Differential and integral equations for Legendre – Laguerre based hybrid polynomials
UDC 517.9 In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial di...
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| author | Khan, S. Riyasat, M. Wani , Sh. A. Khan, S. Riyasat, M. Wani , Sh. A. M. Sh. A. |
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| description | UDC 517.9 In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial differential equations are established. The analogous results for the three-variable Hermite – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, –E uler and – Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by deriving homogeneous Volterra integral equations for these polynomials and for their relatives. |
| doi_str_mv | 10.37863/umzh.v73i3.894 |
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DOI: 10.37863/umzh.v73i3.894
UDC 517.5
S. Khan (Aligarh Muslim Univ., India),
M. Riyasat (Zakir Hussain College Eng. and Technology, Aligarh Muslim Univ., India),
Sh. A. Wani (Univ. Kashmir, Srinagar, India)
DIFFERENTIAL AND INTEGRAL EQUATIONS
FOR LEGENDRE – LAGUERRE BASED HYBRID POLYNOMIALS
ДИФЕРЕНЦIАЛЬНI ТА IНТЕГРАЛЬНI РIВНЯННЯ
ДЛЯ ГIБРИДНИХ ПОЛIНОМIВ НА БАЗI ПОЛIНОМIВ ЛЕЖАНДРА – ЛАГЕРРА
In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properti-
es including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential,
integro-differential and partial differential equations are established. The analogous results for the three-variable Hermi-
te – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, – Euler and
– Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by
deriving homogeneous Volterra integral equations for these polynomials and for their relatives.
Розглянуто гiбридну сiм’ю полiномiв Лежандра – Лагерра – Аппеля та встановлено їхнi властивостi, якi включають
розклади рядiв, форми детермiнантiв, рекурентнi спiввiдношення, оператори зсуву, за якими йдуть диференцiальнi
та iнтегро-диференцiальнi рiвняння, а також диференцiальнi рiвняння з частинними похiдними. Подiбнi результати
отримано для полiномiв Ермiта – Лагерра – Аппеля з трьома змiнними. У термiнах полiномiв Лежандра – Лагерра –
Бернуллi, – Ейлера та – Дженоккi побудовано деякi приклади, щоб показати застосування основних результатiв.
Далi, для цих та пов’язаних з ними полiномiв отримано однорiдне iнтегральне рiвняння Вольтерра.
1. Introduction and preliminaries. The study of differential equations is a wide field in pure and
applied mathematics, physics and engineering. The mathematical theory of differential equations first
developed together with the sciences where the equations had originated and where the results found
applications. Differential equations play an important role in modeling virtually every physical,
technical, or biological process, from celestial motion to bridge design, to interactions between
neurons. We recall the following definitions.
Let \{ pn(x)\} \infty n=0 be a sequence of polynomials such that \mathrm{d}\mathrm{e}\mathrm{g}(pn(x))=n, n \in \BbbN 0 := \{ 0, 1, 2, . . .\} .
The differential operators \Theta -
n and \Theta +
n satisfying the properties
\Theta -
n \{ pn(x)\} = pn - 1(x), \Theta +
n \{ pn(x)\} = pn+1(x), (1.1)
are called derivative and multiplicative operators, respectively. The polynomial sequence \{ pn(x)\} \infty n=0
satisfying equation (1.1) is then called quasimonomial.
The derivative and multiplicative operators for a given family of polynomials give rise to some
useful properties such as
(\Theta -
n+1\Theta
+
n )\{ pn(x)\} = pn(x), (\Theta +
n - 1\Theta
+
n - 2 . . .\Theta
+
2 \Theta
+
1 \Theta
+
0 )\{ p0(x)\} = pn(x). (1.2)
The technique used in obtaining differential equations via (1.2) is known as the factorization
method [12, 13]. The main idea of the factorization method is to find the derivative and multiplicative
operators such that equation (1.2) holds. The factorization method can be equivalently treated as
monomiality principle. The monomiality principle [7] and the associated operational rules are used
c\bigcirc S. KHAN, M. RIYASAT, SH. A. WANI, 2021
408 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
DIFFERENTIAL AND INTEGRAL EQUATIONS FOR LEGENDRE – LAGUERRE BASED HYBRID . . . 409
in [8] to explore new classes of isospectral problems leading to nontrivial generalizations of special
functions.
The Appell polynomial sequences [2] arise in numerous problems of mathematics, physics, and
engineering. The set of all Appell sequences is closed under the operation of umbral compositions
of polynomial sequences and forms an abelian group. The Appell polynomial sequences are defined
by the generating function
\scrR (t)eyt =
\infty \sum
n=0
\scrR n(y)
tn
n!
. (1.3)
The power series \scrR (t) is given by
\scrR (t) = \scrR 0 +
t
1!
\scrR 1 +
t2
2!
\scrR 2 + . . .+
tn
n!
\scrR n + . . . =
\infty \sum
n=0
\scrR n
tn
n!
, \scrR 0, \not = 0,
with \scrR i, i = 0, 1, 2, . . . , real coefficients. The function \scrR (t) is an analytic function at t = 0 and
for any \scrR (t), the derivative of \scrR n(y) satisfies
\scrR \prime
n(y) = n\scrR n - 1(y).
The Appell polynomial sequences are defined by the series expansion
\scrR n(y) =
n\sum
k=0
\biggl(
n
k
\biggr)
\scrR k yn - k. (1.4)
For the suitable choices of the function \scrR (t), different members belonging to the family of Appell
polynomials can be obtained. These members and their related numbers are given in Table 1.1.
The Bernoulli and Euler numbers appear in the Taylor series expansions of trigonometric and
hyperbolic tangent and cotangent and trigonometric and hyperbolic secant functions, respectively.
The Genocchi numbers appear in counting the number of up-down ascent sequences and graph and
automata theories.
We know that the generalized special polynomials provide new means of analysis for the solution
of large classes of partial differential equations often encountered in physical problems. Most of the
special functions of mathematical physics and their generalizations have been suggested by physical
problems. Some of these special polynomials are listed below.
The two-variable Laguerre polynomials (2VLP) Ln(x, y) [10] are defined by means of the gene-
rating equation
eytC0(xt) =
\infty \sum
n=0
Ln(x, y)
tn
n!
, (1.5)
where C0(xt) is the 0th Tricomi function [1] defined by the operational definition
C0(\alpha x) = \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- \alpha D - 1
x
\bigr)
\{ 1\} , D - n
x \{ 1\} :=
xn
n!
is inverse derivative operator. (1.6)
The Tricomi function Cn(x) is defined by the series expansion
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
410 S. KHAN, M. RIYASAT, SH. A. WANI
Table 1.1. Certain members belonging to the Appell family
S.No.
Name of the
\scrR (t) Generating function Series expansionpolynomial and
related number
I
Bernoulli
polynomials
and numbers
[11]
t
et - 1
\biggl(
t
et - 1
\biggr)
eyt =
\infty \sum
n=0
Bn(y)
tn
n!
