , γq;
λ1 + λ2 cosψ + λ cos θ cosψ

 dψ dθ

=
π

R


q∏

j=1
(γj − 1)

p∏
j=1

(αj − 1)


 pFq

 α1 − 1, · · · , αp − 1;

γ1 − 1, · · · , γq − 1;
λ1 +R



− pF q

 α1 − 1, · · · , αp − 1;

γ1 − 1, · · · , γq − 1;
λ1 −R

 , (6)

where R is given, as before, by (5) and, for convergence of the hypergeometric series
involved, we require that p ≦ q or p = q + 1 and

max{|λ1|+ |λ2|+ |λ1 ±R|} < 1,

by appealing to the principle of analytic continuation.
Our present investigation is motivated essentially by the aforementioned importance

of the double integral (4), as well as by its potentially useful generalization (6). It seems
to be worthwhile to explore the possibility of evaluation of some further extended versions
of the double integrals (4) and (6).

2. The Hurwitz-Lerch Zeta Function and the Mittag-Leffler Type
Functions

I choose first to mention my having met many times and having discussed mathematical
researches, especially on various families of higher transcendental functions and related
topics (including, of course, about the widely- and extensively-investigated Fox H-function
and the Fox-Wright function pΨq, ) with my Canadian colleague, Charles Fox (1897–1977)
of birth and education in England, both at McGill University and Sir George Williams
University (now Concordia University) in Montréal, mainly during the 1970s (see, for
details, [6] and [17]). Another remarkable mathematical scientist of modern times happens
to be Sir Edward Maitland Wright (1906–2005), with whom I had the privilege to meet and
discuss researches emerging from his publications on hypergeometric and related higher



H. M. Srivastava / Eur. J. Pure Appl. Math, 15 (3) (2022), 810-820 813

transcendental functions during my visit to the University of Aberdeen in Scotland in the
year 1976. We recall here a series of monumental works by Wright (see, for example, [28],
[29] and [30]), in which he introduced and systematically studied the asymptotic expansion
of the following Taylor-Maclaurin series (see [28, p. 424]):

Eα,β(ϕ; z) :=

∞∑
n=0

ϕ(n)

Γ(αn+ β)
zn

(
α, β ∈ C; ℜ(α) > 0

)
, (7)

where ϕ(t) is a function satisfying suitable sufficient conditions.
The general Wright function Eα,β(ϕ; z), defined by (7), not only extends the familiar

Mittag-Leffler function Eα(z) and its two-parameter version Eα,β(z), which are defined,
respectively, by (see [13], [26] and [27])

Eα(z) :=
∞∑
k=0

zk

Γ(αk + 1)
and Eα,β (z) :=

∞∑
k=0

zk

Γ(αk + β)
(8)

(
z, α, β ∈ C; ℜ(α) > 0

)
,

but also the above-mentioned Fox-Wright function pΨq, defined by (see, for details, [4,
p. 183] and [24, p. 21]; see also [9, p. 56], [8, p. 65] and [23, p. 19])

pΨ
∗
q

 (a1, A1) , · · · , (ap, Ap) ;

(b1, B1) , · · · , (bq, Bq) ;
z


:=

∞∑
n=0

(a1)A1n
· · · (ap)Apn

(b1)B1n
· · · (bq)Bqn

zn

n!

=:
Γ (b1) · · ·Γ (bq)

Γ (a1) · · ·Γ (ap)
pΨq

 (a1, A1) , · · · , (ap, Ap) ;

(b1, B1) , · · · , (bq, Bq) ;
z

 (9)

(
ℜ(Aj) > 0 (j = 1, · · · , p) ; ℜ(Bj) > 0 (j = 1, · · · , q) ; ℜ

( q∑
j=1

Bj −
p∑

j=1

Aj

)
≧ −1

)
,

where (κ)ν denotes the general Pochhammer symbol or the shifted factorial, which we
have defined already by (2), and the equality in the convergence condition holds true only
for suitably-bounded values of |z| given by

|z| < ∇ :=

 p∏
j=1

A
−Aj

j

 ·

 q∏
j=1

B
Bj

j

 .

