On closed-form solutions of triple series equations involving Laguerre polynomials
We consider some triple series equations involving generalized Laguerre polynomials. These equations are reduced to triple integral equations for Bessel functions. The closed-form solutions of the triple integral equations for Bessel functions are obtained and, finally, we get the closed-form soluti...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508847061336064 |
|---|---|
| author | Dhaliwal, R.S. Rokne, J. Singh, B. M. Дналивал, Р.С. Рокне, Ж. Сінгх, Б. М. |
| author_facet | Dhaliwal, R.S. Rokne, J. Singh, B. M. Дналивал, Р.С. Рокне, Ж. Сінгх, Б. М. |
| author_sort | Dhaliwal, R.S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:39:03Z |
| description | We consider some triple series equations involving generalized Laguerre polynomials. These equations are reduced to triple integral equations for Bessel functions. The closed-form solutions of the triple integral equations for Bessel functions are obtained and, finally, we get the closed-form solutions of triple series equations for Laguerre polynomials. |
| first_indexed | 2026-03-24T02:31:42Z |
| format | Article |
| fulltext |
UDC 517.9
B. M. Singh, J. Rokne, R. S. Dhaliwal* (Univ. Calgary, Canada)
ON CLOSED FORM SOLUTIONS OF TRIPLE SERIES
EQUATIONS INVOLVING LAGUERRE POLYNOMIALS
PRO ROZV’QZKY ZAMKNENO} FORMY DLQ RIVNQN|
POTRIJNYX RQDIV, WO MISTQT| POLINOMY LAHERRA
We consider some triple series equations involving generalized Laguerre polynomials. The equations
are reduced to triple integral equations of Bessel functions. Closed form solutions for the triple integral
equations of Bessel functions are obtained and finally closed form solutions of triple series equations of
Laguerre polynomials are obtained.
Rozhlqnuto deqki rivnqnnq potrijnyx rqdiv, wo mistqt\ uzahal\neni polinomy Laherra. Rivnqn-
nq zvedeno do potrijnyx intehral\nyx rivnqn\ funkcij Besselq. Otrymano rozv’qzky zamkneno]
formy dlq potrijnyx intehral\nyx rivnqn\ funkcij Besselq, a takoΩ rozv’qzky zamkneno]
formy dlq rivnqn\ potrijnyx rqdiv z polinomamy Laherra.
1. Introduction. Srivastava [1] was the first mathematician to solve dual series
equations of Laguerre polynomials. Later on Lowndes [2], Srivastava [3 – 6] and
Srivastava and Panda [7] generalized the dual series equations discussed by Srivastava
[1]. In recent years Singh, Rokne and Dhaliwal [8, 9] have also discussed dual series
equations of Laguerre polynomials and have obtained solutions in closed form.
Lowndes [10] and Dwivedi and Trivedi [11] solved triple series equations involving
Laguerre polynomials and obtained solutions through Fredholm integral equations of
the second kind which can be solved numerically. Further Lowndes and Srivastava
[12] solved triple series equations of Laguerre polynomials and obtained a closed form
solution by using the Erdélyi – Kober operator of fractional integration. In this paper
we consider the triple series equations of Laguerre polynomials given by
A L x
n
f xn n
n
α
α
( )
( + + )
= ( )
=
∞
∑
2
0
1
2
1
/
Γ
, 0 < x < a, (1)
A L x
n
g xn n
n
β
β
( )
( + + )
= ( )
=
∞
∑
2
0
2
2
1
/
Γ
, a < x < b, (2)
A L x
n
f xn n
n
γ
γ
( )
( + + )
= ( )
=
∞
∑
2
0
3
2
1
/
Γ
, b < x < ∞, (3)
with three given functions f x1( ) , g x2( ) and f x3( ) of sufficient smoothness and
three given parameters α, β , γ > – 1. Here L xn
p( ) , p > – 1, x ∈ R, n ≥ 0, are
Laguerre polynomials.