Bn(y) =
n\sum
k=0
\biggl(
n
k
\biggr)
Bky
n - k
\biggl(
t
et - 1
\biggr)
=
\infty \sum
n=0
Bn(:= Bn(0) = Bn(1))
tn
n!
B0 = 1, B1 = \pm 1
2
, B2 =
1
6
II
Euler
polynomials
and numbers
[11]
2
et + 1
\biggl(
2
et + 1
\biggr)
eyt =
\infty \sum
n=0
En(y)
tn
n! En(y) =
n\sum
k=0
\biggl(
n
k
\biggr)
\times
\times Ek
2k
\biggl(
y - 1
2
\biggr) n - k
2et
e2t + 1
=
\infty \sum
n=0
En
\biggl(
:= 2nEn
\biggl(
1
2
\biggr) \biggr)
tn
n!
E0 = 1, E1 = 0, E2 = - 1
III
Genocchi
polynomials
and numbers
[4, 14]
2t
et + 1
\biggl(
2t
et + 1
\biggr)
eyt =
\infty \sum
n=0
Gn(y)
tn
n!
Gn(y) =
n\sum
k=0
\biggl(
n
k
\biggr)
Gky
n - k
2t
et + 1
=
\infty \sum
n=0
Gn(:= Gn(0))
tn
n!
G0 = 0, G1 = 1, G2 = - 1
Cn(x) =
\infty \sum
k=0
( - 1)k xk
k! (n+ k)!
. (1.7)
The series expansion and operational representation for the 2VLP Ln(x, y) are given by [10]
Ln(x, y) = n!
n\sum
k=0
( - 1)k xk yn - k
(k!)2(n - k)!
, (1.8)
Ln(x, y) = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- D - 1
x
\partial
\partial y
\biggr)
\{ yn\} . (1.9)
Next, the two-variable Legendre polynomials (2VLeP) Sn(z, y) [9] are specified by means of the
generating equation
eyt C0( - zt2) =
\infty \sum
n=0
Sn(z, y)
tn
n!
. (1.10)
The series expansion and operational representation for the 2VLeP Sn(z, y) are given by [9]
Sn(z, y) = n!
[n
2
]\sum
k=0
zk yn - 2k
(k!)2 (n - 2k)!
, (1.11)
Sn(z, y) = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
D - 1
z
\partial 2
\partial y2
\biggr)
\{ yn\} . (1.12)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
DIFFERENTIAL AND INTEGRAL EQUATIONS FOR LEGENDRE – LAGUERRE BASED HYBRID . . . 411
Remark 1.1. In fact, from equations (1.6) and (1.10), we find that Sn(z, y) = Hn(y,D
- 1
z ),
where Hn(y,D
- 1
z ) are the two-variable Hermite Kampé de Feriet polynomials defined by [3]
eyt+D - 1
z t2 =
\infty \sum
n=0
Hn(y,D
- 1
z )
tn
n!
.
To introduce the multivariable hybrid polynomials and to characterize their properties via dif-
ferent generating function methods is an interesting approach. These polynomials may be useful in
certain problems of number theory, combinatorics, numerical analysis, theoretical physics, approxi-
mation theory and other fields of pure and applied mathematics. This gives motivation to introduce
a new hybrid family of three-variable Legendre – Laguerre – Appell polynomials (3VLeLAP). The
series expansion, determinant form, recurrence relations, shift operators and differential equations
for these polynomials are derived. Certain applications are framed in order to give the results for
the three-variable Legendre – Laguerre – Bernoulli, – Euler and – Genocchi polynomials. The integral
equations for the Legendre – Laguerre – Appell and other hybrid polynomials are also established.
2. Legendre – Laguerre based hybrid polynomials. First, we introduce a hybrid family of the
three-variable Legendre – Laguerre polynomials (3VLeLP) by making use of replacement technique
and slightly focus on proving some properties related to these polynomials.
Expanding the exponential function and replacing the powers of y, that is yn, n = 0, 1, 2, . . . ,
by the polynomials Sn(z, y), n = 0, 1, 2, . . . , in equation (1.5) and then using equation (1.10), we
get the following generating function for the 3VLeLP:
The 3VLeVP SLn(x, z, y) are defined by means of the generating function
eytC0(xt)C0( - zt2) =
\infty \sum
n=0
SLn(x, z, y)
tn
n!
. (2.1)
Using equations (1.5) and (1.7) or (1.10) and (1.7) appropriately in equation (2.1) and after
simplification, we get the following series expansions for the 3VLeLP SLn(x, y, z):
SLn(x, z, y) = n!
[n/2]\sum
k=0
Ln - 2k(x, y)z
k
(n - 2k)! (k!)2
\Biggl(
or = n!
n\sum
k=0
( - 1)kxkSn - k(z, y)
(n - k)! (k!)2
\Biggr)
, (2.2)
which in view of equations (1.8) or (1.11) can also be expressed as
SLn(x, z, y) = n!
k+2l\leq n\sum
k, l=0
zl( - x)k yn - k - 2l
(n - k - 2l)! (k!)2(l!)2
.
Using equations (1.9) and (1.11) or (1.12) and (1.8) appropriately in equation (2.2) gives the
following operational representations for the 3VLeLP SLn(x, z, y):
SLn(x, z, y) = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- D - 1
x
\partial
\partial y
\biggr)
\{ Sn(z, y)\}
\biggl(
or = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
D - 1
z
\partial 2
\partial y2
\biggr)
\{ Ln(x, y)\}
\biggr)
,
which on using equations (1.12) or (1.9) can also be expressed as
SLn(x, z, y) = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
D - 1
z
\partial 2
\partial y2
- D - 1
x
\partial
\partial y
\biggr)
\{ yn\} .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
412 S. KHAN, M. RIYASAT, SH. A. WANI
Now, we introduce a hybrid family of 3VLeLAP via generating function, series expansions and
determinant definition. For this, we prove the following results.
Theorem 2.1. The 3VLeLAP are defined by the generating function
\scrR (t)eytC0(xt)C0( - zt2) =
\infty \sum
n=0
SL\scrR n(x, z, y)
tn
n!
. (2.3)
Proof. Expanding the exponential function eyt and then replacing the powers of y, i.e.,
y0, y1, y2, . . . , yn by the polynomials SL0(x, z, y), SL1(x, z, y), . . . , SLn(x, z, y) in the left-hand
side and y by the polynomial SL1(x, z, y) in the right-hand side of equation (1.3) and after summing
up the terms in the left-hand side of the resultant equation, we have
\scrR (t)
\infty \sum
n=0
SLn(x, z, y)
tn
n!
=
\infty \sum
n=0
\scrR n(SL1(x, z, y))
tn
n!
,
which on using equation (2.1) in the left-hand side and denoting the resultant 3VLeLAP in the
right-hand side by
SL\scrR n(x, z, y) that is
SL\scrR n(x, z, y) := \scrR n\{ SL1(x, z, y)\} , (2.4)
we get generating function (2.3).
