Some Generating Functions of Modified Gegenbauer
Polynomials by Lie Algebraic Method
K.P. Samanta1,B. C. Chandra2,∗,C.S.Bera3
1Department of Mathematics, Bengal Engineering and Science University, Shibpur P.O. Botanic Garden, Howrah - 711 103, India 2Sree Vidyaniketan; 9A, J. M. Avenue, Kolkata - 700006, W.B, India
3Department of Mathematics, Bagnan College, Bagnan, Howrah-711303, W.B., India
∗Corresponding Author: biplab [email protected]
Copyright c⃝2014 Horizon Research Publishing All rights reserved.
Abstract In this paper we have obtained some novel generating functions of Cλ+n
n (x)-a modification of Gegenbauer
polynomials,Cλ
n(x)by utilizing L. Wesiner’s group-theoretic method. By giving suitable interpretations to both the index
(n)and the parameter(λ)of the polynomial under consideration, we obtain, in section 2, a set of infinitesimal operators known as raising and the lowering operators which generates a four dimensional Lie algebra. Finally, in Section 3, a novel generating function of the modified Gegenbauer polynomials which in turn yields a number of new and known results on generating functions
Keywords Gegenbauer Polynomial, Generating Functions
AMS-2010 Classification Code:33C45;33C47
1
Introduction
The Gegenbauer polynomials,Cnλ(x)defined by [ 6 ]:
Cnλ(x) =
[n2]
∑
p=0
(−1)p (λ)n−p(2x)
n−2p
p!(n−2p)!
is a solution of the following ordinary differential equation:
[(1−x2) d
2
dx2 −(2λ+ 1)x
d
dx+n(2λ+n)]u= 0. (1.1)
In this paper we consider the modified Gegenbauer polynomials, Cλ+n
n (x)which satisfy the following ordinary differential equation:
[(1−x2) d
2
dx2 −(2λ+ 2n+ 1)x
d
dx+n(2λ+ 3n)]u= 0. (1.2)
Several generating functions for Gegenbauer polynomials have been derived by different method namely classical, theory of Lie groups etc. Here we are mainly interested in group theoretic method as intro-duced by L. Weisner[5]. With the help of this method McBride[4], Chongdar[2], Ghosh[3], Das and Chatterjea[7], Sultan[8], Majumder[9], Viswanathan[1] and others have derived a large number of gen-erating functions for Gegenbauer polynomials.
2
Group-Theoretic Discussion and Lie-Algebra
Replacing dxd by ∂x∂, n byy∂y∂ , λbyz∂z∂ andubyv(x, y, z)in (1.2), we get the following partial differential equation:
(1−x2)∂
2v
∂x2 −2xy
∂2v
∂x∂y −2xz ∂2v
∂x∂z + 2yz ∂2v
∂y∂z + 3y
2∂2v
∂y2 −x
∂v ∂x + 3y
∂v
∂y = 0 (2.1)
Thus we see thatv(x, y, z) = Cλ+n
n (x)ynzλ is a solution of (2.1), sinceCnλ+n(x)is a solution of (1.2). Now by using the following differential recurrence relations:
(x d
dx −n)C
λ+n
n (x) = 2(λ+n)C λ+n+1
n−2 (x) (2.2)
and
[
x(1−x2) d
dx −(2λ+ 3n)x
2+ (n+ 1)]Cλ+n n (x) =
(n+ 1)(n+ 2) 2(1−λ−n) C
λ+n−1
n+2 (x) (2.3)
Now, we define the following set of infinitesimal partial differential operators Ai(i = 1,2,3,4)as follows:
A1 =y
∂
∂y; A2 =z ∂
∂z; A3 = xz3
y2
∂ ∂x −
z3
y ∂ ∂y;
and
A4 =x(1−x2)
y2 z3
∂
∂x + (1−3x
2)y 3
z3
∂ ∂y −
2x2 y2 z2
∂ ∂z +
y2 z3.
such that
A1
(
Cnλ+n(x)ynzλ
)
=nCnλ+n(x)ynzλ, A2
(
Cnλ+n(x)ynzλ
)
=λCnλ+n(x)ynzλ, A3
(
Cnλ+n(x)ynzλ
)
= 2(n+λ)Cnλ−+2n+1(x)yn−2zλ+3, A4
(
Cnλ+n(x)ynzλ
)
= (n+ 1)(n+ 2) 2(1−λ−n) C
λ+n−1
n+2 (x)y
n+2zλ−3.
We now proceed to find the commutator relations. Using the notation:
[A , B]u= (AB−BA)u,
we have the following commutator relations:
[A1 , A2] = 0; [A1 , A3] = −2A3; [A1 , A4] = 2A4; [A2 , A3] = 3A3; [A2 , A4] = −3A4;
and [A3 , A4] =−2(2A1+ 1).