The series equations (l) – (3) have a more general form than the series equations
discussed by Lowndes and Srivasava [12]. Particular cases of the series equations of
the paper are discussed in Section 6. The aim of this paper is to find a closed form for
the series equations (l) – (3).
2. Useful results. We need the following results for studying of the triple series
equations (1), (2) and (3).
From the book of Bateman and Erdély ([13], Ch. 10, 10.12, (18)) we find the
following result:
t
p n
L x e
xtn
n n
p
n
t
2
2
0
2
2 1
2
2
2
Γ( + + )
( ) =
=
∞ −
∑ / /
pp
pJ xt( ) , p > – 1, (4)
*
Dr. Dhaliwal, professor emeritus, passed away suddenly on October, 10, 2007.
© B. M. SINGH, J. ROKNE, R. S. DHALIWAL , 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2 231
232 B. M. SINGH, J. ROKNE, R. S. DHALIWAL
where J p(⋅) is the Bessel function of the first kind.
From Bateman and Erdély ([13], Ch. 10, 10.12, (21)) we find the integral
t e J xt dt n x e Lp n t
p
n p x
n
p1
2
1
1
22 1 2
0
22+ + −
∞
−( ) =∫ / /! 11
2
2
x
, p1 > – 1. (5)
If λ1 > µ1 > – 1 then the integral
J a J b d
a b
b aλ µ
µ λ µξ ξ ξ ξ
1 1
1 1 1
1
0
2
0 0
( ) ( ) =
< <
( −+ −
∞
∫
, ,
bb
b a
2 1
1 1
1 1
1 1 12
0
)
( − )
< <
− −
− −
λ µ
λ µ λ λ µΓ
, ,
(6)
is a well-known result in the theory of Bessel function of Watson ([14, p. 401],
equations (1) and (3)).
3. Reduction of triple series equations to triple integral equations. In this
section we shall reduce the triple series equations (1), (2) and (3) to triple integral
equations of Bessel functions.
Following Lowndes and Srivastava [12] the coefficients An in equations (1), (2)
and (3) can be written in terms of an unknown function A t( ) :
A t e A t dtn
n n t= ( )− −
∞
∫2 2 2
0
2 / , n = 0, 1, 2, 3, … . (7)
Substituting equation (7) into equations (1), (2) and (3) and interchanging the order
of summation and integrations and using equations (4) we find the following form of
the triple integral equations for α, β , γ > – 1:
t A t J xt dt
x
f x−
∞
( ) ( ) =
( )∫ α
α
α
0
12
, 0 < x < a, (8)
t A t J xt dt
x
g x−
∞
( ) ( ) =
( )∫ β
β
β
0
22
, a < x < b, (9)
t A t J xt dt
x
f x−
∞
( ) ( ) =
( )∫ γ
γ
γ
0
32
, b < x < ∞. (10)
In Sections A and B we find two different explict integral formulae for A x( ) for
the two following cases:
0 < α – γ < 1, 0 < β – γ < 1, α, β, γ > – 1, (11)
0 < β – γ < 1, 0 < γ – α < 1, α, β, γ > – 1. (12)
4. Section A. In this section we shall find the solution of the triple integral
equations (8), (9) and (10) subject to conditions (11).
We assume that
t A x J xt dt x−
∞
( ) ( ) = ( )∫ γ
γ ψ
0
, 0 < x < b. (13)
Using Hankel transforms and equations (10) and (13) we obtain
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ON CLOSED FORM SOLUTIONS OF TRIPLE SERIES EQUATIONS … 233
A t t u u J ut du
t
u f u J
b
( ) = ( ) ( ) + ( )+
+
+∫1
0
1
1
3
2
γ
γ
γ
γ
γ
γψ (( )
∞
∫ ut du
b
, 0 < t < ∞. (14)
Substituting the value of A t( ) from equation (14) into equation (8) and interchanging
the order of integrations, using the integral defined by equation (6) we get
u u du
x u
x f xx 1
2 2 1
0
2
1
2
+
+ − +
( )
( − )
=
( − ) ( )
∫
γ
γ α
α
γ
ψ α γΓ
11
, 0 < x < a, α > γ > – 1. (15)
The above equation is an Abel’s type integral equation if γ > α – 1. Hence the
solution of equation (15) can be written in the form
ψ α γ π α γ
π
γ
γ
α
( ) = ( − ) ( − ) ( )− − +[ ]
u
u d
du
x f x1 1 2
1
2
sin Γ ddx
u x
u F uu
( − )
=
( )
−∫ 2 2
0
1
2α γ
γ
γ
, 0 < x < a.