Theorem 2.2. The 3VLeLAP are defined by the series expansion
SL\scrR n(x, z, y) = n!
n\sum
k=0
[n/2]\sum
l=0
\scrR n - k - 2l(y)( - x)kzl
(n - k - 2l)! (k!)2 (l!)2
. (2.5)
Proof. Using equations (1.3) and (1.7) in the left-hand side of equation (2.3) and applying
the Cauchy-product rule and then comparing the coefficients of like powers of tn/n! gives series
expansion (2.5).
Theorem 2.3. The 3VLeLAP of degree n are defined by
SL\scrR 0(x, z, y) =
1
\beta 0
,
SL\scrR n(x, z, y) =
=
( - 1)n
(\beta 0)
n+1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1 SL1(x, z, y) SL2(x, z, y) . . . SLn - 1(x, z, y) SLn(x, z, y)
\beta 0 \beta 1 \beta 2 . . . \beta n - 1 \beta n
0 \beta 0
\biggl(
2
1
\biggr)
\beta 1 . . .
\biggl(
n - 1
1
\biggr)
\beta n - 2
\biggl(
n
1
\biggr)
\beta n - 1
0 0 \beta 0 . . .
\biggl(
n - 1
2
\biggr)
\beta n - 3
\biggl(
n
2
\biggr)
\beta n - 2
. . . . . . . .
. . . . . . . .
0 0 0 . . . \beta 0
\biggl(
n
n - 1
\biggr)
\beta 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
, (2.6)
where n = 1, 2, . . . , \beta 0, \beta 1, . . . , \beta n \in \BbbR , \beta 0 \not = 0.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
DIFFERENTIAL AND INTEGRAL EQUATIONS FOR LEGENDRE – LAGUERRE BASED HYBRID . . . 413
Proof. Replacing the powers y0, y1, y2, . . . , yn by the polynomials SL0(x, z, y),
SL1(x, z, y), . . . , SLn(x, z, y) in the right-hand side and y by the polynomial SL1(x, z, y) in the
left-hand side of determinant definition of the Appell polynomials ([6, p. 1533], (29), (30)) and then
using equation (2.4) in the left-hand side of resultant equation, we obtain determinant definition (2.6).
Further, we focus on obtaining recurrence relations and shift operators for the 3VLeLAP
SL\scrR n(x, z, y). For this, we prove the following results.
Theorem 2.4. The 3VLeLAP
SL\scrR n(x, z, y) satisfy the recurrence relation
SL\scrR n+1(x, z, y) =
\bigl(
y + \alpha 0 - D - 1
x
\bigr)
SL\scrR n(x, z, y)+
+2nD - 1
z SL\scrR n - 1(x, z, y) +
n\sum
k=1
\biggl(
n
k
\biggr)
\alpha k SL\scrR n - k(x, z, y), (2.7)
where the coefficients \{ \alpha k\} k\in \BbbN 0 are given by expansions
\scrR \prime (t)
\scrR (t)
=
\infty \sum
k=0
\alpha k
tk
k!
. (2.8)
Proof. We consider generating function (2.3) in the form
\scrR (t)e(y - D - 1
x )t+D - 1
z t2 =
\infty \sum
n=0
SL\scrR n(x, z, y)
tn
n!
,
which on differentiating both sides with respect to t and using equations (2.3) and (2.8) and then
applying the Cauchy-product rule in the left-hand side of the resultant equation, it follows that
\infty \sum
n=0
SL\scrR n+1(x, z, y)
tn
n!
=
\infty \sum
n=0
\biggl(
(y + \alpha 0 - D - 1
x )
SL\scrR n(x, z, y)+
+2nD - 1
z SL\scrR n - 1(x, z, y) +
n\sum
k=1
\biggl(
n
k
\biggr)
\alpha k SL\scrR n - k(x, z, y)
\biggr)
tn
n!
.
Equating the coefficients of like powers of tn/n! on both sides of the above equation yields
recurrence relation (2.7).
Theorem 2.5. The shift operators for the 3VLeLAP
SL\scrR n(x, z, y) are given by
y\$
-
n :=
1
n
Dy, (2.9)
x\$
-
n := - 1
n
Dx, (2.10)
z\$
-
n :=
1
n
D - 1
y Dz, (2.11)
y\$
+
n := y + \alpha 0 - D - 1
x + 2D - 1
z Dy +
n\sum
k=1
\alpha k
k!
Dk
y , (2.12)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
414 S. KHAN, M. RIYASAT, SH. A. WANI
x\$
+
n := y + \alpha 0 - D - 1
x - 2D - 1
z Dx +
n\sum
k=1
( - 1)k
\alpha k
k!
Dk
x, (2.13)
z\$
+
n := y + \alpha 0 - D - 1
x + 2D - 1
y +
n\sum
k=1
\alpha k
k!
D - k
y Dk
z . (2.14)
Proof. Differentiating both sides of generating relation (2.3) with respect to y and then simpli-
fying it follows that
y\$
-
n
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
1
n
Dy
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
SL\scrR n - 1(x, z, y), (2.15)
which proves assertion (2.9).
Again, differentiating both sides of equation (2.3) with respect to x and on simplification, we
find
x\$
-
n
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
- 1
n
Dx
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
SL\scrR n - 1(x, z, y), (2.16)
which gives assertion (2.10).
Further, differentiating both sides of generating function (2.3) with respect to z and after simpli-
fication of the resultant equation, we get
z\$
-
n
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
1
n
D - 1
y Dz
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
SL\scrR n - 1(x, z, y), (2.17)
which yields assertion (2.11).
Using equation (2.15) in the relation
SL\scrR n - k(x, z, y) =
\bigl(
\$ -
n - k+1\$
-
n - k+2 . . .\$
-
n - 1\$
-
n
\bigr) \bigl\{
SL\scrR n(x, z, y)
\bigr\}
, (2.18)
gives
SL\scrR n - k(x, z, y) =
(n - k)!
n!
Dk
y
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
. (2.19)
Making use of equation (2.19) in recurrence relation (2.7) and in view of the fact that
\$+
n
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
SL\scrR n+1(x, z, y), (2.20)
we obtain
y\$
+
n
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
\Biggl(
y + \alpha 0 - D - 1
x + 2D - 1
z Dy +
n\sum
k=1
\alpha k
k!
Dk
y
\Biggr) \bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
=
SL\scrR n+1(x, z, y),
which proves assertion (2.12).
In order to derive the expression for raising operator (2.13), we use equation (2.16) in rela-
tion (2.18) and on simplification, we have
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DIFFERENTIAL AND INTEGRAL EQUATIONS FOR LEGENDRE – LAGUERRE BASED HYBRID . . . 415
SL\scrR n - k(x, z, y) = ( - 1)k
(n - k)!
n!