From the above commutator relations, we state the following theorem:
Theorem: The set of operators{1, Ai(i= 1,2,3,4)}, where 1 stands for the identity operator, generates a Lie-AlgebraL.
It can be shown that the partial differential operator L given by
L= (1−x2)∂2
∂x2 −2xy ∂ 2
∂x∂y −2xz ∂2
∂x∂z + 2yz ∂2 ∂y∂z + 3y
2 ∂2
∂y2 −x∂x∂ + 3y∂y∂
can be expressed as follows:
x2 L=A3A4+A21+ 3A1 + 2. (2.4)
From the above commutator relations, it can be easily verified that L commutes withAi(i= 1,2,3,4):
The extended form of the group generated by each of the operatorsAi(i= 1,2,3,4)are as follows:
ea1A1f(x, y, z) = f(x, ea1y, z), (2.6)
ea2A2f(x, y, z) = f(x, y, ea2z), (2.7)
ea3A3f(x, y, z) = f
(
x
(1− 2a3z3
y2 ) 1 2
, y(1−2a3z
3
y2 ) 1 2, z
)
, (2.8)
ea4A4f(x, y, z) = {1−2a 4
y2
z3}
−1 2
×f
(
x
{1−2a4(1−x2)y 2
z3} 1 2
, y(1−2a4
y2
z3)
{1−2a4(1−x2)y 2
z3} 3 2
, z(1−2a4
y2
z3)
{1−2a4(1−x2)y 2
z3}
)
.
(2.9)
where allai(i= 1,2,3,4)are arbitrary constant andf(x, y, z)is arbitrary function. From the above we notice that
ea4A4ea3A3ea2A2ea1A1f(x, y, z)
={1−2a4
y2
z3}
−1
2f(ζ, η, θ), (2.10)
where
ζ = x
{1−2a4(1−x2)y 2
z3} 1
2{1−2a3(1−2a4y 2
z3)
z3
y2} 1 2
,
η=ea1 y(1−2a4
y2
z3)
{1−2a4(1−x2)y 2
z3} 3 2
{1−2a3(1−2a4
y2
z3)
z3
y2} 1 2,
θ=ea2 z(1−2a4
y2
z3)
{1−2a4(1−x2)y 2
z3}
.
3
Generating Functions
From (2.1),v(x , y, z) = Cλ+n
n (x)ynzλ is a solution of the system:
Lv = 0 Lv = 0 Lv = 0
(A1−n)v = 0 (A2−λ)v = 0 (A1 +A2−n−λ)v = 0.
Also from (2.5), we easily get,
S x2L
[
Cnλ+n(x)ynzλ
]
=x2L S
[
Cnλ+n(x)ynzλ
]
= 0;
where S =ea4A4ea3A3ea2A2ea1A1.
So the transformation S
[
Cnλ+n(x)ynzλ
]
is annihilated byx2L.
Now puttinga1 =a2 = 0and replacingf(x, y, z)byCnλ+n(x)ynzλ in (2.10), we get
ea4A4ea3A3
[
Cnλ+n(x)ynzλ
]
={1−2a4
y2
z3}
n+λ−1
2{1−2a3(1−2a4y 2
z3)
z3
y2}
n
2
× {1−2a4(1−x2)
y2
z3}
−3n
2 −λ Cλ+n
n (ζ)y nzλ,
where
ζ = x
{1−2a4(1−x2)y 2
z3} 1
2{1−2a3(1−2a4y 2
z3)
z3
y2} 1 2
.
On the other hand, we get
ea4A4ea3A3
[
Cnλ+n(x)ynzλ
]
=
∞
∑
k=0
n+2k
2
∑
p=0
(2a3)p (
a4
2 )
k (n+λ+p−1 p
)
× (n−2p+ 1)2k
k!(1−λ−p−n)k
Cnλ−+2np++2p−kk(x)yn−2p+2kzλ+3p−3k. (3.2)
Equating (3.1) and (3.2), we get
{1−2a4
y2 z3}
n+λ−12{1−2a
3(1−2a4
y2 z3)
z3 y2}
n
2 {1−2a
4(1−x2)
y2 z3}
−3n
2 −λ
× Cnλ+n
(
x
{1−2a4(1−x2)y 2
z3} 1
2{1−2a3(1−2a4y 2
z3)
z3
y2} 1 2
)
=
∞
∑
k=0
n+2k
2
∑
p=0
(2a3)p(
a4
2)
k(n+λ+p−1 p
) (n−2p+ 1)
2k
k!(1−λ−p−n)k
× Cnλ−+2np++2p−kk (x)y−2p+2kz3p−3k (3.3) and this may be regarded as a new generating relation which in turn yields a good number of particular generating relation (new/ known) by attributing different values ofai(i= 1,2,3,4).