In some recent developments, which are based upon the general Wright function
Eα,β(ϕ; z), defined by (7), Srivastava [21] introduced the following function and applied



H. M. Srivastava / Eur. J. Pure Appl. Math, 15 (3) (2022), 810-820 814

it in his study of a family of fractional-order kinetic equations (see, for details, [19] and
[20]):

Eα,β(φ; z, s, κ) :=
∞∑
n=0

φ(n)

(n+ κ)s Γ(αn+ β)
zn

(
α, β ∈ C; ℜ(α) > 0

)
, (10)

where the function φ(τ) and the parameters α, β, s and κ are appropriately constrained.
It is not difficult to see that Srivastava’s function Eα,β(φ; z), defined by (10), provides a
hybrid form of the Mittag-Leffler type functions, the Hurwitz-Lerch zeta function Φ(z, s, κ)
defined by

Φ(z, s, a) :=
∞∑
n=0

zn

(n+ κ)s
(11)

(
κ ∈ C \ Z−

0 ; s ∈ C when |z| < 1; ℜ(s) > 1 when |z| = 1
)
,

as well as the following interesting and potentially useful family of the multi-parameter
Hurwitz-Lerch Zeta functions

Φ
(ρ1, ··· ,ρp;σ1, ··· ,σq)
λ1, ··· ,λp;µ1, ··· ,µq

(z, s, κ),

which is defined by (see [25, p. 503, Eq. (6.2)]; see also [18] and [22])

Φ
(ρ1, ··· ,ρp,σ1, ··· ,σq)
λ1, ··· ,λp;µ1, ··· ,µq

(z, s, κ) :=
∞∑
n=0

p∏
j=1

(λj)nρj

n! ·
q∏

j=1
(µj)nσj

zn

(n+ κ)s
(12)

(
p, q ∈ N0; λj ∈ C (j = 1, · · · , p); κ, µj ∈ C \ Z−

0 (j = 1, · · · , q);

ρj , σk ∈ R+ (j = 1, · · · , p; k = 1, · · · , q); ∆∗∗ > −1 when s, z ∈ C;

∆∗∗ = −1 and s ∈ C when |z| < ∇∗;

∆∗∗ = −1 and ℜ(Ξ) > 1

2
when |z| = ∇∗

)
,

where, for convenience,

∆∗∗ :=

q∑
j=1

σj −
p∑

j=1

ρj and Ξ := s+

q∑
j=1

µj −
p∑

j=1

λj) +
p− q

2
(13)

and

∇∗ :=

 p∏
j=1

ρ
−ρj
j

 ·

 q∏
j=1

σ
σj

j

 , (14)



H. M. Srivastava / Eur. J. Pure Appl. Math, 15 (3) (2022), 810-820 815

3. A General Family of Double Integrals

Before presenting an extended version of the double integrals (4) and (6), we list here
each of the following elementary results which will be needed in the derivation of our
general double integral.

I. A Multiple Series Identity

∞∑
m1, ··· ,mr=0

f (m1 + · · ·+mr)
zm1
1

m1!
· · · z

mr
r

mr!
=

∞∑
m=0

f(m)
(z1 + · · ·+ zr)

m

m!
, (15)

provided that the series involved are absolutely convergent.

II. An Integral Identity

∫ π

0
cosm t g(sin t) dt = [1 + (−1)m]

∫ π
2

0
cosm t g(sin t) dt (m ∈ N0)

=


2

∫ π
2

0
cos2n t g(sin t) dt (m = 2n; n ∈ N0)

0 (m = 2n+ 1; n ∈ N0),

(16)

provided that each of the integrals exists.

III. A Simple Series Identity

∞∑
m=0

[1− (−1)m]h(m) = 2

∞∑
m=0

h(2m+ 1), (17)

provided that each of the series exists.

IV. A Trigonometric Integral

∫ π
2

0
cosµ t sinν t dt =

Γ
(
µ+1
2

)
Γ
(
ν+1
2

)
2 Γ
(
µ+ν+2

2

) (
min{ℜ(µ),ℜ(ν)} > −1

)
. (18)

V. Legendre’s Duplication Formula and Its Consequences

Γ(2z) =
22z−1

√
π

Γ(z) Γ

(
z +

1

2

)
, (19)



H. M. Srivastava / Eur. J. Pure Appl. Math, 15 (3) (2022), 810-820 816

which readily yields the following simpler consequences:

(2m)! = 22m m!

(
1

2

)
m

and (2m+ 1)! = 22m m!

(
3

2

)
m

(m ∈ N0). (20)

With a view to presenting our proposed generalization of the double integrals in (4)
and (6), we first slightly modify the definition (10) as follows:

E∗
α,β(φ

∗; z, s, κ) :=
∞∑
n=0

φ∗(n)

(n+ κ)s Γ(αn+ β)

zn

n!

(
α, β ∈ C; ℜ(α) > 0

)
, (21)

where, just as in the definition (10), the function φ∗(τ) and the parameters α, β, s and κ
are appropriately constrained.