Dk
x
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
,
which on using in recurrence relation (2.7) and taking help of relation (2.20) gives
x\$
+
n
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
=
\Biggl(
y + \alpha 0 - D - 1
x - 2D - 1
z Dx +
n\sum
k=1
( - 1)k
\alpha k
k!
Dk
x
\Biggr) \bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
SL\scrR n+1(x, z, y),
which proves assertion (2.13).
Similarly, using equation (2.17) in relation (2.18) and after simplification it follows that
SL\scrR n - k(x, z, y) =
(n - k)!
n!
D - k
y Dk
z
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
. (2.21)
Further, in view of equations (2.21), (2.7) and (2.20), we get
z\$
+
n
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
=
\Biggl(
y + \alpha 0 - D - 1
x + 2D - 1
y +
n\sum
k=1
\alpha k
k!
D - k
y Dk
z
\Biggr) \bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
SL\scrR n+1(x, z, y),
which led to assertion (2.14).
Next, we establish the differential, integro-differential and partial differential equations for the
3VLeLAP
SL\scrR n(x, z, y).
Theorem 2.6. The 3VLeLAP
SL\scrR n(x, z, y) satisfy the differential equation\Biggl(
xyD2
x - (x - y)Dx - \alpha 0Dy - 2zDz -
n\sum
k=1
\alpha k
k!
Dk+1
y + n
\Biggr)
SL\scrR n(x, z, y) = 0. (2.22)
Proof. Consider the factorization relation
\scrL -
n+1\scrL
+
n
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
SL\scrR n(x, z, y). (2.23)
Now, making use of operators (2.9) and (2.12) in above equation and taking help of the relation
(y - D - 1
x )Dy = - xyD2
x + (x - y)Dx, D2
y = DzzDz,
we are led to differential equation (2.22).
Theorem 2.7. The 3VLeLAP
SL\scrR n(x, z, y) satisfy the integro-differential equations\Biggl(
(y + \alpha 0)Dx - 2D - 1
z +
n\sum
k=1
( - 1)k
\alpha k
k!
Dk+1
x + n
\Biggr)
SL\scrR n(x, z, y) = 0, (2.24)
\Biggl( \Bigl(
y + \alpha 0 - D - 1
x + 2D - 1
y
\Bigr)
Dz +
n\sum
k=1
\alpha k
k!
D - k
y Dk+1
z - (n+ 1)Dy
\Biggr)
SL\scrR n(x, z, y) = 0. (2.25)
Proof. Using expressions (2.10), (2.13) and (2.11), (2.14), respectively, in relation (2.23) give
integro-differential equations (2.24) and (2.25).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
416 S. KHAN, M. RIYASAT, SH. A. WANI
Table 2.1. Result for
HL\scrR n(x, y,D
- 1
z )
S.No. Result Expression
I generating
function
eyt+D - 1
z t2C0(xt) = HLn(x, y,D
- 1
z )
tn
n!
II series
definition HL\scrR n(x, y,D
- 1
z ) = n!
n\sum
k=0
[n/2]\sum
l=0
\scrR n - k - 2l(D
- 1
z )( - x)kyl
(n - k - 2l)! (k!)2 (l!)2
III
recurrence
relation
HL\scrR n+1(x, y,D
- 1
z ) =
\bigl(
D - 1
z + \alpha 0 - D - 1
x
\bigr)
HL\scrR n(x, y,D
- 1
z ) +
+ 2nD - 1
y HL\scrR n - 1(x, y,D
- 1
z ) +
n\sum
k=1
\biggl(
n
k
\biggr)
\alpha k HL\scrR n - k(x, y,D
- 1
z )
IV shift operators
D - 1
z
\$ -
n :=
1
n
DD - 1
z
x\$
-
n := - 1
n
Dx
y\$
-
n :=
1
n
D - 1
D - 1
z
Dy,
D - 1
z
\$+
n := D - 1
z + \alpha 0 - D - 1
x + 2D - 1
y DD - 1
z
+
n\sum
k=1
\alpha k
k!
Dk
D - 1
z
x\$
+
n := y + \alpha 0 - D - 1
x - 2D - 1
z Dx +
n\sum
k=1
( - 1)k
\alpha k
k!
Dk
x
y\$
+
n := D - 1
z + \alpha 0 - D - 1
x + 2D - 1
D - 1
z
+
n\sum
k=1
\alpha k
k!
D - k
D - 1
z
Dk
y
V
differential
equation
\biggl(
xD - 1
z D2
x - (x - D - 1
z )Dx - \alpha 0DD - 1
z
- 2yDy -
-
n\sum
k=1
\alpha k
k!
Dk+1
D - 1
z
+ n
\biggr)
HL\scrR n
\bigl(
x, y,D - 1
z
\bigr)
= 0
VI
integro-
differential
equations
\biggl( \bigl(
D - 1
z + \alpha 0
\bigr)
Dx - 2D - 1
y +
n\sum
k=1
( - 1)k
\alpha k
k!
Dk+1
x + n
\biggr)
HL\scrR n(x, y,D
- 1
z ) = 0\biggl( \bigl(
D - 1
z + \alpha 0 - D - 1
x + 2D - 1
D - 1
z
\bigr)
Dy +
n\sum
k=1
\alpha k
k!
D - k
D - 1
z
Dk+1
y -
- (n+ 1)DD - 1
z
\biggr)
HL\scrR n
\bigl(
x, y,D - 1
z
\bigr)
= 0
VII
partial-
differential
equations
\biggl( \bigl(
D - 1
z + \alpha 0
\bigr)
Dn
yDx - 2Dn - 1
y +
+
n\sum
k=1
( - 1)k
\alpha k
k!
Dn
yD
k+1
x + nDn
y
\biggr)
HL\scrR n(x, y,D
- 1
z ) = 0\biggl( \bigl(
D - 1
z + \alpha 0 - D - 1
x
\bigr)
Dn
D - 1
z
Dy + (n+ 2)Dn - 1
D - 1
z
Dy + 2Dn - 1
D - 1
z
Dy +
+
n\sum
k=1
\alpha k
k!
Dn - k
D - 1
z
Dk+1
y -
- (n+ 1)Dn+1
D - 1
z
\biggr)
HL\scrR n(x, y,D
- 1
z ) = 0
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DIFFERENTIAL AND INTEGRAL EQUATIONS FOR LEGENDRE – LAGUERRE BASED HYBRID . . . 417
Theorem 2.8. The 3VLeLAP
SL\scrR n(x, z, y) satisfy the partial-differential equations\Biggl(
(y + \alpha 0)D
n
zDx - 2Dn - 1
z +
n\sum
k=1
( - 1)k
\alpha k
k!
Dn
zD
k+1
x + nDn
z
\Biggr)
SL\scrR n(x, z, y) = 0, (2.26)
\biggl( \bigl(
y + \alpha 0 - D - 1
x
\bigr)
Dn
yDz + (n+ 2)Dn - 1
y Dz + 2Dn - 1
y Dz+
+
n\sum
k=1
\alpha k
k!