(16)
For obtaining equation (16) we have the condition
0 < 1 – ( α – γ ) < 1. (17)
Substituting equation (14) into (9) and interchanging the order of integrations and using
the result defined by equation (6) and assuming β > γ we find that
u u du
x u
x g xx 1
2 2 1
0
2
2
2
+
+ − +
( )
( − )
=
( − ) ( )
∫
γ
γ β
β
γ
ψ β γΓ
11
, a < x < b, (18)
where
0 < 1 + γ – β < 1. (19)
With equation (16) we can write equation (18) in the form
u u du
x u
x g x
a
x 1
2 2 1
2
2
2
+
+ − +
( )
( − )
=
( − ) ( )
∫
γ
γ β
β
γ
ψ β γΓ
11
1 2
1
2 2 1
0
1
2
−
( )
( − )
+
+ −∫γ
γ
γ β
p F p dp
x p
a
, a < x < b. (20)
Equation (20) is an Abel’s type integral equation if γ > β – 1 and the right-hand side
of equation (20) is known function of x. Hence the solution of equation (20) can be
written as
ψ β γ β γ π
π
γ
γ
β
( ) = ( − ) ( − ) ( )− − +[ ]
u
u d
du
x g x1 1 2
2
2
Γ sin ddx
u xa
u
( − ) −∫ 2 2 β γ
–
–
u d
du
x dx
u x
p
a
u− −
− −
[ ]( − )
( − )∫
γ
γ β γ
γ
π
β γ π
1
1 2 2
2
2
sin
++
+ −
( )
( − )∫
1
1
2 2 1
0
F p dp
x p
a
γ β
, a < u < b. (21)
From the integral
d
du
x dx
u x x p
u a p
a
u
( − ) ( − )
= ( − )
− −( − )∫ 2 2 2 2 1
2 2
β γ β γ
β−−
−( − )( − )
γ
β γu p u a2 2 2 2
, p < a < u, (22)
and interchanging the order of integrations in integrals of equation (21) we find that
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
234 B. M. SINGH, J. ROKNE, R. S. DHALIWAL
ψ β γ β γ π
π
γ
γ
β
( ) = ( − ) ( − ) ( )− − +[ ]
u
u d
du
x g x1 1 2
2
2
Γ sin ddx
u xa
u
( − ) −∫ 2 2 β γ
–
–
u
u a
p F p a−
− −
+[ ]( − )
( − )
( )( −γ
γ β γ
γ
π
β γ π
2 1 2 2
2 1
1
2sin pp dp
u p
a 2
2 2
0
)
−
−
∫
β γ
= (23)
=
u G uγ
γ
2
2
( )
, a < u < b. (24)
Equation (14) together with equations (16) and (24) lead to
A t t p F p J pt dp
a
( ) = ( ) ( )
− + +∫2 1 1
1
0
γ γ γ
γ +
+ p G p J pt dp f p p J pt dp
a
b
b
γ
γ
γ
γ
+ +
∞
( ) ( ) + ( ) ( )
∫ ∫1
2 3
1 . (25)
From equation (25) and the integral (5) we can write equation (7) in the form
A n F p G p f p pn
a
a
b
b
= ( ) + ( ) + ( )
−
∞
∫ ∫ ∫! 2 1
0
2 3
γ 11 2 2
22
2
+ −
γ γe L
p
dpp
n
/ , n = 0, 1, 2, … .