Now, by applying the definition (21) and the case r = 3 of the multiple series identity
(15), we find that

Ω :=

∫ π

0

∫ π

0
E∗
α,β

(
φ∗;λ1 + λ2 cosψ + λ cos θ cosψ, s, κ

)
sinψ dψ dθ

=

∫ π

0

∫ π

0

∞∑
n=0

φ∗(n)

(n+ κ)s Γ(αn+ β)

(
λ1 + λ2 cosψ + λ cos θ cosψ

)n
n!

sinψ dψ dθ

=

∞∑
ℓ,m,n=0

φ∗(ℓ+m+ n)

(ℓ+m+ n+ κ)s Γ
(
α(ℓ+m+ n) + β

) λℓ1
ℓ!

λm2
m!

λn

n!

·
(∫ π

0
cosm ψ sinn+1 ψ dψ

)(∫ π

0
cosn θ dθ

)
, (22)

which, in view of the integral formulas (16), (18), (19) and (20), readily yields

Ω =
∞∑

ℓ,m,n=0

φ∗(ℓ+ 2m+ 2n)

(ℓ+ 2m+ 2n+ κ)s Γ
(
α(ℓ+ 2m+ 2n) + β

) λℓ1
ℓ!

λ2m2
(2m)!

λ2n

(2n)!

·

(
2

∫ π
2

0
cos2m ψ sin2n+1 ψ dψ

)(
2

∫ π
2

0
cos2n θ dθ

)

= 2π
∞∑

ℓ,m,n=0

φ∗(ℓ+ 2m+ 2n)(
3
2

)
m+n

(ℓ+ 2m+ 2n+ κ)s Γ
(
α(ℓ+ 2m+ 2n) + β

)

· λ
ℓ
1

ℓ!

(
λ2
2

)2m

m!

(
λ

2

)2n

n!
. (23)

Upon replacing n in (23) by n−m (0 ≦ m ≦ n), we sum the resulting binomial series
and simplify the outcome by using the identity (20) once again. We thus find that

Ω =
2π

R

∞∑
ℓ,n=0

φ∗(ℓ+ 2n)

(ℓ+ 2n+ κ)s Γ
(
α(ℓ+ 2n) + β

) λℓ1
ℓ!

R2n+1

(2n+ 1)!



H. M. Srivastava / Eur. J. Pure Appl. Math, 15 (3) (2022), 810-820 817

or, equivalently,

Ω =
2π

R

∞∑
ℓ,n=0

φ∗(ℓ− 1 + (2n+ 1)
)

(
ℓ− 1 + (2n+ 1) + κ

)s
Γ

(
α
(
ℓ− 1 + (2n+ 1)

)
+ β

) λℓ1
ℓ!

R2n+1

(2n+ 1)!
, (24)

where R is given by (5).
Finally, we apply the elementary series identity (17), together with the case r = 2 of

the multiple series identity (15). We are thus led from (24) to the following result:

Ω :=

∫ π

0

∫ π

0
E∗
α,β

(
φ∗;λ1 + λ2 cosψ + λ cos θ cosψ, s, κ

)
sinψ dψ dθ

=
π

R

( ∞∑
n=0

φ∗(n− 1)

(n+ κ− 1)s Γ
(
α(n− 1) + β

) (λ1 +R)n

n!

−
∞∑
n=0

φ∗(n− 1)

(n+ κ− 1)s Γ
(
α(n− 1) + β

) (λ1 −R)n

n!

)
, (25)

provided that each member of (25) exists.

Remark. By suitably specializing the sequence φ∗(n), one can deduce from the general
result (25) the corresponding double integrals involving simpler functions of the Mittag-
Leffler and Hurwitz-Lerch types. In a particular case of (6), if we first set

φ∗(n− 1) = (n+ κ− 1)s Γ
(
α(n− 1) + β

) p∏
j=1

(αj − 1)n−1

q∏
j=1

(γj − 1)n−1

and then note, in view of the definition (2), that

p∏
j=1

(αj − 1)n−1

q∏
j=1

(γj − 1)n−1

=


q∏

j=1
(γj − 1)

p∏
j=1

(αj − 1)




p∏
j=1

(αj − 1)n

q∏
j=1

(γj − 1)n

 ,

we arrive at the double integral formula (6).

Acknowledgements

The author expresses his appreciation to Prof. Dr. Eyüp Çetin for his kind invitation
to submit this paper to the European Journal of Pure and Applied Mathematics.



REFERENCES 818

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