Dn - k
y Dk+1
z - (n+ 1)Dn+1
y
\biggr)
SL\scrR n(x, z, y) = 0. (2.27)
Proof. Differentiating equation (2.24) n times with respect to z and equation (2.25) n times
with respect to y, respectively, yields assertions (2.26) and (2.27).
Remark 2.1. From Remark 1.1 we conclude that the 3VLeLP SLn(x, z, y) reduce to the three-
variable Hermite – Laguerre polynomials (3VHLP) HLn(x, y,D
- 1
z ). In view of this fact, we find that
the 3VLeLAP
SL\scrR n(x, z, y) reduce to the three-variable Hermite – Laguerre – Appell polynomials
(3VHLAP)
HL\scrR n(x, y,D
- 1
z ). We present the results for 3VHLAP in Table 2.1.
We note that by taking \alpha k = 0, k = 0, 1, . . . , n, in the results derived above, we can easily find
the corresponding results for the 3VLeLP SLn(x, z, y) and 3VHLP HLn(x, y,D
- 1
z ). Thus, we omit
them.
In the next section, certain examples are constructed as applications of the results derived above.
3. Applications. We study the analogous results for some members of the 3VLeLAP
SL\scrR n(x, z, y) by considering the following examples.
Example 3.1. Taking \scrR (t) =
\biggl(
t
et - 1
\biggr)
and \scrR n(y) = Bn(y) in generating function (2.3) of
the 3VLeLAP
SL\scrR n(x, z, y), we find the three-variable Legendre – Laguerre – Bernoulli polynomials
(3VLeLBP)
SLBn(x, z, y), which are defined by the generating function\biggl(
t
et - 1
\biggr)
eytC0(xt)C0( - zt2) =
\infty \sum
n=0
SLBn(x, z, y)
tn
n!
.
The other results for the 3VLeLBP
SLBn(x, z, y) can be obtained by making the substitutions
\scrR n(y) = Bn(y), \scrR (t) =
t
et - 1
so that
\scrR \prime
(t)
\scrR (t)
= -
\infty \sum
n=0
Bn+1(1)
n+ 1
tn
n!
\Rightarrow \alpha n = - Bn+1(1)
n+ 1
(n \geq 1), \alpha 0 = - 1
2
, \alpha 1 = - 1
12
in equations (2.5), (2.7), (2.9) – (2.14), (2.22) and (2.24) – (2.27). We present these results in Table 3.1.
The determinant definition of the 3VLeLBP
SLBn(x, z, y) can be obtained by substituting \beta 0 = 1
and \beta i =
1
i+ 1
, i = 1, 2, . . . , n, (for which the determinant definition of the Appell polynomi-
als reduce to the Bernoulli polynomials [5, 6]) in determinant definition (2.6) of the 3VLeLAP
SL\scrR n(x, z, y).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
418 S. KHAN, M. RIYASAT, SH. A. WANI
Table 3.1. Results for
SLBn(x, z, y)
S.No. Result Expression
I series definition
SLBn(x, z, y) = n!
n\sum
k=0
[n/2]\sum
l=0
Bn - k - 2l(y)( - x)kzl
(n - k - 2l)! (k!)2 (l!)2
II recurrence relation
SLBn+1(x, z, y) =
\biggl(
y - 1
2
- D - 1
x
\biggr)
SLBn(x, z, y) +
+ 2nD - 1
z SLBn - 1(x, z, y) -
-
n\sum
k=1
\biggl(
n
k
\biggr)
Bk+1(1)
k + 1 SLBn - k(x, z, y)
III shift operators
y\$
-
n :=
1
n
Dy
x\$
-
n := - 1
n
Dx
z\$
-
n :=
1
n
D - 1
y Dz
y\$
+
n := y - 1
2
- D - 1
x + 2D - 1
z Dy -
n\sum
k=1
Bk+1(1)
(k + 1)!
Dk
y
x\$
+
n := y - 1
2
- D - 1
x - 2D - 1
z Dx -
n\sum
k=1
( - 1)k
Bk+1(1)
(k + 1)!
Dk
x
z\$
+
n := y - 1
2
- D - 1
x + 2D - 1
y -
n\sum
k=1
Bk+1(1)
(k + 1)!
D - k
y Dk
z
IV differential equation
\biggl(
xyD2
x - (x - y)Dx +
1
2
Dy - 2zDz +
+
n\sum
k=1
Bk+1(1)
(k + 1)!
Dk+1
y + n
\biggr)
SLBn(x, z, y) = 0
V integro-differential equations
\biggl( \biggl(
y - 1
2
\biggr)
Dx - 2D - 1
z -
-
n\sum
k=1
( - 1)k
Bk+1(1)
(k + 1)!
Dk+1
x + n
\biggr)
SLBn(x, z, y) = 0\biggl( \biggl(
y - 1
2
- D - 1
x + 2D - 1
y
\biggr)
Dz -
-
n\sum
k=1
Bk+1(1)
(k + 1)!
D - k
y Dk+1
z - (n+ 1)Dy
\biggr)
SLBn(x, y, z) = 0
VI partial-differential equations
\biggl( \biggl(
y - 1
2
\biggr)
Dn
zDx - 2Dn - 1
z -
n\sum
k=1
( - 1)k
Bk+1(1)
(k + 1)!
Dn
zD
k+1
x +
+ nDn
z
\biggr)
SLBn(x, z, y) = 0\biggl( \biggl(
y - 1
2
- D - 1
x
\biggr)
Dn
yDz + (n+ 2)Dn - 1
y Dz + 2Dn - 1
y Dz -
-
n\sum
k=1
Bk+1(1)
(k + 1)!
Dn - k
y Dk+1
z - (n+ 1)Dn+1
y
\biggr)
SLBn(x, z, y) = 0
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DIFFERENTIAL AND INTEGRAL EQUATIONS FOR LEGENDRE – LAGUERRE BASED HYBRID . . . 419
Example 3.2. Taking \scrR (t) =
\biggl(
2
et + 1
\biggr)
and \scrR n(y) = En(y) in generating function (2.3) of
the 3VLeLAP
SL\scrR n(x, z, y), we find the three-variable Legendre – Laguerre – Euler polynomials
(3VLeLEP)
SLEn(x, z, y), which are defined by the generating function
\Bigl( 2
et + 1
\Bigr)
eytC0(xt)C0( - zt2) =
\infty \sum
n=0
SLEn(x, z, y)
tn
n!
.
The other results for the 3VLeLEP
SLEn(x, z, y)) can be obtained by making the substitutions
\scrR n(y) = En(y), \scrR (t) =
2
et + 1
so that
\scrR \prime
(t)
\scrR (t)
=
\infty \sum
n=0
\scrE n
2
tn
n!
\Rightarrow \alpha n =
\scrE n
2
(n \geq 1), \alpha 0 = - 1
2
, \alpha 1 = - 1
2
\Biggl(
\scrE n =
- 1
2n
n\sum
k=0
\biggl(
n
k
\biggr)
En - k
\Biggr)
in equations (2.5), (2.7), (2.9) – (2.14), (2.22) and (2.24) – (2.27). We present these results in Table 3.2.