(26)
Conditions (17) and (19) can be written as:
0 < α – γ < 1, 0 < β – γ < 1. (27)
Equation (26) represents the solution of equations (1), (2) and (3) subject to the
conditions (27).
5. Section B. In this section we find the solution of the triple series equations (1),
(2) and (3) when β > α, γ > α.
For solving the triple series equations we will solve the triple integral equations (8),
(9) and (10) when β > α, γ > α. Making use of equation (8), we assume that
t A t J xt dt
f x
x
x a
x
−
∞
( ) ( ) = ( )
< <
( )
∫ α
α
α
ϕ0
1 2
0, ,
, aa x< < ∞
,
(28)
where ϕ( )x is an unknown function to be determined.
Using Hankel transforms we get from equation (28) that
A t
t
f u u J ut du t u u J
a
( ) = ( ) ( ) + ( )
+
+ +∫
1
1
1
0
1
2
α
α
α
α
α
αϕ (( )
∞
∫ ut du
a
, 0 < t < ∞. (29)
Substituting equation (29) into equation (9), interchanging the order of integrations and
using the integral (6) we find that
u u du
x u
x
g
a
x α
β α
β
α
ϕ β α+
−( − ) +
( )
( − )
= ( − )
∫
1
2 2 1
2
1 2
2
Γ (( ) −
( )
( − )
+
−( − )∫x
u f u du
x u
a
1
2
2 1
1
2 2 1
0
α
α
β α
, a < x < b.
(30)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ON CLOSED FORM SOLUTIONS OF TRIPLE SERIES EQUATIONS … 235
The left-hand side of equation (30) is an Abel’s type integral equation with
corresponding condition β – α < 1 and the right-hand side is a known function of x
hence we get the solution of the above equation as
ϕ β α π β α
π
α
α
β
( ) = ( − ) ( − ) ( )− − +[ ]
u
u d
du
x g x1 1 2
2
2
sin Γ ddx
u xa
u
( − ) −∫ 2 2 β α
–
–
sin[ ]( − )
( − )
− −
− −∫
β α π
π
α
α β α
αu d
du
x dx
u x
p
a
u1
1 2 2
2
2
++
−( − )
( )
( − )∫
1
1
2 2 1
0
f p dp
x p
a
β α
, a < u < b, (31)
where
0 < 1 – ( β – α ) < 1. (32)
Interchanging the order of integrations in the second integral of equation (31) and
using the result (22) we can write equation (31) in the following form:
ϕ β α β α π
π
α
α
β
( ) = ( − ) ( − ) ( )− − +[ ]
u
u d
du
x g xΓ 1 1 2
2
2
sin ddx
u xa
u
( − ) −∫ 2 2 β α
–
–
sin[ ]( − )
( − )
( − )−
− −
+ −β α π
π
α
α β α
α βu
u a
p a p
2 1 2 2
2 1 2 2 αα f p dp
u p
a
1
2 2
0
( )
−∫ =
=
u l uα
α
2
2
( )
, a < u < b. (33)
Substituting equation (29) into equation (10) and interchanging the order of
integrations we can find that as in equation (33) that
ϕ γ α γ α π
π
α
α
γ
( ) = ( − ) ( − ) ( )− − +[ ]
u
u d
du
x f xΓ 1 1 2
3
2
sin ddx
u xa
u
( − ) −∫ 2 2 γ α
–
–
sin[ ]( − )
( − )
( − )−
− −
+ −γ α π
π
α
α γ α
α γu
u a
p a p
2 1 2 2
2 1 2 2 αα f p dp
u p
a
1
2 2
0
( )
−∫ = (34)
=
u l uα
α
3
2
( )
, b < u < ∞, (35)
where
0 < 1 – ( γ – α ) < 1. (36)
From equations (33) and (35), equation (29) can be written as
A t
t
f u l u l u
a
a
b
b
( ) = ( ) + ( ) + ( )
+ ∞
∫ ∫ ∫
1
1
0
2 3
2
α
α
( )+u J ut duα
α
1 . (37)
With equations (37) equation (7) can be written as
A n f u l u l un
a
a
b
b
= ( ) + ( ) + ( )
−
∞
∫ ∫ ∫! 2 1
0
2 3
α
+ −u e L
u
duu
n
1 2 2
22
2
α α/ , n = 0, 1, 2, … .