The determinant definition of the 3VLeLEP
SLEn(x, z, y) can be obtained by substituting \beta 0 = 1
and \beta i =
1
2
, i = 1, 2, . . . , n (for which the determinant definition of the Appell polynomials reduce
to the Euler polynomials [6]) in determinant definition (2.6) of the 3VLeLAP
SL\scrR n(x, z, y).
Example 3.3. Taking \scrR (t) =
\biggl(
2t
et + 1
\biggr)
and \scrR n(y) = Gn(y) in generating function (2.3) of the
3VLeLAP
SL\scrR n(x, z, y), we find the three-variable Legendre – Laguerre – Genocchi polynomials
(3VLeLGP)
SLGn(x, z, y), which are defined by the generating function
\Bigl( 2t
et + 1
\Bigr)
eytC0(xt)C0( - zt2) =
\infty \sum
n=0
SLGn(x, z, y)
tn
n!
.
The other results for the 3VLeLGP
SLGn(x, z, y) can be obtained by making the substitutions
\scrR n(y) = Gn(y), \scrR (t) =
2t
et + 1
so that
\scrR \prime
(t)
\scrR (t)
=
\infty \sum
n=0
Gn
2
tn
n!
\Rightarrow \alpha n =
Gn
2
(n \geq 2), \alpha 0 = 1, \alpha 1 = - 1
in equations (2.5), (2.7), (2.9) – (2.14), (2.22) and (2.24) – (2.27). We present these results in Table 3.3.
The determinant definition of the 3VLeLGP
SLGn(x, z, y) can be obtained by substituting \beta 0 = 1
and \beta i =
1
2(i+ 1)
, i = 1, 2, . . . , n (for which the determinant definition of the Appell polynomials
reduce to the Genocchi polynomials) in determinant definition (2.6) of the 3VLeLAP
SL\scrR n(x, z, y).
In view of Remark 2.1, we note that the corresponding results for the three-variable Her-
mite – Laguerre – Bernoulli, – Euler and – Genocchi polynomials
HLBn(x, z, y), HLEn(x, z, y) and
HLGn(x, z, y), respectively, can be obtained easily. Thus, we omit them.
In the next section, we derive the Volterra integral equations for the 3VLeLAP and for their
relatives.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
420 S. KHAN, M. RIYASAT, SH. A. WANI
Table 3.2. Results for
SLEn(x, z, y)
S.No. Result Expression
I series definition
SLEn(x, z, y) = n!
n\sum
k=0
[n/2]\sum
l=0
En - k - 2l(y)( - x)kzl
(n - k - 2l)! (k!)2 (l!)2
II recurrence relation
SLEn+1(x, z, y) =
\biggl(
y - 1
2
- D - 1
x
\biggr)
SLEn(x, z, y) +
+ 2nD - 1
z SLEn - 1(x, z, y) +
n\sum
k=1
\biggl(
n
k
\biggr)
\scrE k
2 SLEn - k(x, z, y)
III shift operators
y\$
-
n :=
1
n
Dy
x\$
-
n := - 1
n
Dx
z\$
-
n :=
1
n
D - 1
y Dz
y\$
+
n := y - 1
2
- D - 1
x + 2D - 1
z Dy +
n\sum
k=1
\scrE k
2 k!
Dk
y
x\$
+
n := y - 1
2
- D - 1
x - 2D - 1
z Dx +
n\sum
k=1
( - 1)k
\scrE k
2 k!
Dk
x
z\$
+
n := y - 1
2
- D - 1
x + 2D - 1
y +
n\sum
k=1
\scrE k
2 k!
D - k
y Dk
z
IV differential equation
\biggl(
xyD2
x - (x - y)Dx +
1
2
Dy - 2zDz -
n\sum
k=1
\scrE k
2 k!
Dk+1
y +
+ n
\biggr)
SLEn(x, z, y) = 0
V integro-differential equations
\biggl( \biggl(
y - 1
2
\biggr)
Dx - 2D - 1
z +
n\sum
k=1
( - 1)k
\scrE k
2 k!
Dk+1
x +
+ n
\biggr)
SLEn(x, z, y) = 0\biggl( \biggl(
y - 1
2
- D - 1
x + 2D - 1
y
\biggr)
Dz +
n\sum
k=1
\scrE k
2 k!
D - k
y Dk+1
z -
- (n+ 1)Dy
\biggr)
SLEn(x, z, y) = 0
VI partial-differential equations
\biggl( \biggl(
y - 1
2
\biggr)
Dn
zDx - 2Dn - 1
z +
n\sum
k=1
( - 1)k
\scrE k
2 k!
Dn
zD
k+1
x +
+ nDn
z
\biggr)
SLEn(x, z, y) = 0\biggl( \biggl(
y - 1
2
- D - 1
x
\biggr)
Dn
yDz + (n+ 2)Dn - 1
y Dz + 2Dn - 1
y Dz +
+
n\sum
k=1
\scrE k
2 k!
Dn - k
y Dk+1
z - (n+ 1)Dn+1
y
\biggr)
SLEn(x, z, y) = 0
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
DIFFERENTIAL AND INTEGRAL EQUATIONS FOR LEGENDRE – LAGUERRE BASED HYBRID . . . 421
Table 3.3. Results for
SLGn(x, z, y)
S.No. Result Expression
I series definition
SLGn(x, z, y) = n!
n\sum
k=0
[n/2]\sum
l=0
Gn - k - 2l(y)( - x)kzl
(n - k - 2l)! (k!)2 (l!)2
II recurrence relation
SLGn+1(x, z, y) =
\bigl(
y + 1 - D - 1
x
\bigr)
SLGn(x, z, y) +
+ 2nD - 1
z SL\scrG n - 1(x, z, y)+
+
n\sum
k=1
\biggl(
n
k
\biggr)
Gk
2 SLGn - k(x, z, y)
III shift operators
y\$
-
n :=
1
n
Dy
x\$
-
n := - 1
n
Dx
z\$
-
n :=
1
n
D - 1
y Dz
y\$
+
n := y + 1 - D - 1
x + 2D - 1
z Dy +
n\sum
k=1
Gk
2 k!
Dk
y
x\$
+
n := y + 1 - D - 1
x - 2D - 1
z Dx +
n\sum
k=1
( - 1)k
Gk
2 k!
Dk
x
z\$
+
n := y + 1 - D - 1
x + 2D - 1
y +
n\sum
k=1
Gk
2 k!
D - k
y Dk
z
IV differential equation
\biggl(
xyD2
x - (x - y)Dx - Dy - 2zDz -
n\sum
k=1
Gk
2 k!
Dk+1
y +
+n
\biggr)
SLGn(x, z, y) = 0
V integro-differential equations
\biggl(
(y + 1)Dx - 2D - 1
z +
n\sum
k=1
( - 1)k
Gk
2 k!
Dk+1
x + n
\biggr)
SLGn(x, z, y) = 0\biggl( \bigl(
y + 1 - D - 1
x + 2D - 1
y
\bigr)
Dz +
n\sum
k=1
Gk
2 k!
D - k
y Dk+1
z -
- (n+ 1)Dy
\biggr)
SLGn(x, z, y) = 0
VI partial-differential equations
\biggl(
(y + 1)Dn
zDx - 2Dn - 1
z +
n\sum
k=1
( - 1)k
Gk
2 k!
Dn
zD
k+1
x +
+ nDn
z
\biggr)
SLGn(x, z, y) = 0\biggl( \bigl(
y + 1 - D - 1
x
\bigr)
Dn
yDz + (n+ 2)Dn - 1
y Dz + 2Dn - 1
y Dz +
+
n\sum
k=1
Gk
2 k!
Dn - k
y Dk+1
z - (n+ 1)Dn+1
y
\biggr)
SLGn(x, z, y) = 0
4. Volterra integral equations. Integral equations arise in many scientific and engineering
problems, such as diffraction problems scattering in quantum mechanics, conformal mapping and
water waves etc. In order to further stress the importance of integral equations, we derive the integral
equations for the 3VLeLAP by proving the following result.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
422 S. KHAN, M. RIYASAT, SH. A. WANI
Theorem 4.1. The 3VLeLAP satisfy the homogeneous Volterra integral equation
\phi (y) =
1
\alpha 1
\bigl(
xyD2
x - (x - y)Dx - 2zDz + n
\bigr)
\times
\times
\Biggl(
n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)r\scrR k
xr
r!
\times
\times yn - k - r - 1 +
n\sum
r=0
n - r\sum
k=0
\biggl(
n
r
\biggr) \biggl(
n - r
k
\biggr)
( - 1)r\scrR k
xr
r!
yn - k - r
\Biggr)
-
- \alpha 0
\alpha 1
n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)r\scrR k
xr
r!
yn - k - r - 1+
+
y\int
0
\biggl(
1
\alpha 1
\bigl(
xyD2
x - (x - y)Dx - 2zDz + n
\bigr)
(y - \xi ) +
\alpha 0
\alpha 1
\biggr)
\phi (\xi )d\xi . (4.1)
Proof. We consider the following second order differential equation of the 3VLeLAP:\biggl(
D2
y +
\alpha 0
\alpha 1
Dy -
1
\alpha 1
\bigl(
- xyD2
x + (x - y)Dx + 2zDz - n
\bigr) \biggr)
SL\scrR n(x, z, y) = 0. (4.2)
By taking help of equations (1.3), (1.4) and (2.3), we deduce the initial conditions
SL\scrR n(x, 0, y) = L\scrR n(x, y) =
n\sum
r=0
n - r\sum
k=0
\biggl(
n
r
\biggr) \biggl(
n - r
k
\biggr)
( - 1)r\scrR k
xr
r!
yn - k - r, (4.3)
d
dy
\bigl\{
SL\scrR n(x, 0, y)
\bigr\}
=
= n
SL\scrR n - 1(x, 0, y) = n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)r\scrR k
xr
r!
yn - k - r - 1. (4.4)
Now, consider
D2
y
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
= \phi (y),
which on integrating using initial conditions (4.3) and (4.4) gives
Dy
\bigl\{
SL\scrR n(x, z, y)
\bigr\}
=
=
y\int
0
\phi (\xi )d\xi + n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)r\scrR k
xr
r!
yn - k - r - 1, (4.5)
SL\scrR n(x, z, y) =
y\int
0
\phi (\xi )d\xi 2 +
n\sum
r=0
n - r\sum
k=0
\biggl(
n
r
\biggr) \biggl(
n - r
k
\biggr)
( - 1)r\scrR k
xr
r!
yn - k - r. (4.6)
Use of expressions (4.5) and (4.6) in equation (4.2) led to integral equation (4.1).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
DIFFERENTIAL AND INTEGRAL EQUATIONS FOR LEGENDRE – LAGUERRE BASED HYBRID . . . 423
Remark 4.1. By substituting the values of coefficients \alpha 0 = - 1
2
, \alpha 1 = - B2(1)
2
= - 1
12
and
\scrR k = Bk in integral equation (4.1), we find that, for the 3VLeLBP
SLBn(x, z, y), the following
homogeneous Volterra integral equation holds true:
\phi (y) = - 12
\bigl(
xyD2
x - (x - y)Dx - 2zDz + n
\bigr)
\times
\times
\Biggl(
n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)rBk
xr
r!
yn - k - r - 1+
+
n\sum
r=0
n - r\sum
k=0
\biggl(
n
r
\biggr) \biggl(
n - r
k
\biggr)
( - 1)rBk
xr
r!
yn - k - r
\Biggr)
-
- 6n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)rBk
xr
r!
yn - k - r - 1+
+
y\int
0
\Bigl(
- 12
\Bigl(
xyD2
x - (x - y)Dx - 2zDz + n
\Bigr)
(y - \xi ) + 6
\Bigr)
\phi (\xi )d\xi .
Remark 4.2. By substituting the values of coefficients \alpha 0 = - 1
2
, \alpha 1 =
\scrE 1
2
= - 1
2
and \scrR k = Ek
in integral equation (4.1), we find that, for the 3VLeLEP
SLEn(x, z, y), the following homogeneous
Volterra integral equation holds true:
\phi (y) = - 2
\bigl(
xyD2
x - (x - y)Dx - 2zDz + n
\bigr)
\times
\times
\Biggl(
n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)rEk
xr
r!
yn - k - r - 1+
+
n\sum
r=0
n - r\sum
k=0
\biggl(
n
r
\biggr) \biggl(
n - r
k
\biggr)
( - 1)rEk
xr
r!
yn - k - r
\Biggr)
-
- n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)rEk
xr
r!
yn - k - r - 1+
+
y\int
0
\Bigl(
- 2
\bigl(
xyD2
x - (x - y)Dx - 2zDz + n
\bigr)
(y - \xi ) + 1
\Bigr)
\phi (\xi )d\xi .
Remark 4.3. By substituting \alpha 0 = 1, \alpha 1 = - 1, and \scrR k = Gk in integral equation (4.1), we find
that, for the 3VLeLGP
SLGn(x, z, y), the following homogeneous Volterra integral equation holds
true:
\phi (y) = -
\bigl(
xyD2
x - (x - y)Dx - 2zDz + n
\bigr)
\times
\times
\Biggl(
n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)rGk
xr
r!
n - k - r - 1
yn - k - r - 1+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
424 S. KHAN, M. RIYASAT, SH. A. WANI
+
n\sum
r=0
n - r\sum
k=0
\biggl(
n
r
\biggr) \biggl(
n - r
k
\biggr)
( - 1)rGk
xr
r!
yn - k - r
\Biggr)
+
+n
n - 1\sum
r=0
n - r - 1\sum
k=0
\biggl(
n - 1
r
\biggr) \biggl(
n - r - 1
k
\biggr)
( - 1)rGk
xr
r!
yn - k - r - 1+
+
y\int
0
\Bigl(
-
\bigl(
xyD2
x - (x - y)Dx - 2zDz + n
\bigr)
(y - \xi ) - 1
\Bigr)
\phi (\xi )d\xi .