(38)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
236 B. M. SINGH, J. ROKNE, R. S. DHALIWAL
Conditions (32) and (36) can be writtend in the following form:
0 < β – α < 1, 0 < γ – α < 1. (39)
Equation (38) is a solution of the triple series equations (1), (2) and (3) subject to
conditions (39).
6. Particular cases (((( αααα = γγγγ )))). In this section we shall find the results for particular
cases of Section A and Section B when α = γ. Hence we shall obtain the results of
Lowndes and Srivastava [12].
Making use of equation
sin m
m m1
1 11
π π=
( ) ( − )Γ Γ
, (40)
we find from equation (16) that
F u f u1 1( ) = ( ) , 0 < u < a. (41)
If we assume
β – γ = β – α = λ, 0 < λ < 1, (42)
then we find from equations (23), (24), (41) that
G p
p d
dp
x g x dx
p xa
p
2
1 2 1 2
2
2 21
( ) =
( − )
( )
( − )
− − +α β
λλΓ ∫∫ –
–
2 2
2 2
1 2
1
2 2
2
p
p a
x f x a x dx
p
− +( )
( − )
( )( − )α
λ
α λπλ
π
sin
−−∫
x
a
2
0
, a < p < b. (43)
Making use of equation (41) we can write equation (26) in the following manner:
A n f p G p f p pn
a
a
b
b
= ( ) + ( ) + ( )
−
∞
∫ ∫ ∫! 2 1
0
2 3
γ 11 2 2
22
2
+ −
γ γe L
p
dpp
n
/ . (44)
Equation (44) is the solution the triple series equations (1), (2) and (3) when α = γ,
β ≥ α > – 1.
If we compare this solution defined by equation (44) to ([12, p. 186], 4.11) of
Lowndes and Srivastava solution.
Both are the same if
G p f p2 2( ) = ( ) , (45)
where f p2( ) is defined by equation (4.9) in reference [12, p. 186]. Making m = 0 in
equation (2.8) in [12, p. 183], then the relation (45) is correct. Without using the
derivative formula in [12, p. 183] ((2.5)) both solutions are same. Otherwise are
different when m is not zero.
In Section B, assming γ = α in equations (33), (34) and (35) we find by making
use of equation (40) that
l u
u d
du
x g u du
u xa
u
2
1 2 1 2
2
2 21
( ) =
( − )
( )
( − )
− − +α β
λλΓ ∫∫ –
–
2 2
2 2
1 2
1
2 2
2
u
u a
p f p a p dp
u
− +( )
( − )
( )( − )α
λ
α λπλ
π
sin
−−∫
p
a
2
0
, a < u < b, (46)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ON CLOSED FORM SOLUTIONS OF TRIPLE SERIES EQUATIONS … 237
l u f u3 3( ) = ( ) , b < u < ∞. (47)
Making use of equation (47), equations (38) can be written in the following form:
A n f u l u l u un
a
a
b
b
= ( ) + ( ) + ( )
−
∞
∫ ∫ ∫! 2 1
0
2 3
α 11 2 2
22
2
+ −
α αe L
u
duu
n
/ , n = 0, 1, 2, … .
(48)
We also find from equations (43), (45) and (46) that
l u f u2 2( ) = ( ) . (49)
Making use of equation (49) the equation (48) converge to equation (44).
Finally solutions of the triple series equations (1), (2) and (3) of Sections A and B
converge to Lowndes and Srivastava [12] with m = 0 when α = γ, β ≥ α > – 1 and
m is defined in the paper [12, p. 183] ((2.4)).