To study the combination of operational representations with the integral transforms and their
applications to the theory of fractional calculus for the 3VLeLAP
SL\scrR n(x, z, y) and for their relatives
will be taken in further investigation.
References
1. L. C. Andrews, Special functions for engineers and applied mathematicians, Macmillan Publ. Comp., New York
(1985).
2. P. Appell, Sur une classe de polynômes, Ann. Sci. École Norm. Supér., 9, № 2, 119 – 144 (1880).
3. P. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques: Polynômes d’ Hermite, Gauthier-
Villars, Paris (1926).
4. S. Araci, M. Acikgoz, H. Jolany, Y. He, Identities involving q-Genocchi numbers and polynomials, Notes Number
Theory and Discrete Math., 20, 64 – 74 (2014).
5. F. A. Costabile, F. Dell’Accio, M. I. Gualtieri, A new approach to Bernoulli polynomials, Rend. Mat. Appl., 26, № 1,
1 – 12 (2006).
6. F. A. Costabile, E. Longo, A determinantal approach to Appell polynomials, J. Comput. and Appl. Math., 234, № 5,
1528 – 1542 (2010).
7. G. Dattoli, Hermite – Bessel and Laguerre – Bessel functions: a by-product of the monomiality principle, Adv. Spec.
Funct. and Appl. (Melfi, 1999), Proc. Melfi Sch. Adv. Top. Math. Phys., 1, 147 – 164 (2000).
8. G. Dattoli, C. Cesarano, D. Sacchetti, A note on the monomiality principle and generalized polynomials, J. Math.
Anal. and Appl., 227, 98 – 111 (1997).
9. G. Dattoli, P. E. Ricci, A note on Legendre polynomials, Int. J. Nonlinear Sci. and Numer. Simul., 2, 365 – 370 (2001).
10. G. Dattoli, A. Torre, Operational methods and two variable Laguerre polynomials, Atti Accad. Sci. Torino Cl. Sci.
Fis. Mat. Natur., 132, 1 – 7 (1998).
11. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, vol. III, McGraw-Hill Book
Comp., New York etc. (1955).
12. M. X. He, P. E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. and Appl.
Math., 139, 231 – 237 (2002).
13. L. Infeld, T. E. Hull, The factorization method, Rev. Mod. Phys., 23, 21 – 68 (1951).
14. J. Sandor, B. Crstici, Handbook of number theory, vol. II, Kluwer Acad. Publ., Dordrecht etc. (2004).
Received 22.02.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
|
| id | umjimathkievua-article-894 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:58Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2b/303c53b586caaacb59725f6a0007902b.pdf |
| spelling | umjimathkievua-article-8942025-03-31T08:48:21Z Differential and integral equations for Legendre – Laguerre based hybrid polynomials Differential and integral equations for Legendre – Laguerre based hybrid polynomials Differential and integral equations for Legendre – Laguerre based hybrid polynomials Khan, S. Riyasat, M. Wani , Sh. A. Khan, S. Riyasat, M. Wani , Sh. A. M. Sh. A. Legendre-Laguerre polynomials Appell polynomials Legendre-Laguerre-Appell polynomials Recurrence relations Differential equations Integral equations Legendre-Laguerre polynomials Appell polynomials Legendre-Laguerre-Appell polynomials Recurrence relations Differential equations Integral equations UDC 517.9 In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial differential equations are established. The analogous results for the three-variable Hermite – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, –E uler and – Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by deriving homogeneous Volterra integral equations for these polynomials and for their relatives. УДК 517.9 Диференцiйнi та iнтегральнi рiвняння для гiбридних полiномiв на базi полiномiв Лежандра– Лагерра Розглянуто гібридну сім'ю поліномів Лежандра – Лагерра – Аппеля та встановлено їхні властивості, які включають розклади рядів, форми детермінантів, рекурентні співвідношення, оператори зсуву, за якими йдуть диференціальні та інтегро-диференціальні рівняння, а також диференціальні рівняння з частинними похідними. Подібні результати отримано для поліномів Ерміта – Лагерра – Аппеля з трьома змінними. У термінах поліномів Лежандра – Лагерра – Бернуллі, – Ейлера та – Дженоккі побудовано деякі приклади, щоб показати застосування основних результатів. Далі, для цих та пов'язаних з ними поліномів отримано однорідне інтегральне рівняння Вольтерра. Institute of Mathematics, NAS of Ukraine 2021-03-19 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/894 10.37863/umzh.v73i3.894 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 3 (2021); 408 - 424 Український математичний журнал; Том 73 № 3 (2021); 408 - 424 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/894/8984 |
| spellingShingle | Khan, S. Riyasat, M. Wani , Sh. A. Khan, S. Riyasat, M. Wani , Sh. A. M. Sh. A. Differential and integral equations for Legendre – Laguerre based hybrid polynomials |
| title | Differential and integral equations for Legendre – Laguerre based hybrid polynomials |
| title_alt | Differential and integral equations for Legendre – Laguerre based hybrid polynomials Differential and integral equations for Legendre – Laguerre based hybrid polynomials |
| title_full | Differential and integral equations for Legendre – Laguerre based hybrid polynomials |
| title_fullStr | Differential and integral equations for Legendre – Laguerre based hybrid polynomials |
| title_full_unstemmed | Differential and integral equations for Legendre – Laguerre based hybrid polynomials |
| title_short | Differential and integral equations for Legendre – Laguerre based hybrid polynomials |
| title_sort | differential and integral equations for legendre – laguerre based hybrid polynomials |
| topic_facet | Legendre-Laguerre polynomials Appell polynomials Legendre-Laguerre-Appell polynomials Recurrence relations Differential equations Integral equations Legendre-Laguerre polynomials Appell polynomials Legendre-Laguerre-Appell polynomials Recurrence relations Differential equations Integral equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/894 |
| work_keys_str_mv | AT khans differentialandintegralequationsforlegendrelaguerrebasedhybridpolynomials AT riyasatm differentialandintegralequationsforlegendrelaguerrebasedhybridpolynomials AT wanisha differentialandintegralequationsforlegendrelaguerrebasedhybridpolynomials AT khans differentialandintegralequationsforlegendrelaguerrebasedhybridpolynomials AT riyasatm differentialandintegralequationsforlegendrelaguerrebasedhybridpolynomials AT wanisha differentialandintegralequationsforlegendrelaguerrebasedhybridpolynomials AT m differentialandintegralequationsforlegendrelaguerrebasedhybridpolynomials AT sha differentialandintegralequationsforlegendrelaguerrebasedhybridpolynomials |