Making use of the derivative formula discussed in [12, p. 183] ((2.5)) to the series
equation (2) we can find the solution of the triple series equations (1), (2) and (3) in the
same form as discussed in [12] for general values of m when α = γ, β ≥ α > – 1.
7. Conclusion. In Sections A and B we have developed a method for finding a
closed form of solutions of triple series equations subject to the conditions (11) and
(12). Results for particular cases α = γ and β ≥ α > – 1 of Sections A and B are
given in Section 6 and they include results of Lowndes and Srivastava [12]. If we put
m = 0 in equation (2.8) of the paper [12, p. 183] then the results of this paper and those
of [12] are the same. We have therefore obtained a simplified solution. When m is
not zero the solution discussed in [13] has a more complex form and can be obtained in
this paper by using the derivative formula ([12, p. 183], (2.5)) to the series equations
(2).
1. Srivastava K. N. On dual series relations involving Laguerre polynomials // Pacif. J. Math. – 1966.
– 19. – P. 529 – 533.
2. Lowndes J. S. Some dual series equations involving Laguerre polynomials // Ibid. – 1968. – 25. –
P. 123 – 127.
3. Srivastava H. M. A note on certain dual series equations involving Laguerre polynomials // Ibid. –
1969. – 30. – P. 525 – 527.
4. Srivastava H. M. Dual series relations involving generalized Laguerre polynomials // J. Math.
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Received 25.12.08,
after revision — 08.07.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
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| id | umjimathkievua-article-2859 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:31:42Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/92/e59c4a55a7f95ec77e35db65761efa92.pdf |
| spelling | umjimathkievua-article-28592020-03-18T19:39:03Z On closed-form solutions of triple series equations involving Laguerre polynomials Про розв'язки замкненої форми для рівнянь потрійних рядів, що містять поліноми Лагерра Dhaliwal, R.S. Rokne, J. Singh, B. M. Дналивал, Р.С. Рокне, Ж. Сінгх, Б. М. We consider some triple series equations involving generalized Laguerre polynomials. These equations are reduced to triple integral equations for Bessel functions. The closed-form solutions of the triple integral equations for Bessel functions are obtained and, finally, we get the closed-form solutions of triple series equations for Laguerre polynomials. Розглянуто деякі рівняння потрійних рядів, що містять узагальнені поліноми Лагерра. Рівняння зпедено до потрійних інтегральних рівнянь функцій Бесселя. Отримано розв'язки замкненої форми для потрійних інтегральних рівнянь функцій Бесселя, а також розв'язки замкненої форми для рівнянь потрійних рядів з поліномами Лагерра. Institute of Mathematics, NAS of Ukraine 2010-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2859 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 2 (2010); 231 – 237 Український математичний журнал; Том 62 № 2 (2010); 231 – 237 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2859/2470 https://umj.imath.kiev.ua/index.php/umj/article/view/2859/2471 Copyright (c) 2010 Dhaliwal R.S.; Rokne J.; Singh B. M. |
| spellingShingle | Dhaliwal, R.S. Rokne, J. Singh, B. M. Дналивал, Р.С. Рокне, Ж. Сінгх, Б. М. On closed-form solutions of triple series equations involving Laguerre polynomials |
| title | On closed-form solutions of triple series equations involving Laguerre polynomials |
| title_alt | Про розв'язки замкненої форми для рівнянь потрійних рядів, що містять поліноми Лагерра |
| title_full | On closed-form solutions of triple series equations involving Laguerre polynomials |
| title_fullStr | On closed-form solutions of triple series equations involving Laguerre polynomials |
| title_full_unstemmed | On closed-form solutions of triple series equations involving Laguerre polynomials |
| title_short | On closed-form solutions of triple series equations involving Laguerre polynomials |
| title_sort | on closed-form solutions of triple series equations involving laguerre polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2859 |
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