Before discussing particular cases of the result (3.3) we would like to point it out that the operators
A3, A4 being non-commutative, as seen from the commutator relation [A3 , A4] = −2(2A1 + 1),the
relation (3.3) will change if we change the order ofA3, A4 inea4A4ea3A3,which is given in section 5.
Now we shall consider the following particular cases:
4
particular cases
Case-1: Puttinga4 = 0and replacing2a3z 3
y2 by t in (3.3), we get
(1−t)n2 Cλ+n
n
(
x
(1−t)12
)
=
n
2
∑
p=0
(n+λ+p−1
p
)
Cnλ−+2np+p(x)tp. (4.1)
Case-2:Puttinga3 = 0and replacing2a4y 2
z3 by t in (3.3), we get
(1−t)n+λ−21 (1−t+x2t)− 3n
2 −λCλ+n
n
(
x
(1−t+x2t)12
)
=
∞
∑
k=0
(n+ 1)2k
k!22k(1−λ−n) k
Sub-Case:If we putn = 0,then
(1−t)λ−12 (1−t+x2t)−λ =
∞
∑
k=0
(1−2p)2k
k! 22k(1−λ−p) k
C2λk−k(x)tk. (4.3)
Case-3:Putting2a3z 3
y2 =tand2a4y 2
z3 =win (3.3), we get
(1−w)n+λ−12 (1−t+wt)
n
2{1−(1−x2)w}− 3n
2 −λ
× Cnλ+n
(
x
{1−(1−x2)w}12(1−t+wt) 1 2
)
=
∞
∑
k=0
n+2k
2
∑
p=0
2−2k
(n+λ+p−1
p
) (n−2p+ 1)
2k
k!(1−λ−p−n)k
Cnλ−+2np++2p−kk(x)tpwk. (4.4)
5
Variants of the result (3.3)
By interchanging the order of operatorsA3andA4inea4A4ea3A3ea2A2ea1A1,we get
ea3A3ea4A4ea2A2ea1A1f(x, y, z) = {1−2a
4(1−2a3
z3
y2)
y2
z3}
−1
2f(ζ, η, θ). (5.1)
where
ζ = x
{1−2a3z 3
y2} 1
2{1 + 4a3a4−2a4(1−x2)y 2
z3} 1 2
,
η=ea1y(1−2a3
z3
y2) 1
2{1−2a4(1−2a3z 3
y2)
y2
z3}
{1 + 4a3a4−2a4(1−x2)y 2
z3} 3 2
,
θ=ea2 z{1−2a4(1−2a3
z3
y2)
y2
z3}
{1 + 4a3a4−2a4(1−x2)y 2
z3}
.
Now puttinga1 =a2 = 0and replacingf(x, y, z)byCnλ+n(x)ynzλ in (5.1), we get
ea3A3ea4A4
[
Cnλ+n(x)ynzλ
]
={1−2a4(1−2a3
z3 y2)
y2 z3}
n+λ−1
2(1−2a3z 3
y2)
n
2
× {1 + 4a3a4−2a4(1−x2)
y2
z3} 3n
2+λ Cλ+n
n (ζ)y nzλ,
(5.2)
where
ζ = x
{1−2a3z 3
y2} 1
2{1 + 4a3a4−2a4(1−x2)y 2
z3} 1 2
.
On the other hand, we get
ea3A3ea4A4
[
Cnλ+n(x)ynzλ
]
=
∞
∑
k=0
n+2k
2
∑
p=0
(a3)p
p!
(a4)k
k!
(n−k+λ)p
2k−p
(n+ 1)2k
(1−λ−n)k
Equating (5.2) and (5.3), we get
{1−2a4(1−2a3
z3 y2)
y2 z3}
n+λ−12(1−2a
3
z3 y2)
n
2{1 + 4a
3a4−2a4(1−x2)
y2 z3}
3n
2 +λ
× Cnλ+n
(
x
{1−2a3z 3
y2} 1
2{1 + 4a3a4−2a4(1−x2)y 2
z3} 1 2
)
=
∞
∑
k=0
n+2k
2
∑
p=0
(a3)p
p!
(a4)k
k!
(n−k+λ)p
2k−p
(n+ 1)2k
(1−λ−n)k
× Cnλ+2+n−kk−2+pp(x)y2k−2pz−3k+3p. (5.4)
6
Application
As an application of our result, we now proceed to derive some novel results on bilateral generating relations of the polynomials under consideration.The main result is stated in the following theorems:
Theorem 1:
If there exists a unilateral generating relation of the form:
G(x, w) =
∞
∑
n=0
anCnλ+n(x)wn (6.1)
then
(1−2w)λ−12
{1−2w+ 2wx2}λ G
(
x
{1−2w+ 2wx2}12,
wt(1−2w)
{1−2w+ 2wx2}32
)
=
∞
∑
n=0
wnσn(x, t), (6.2)
where
σn(x, t) = n
∑
p=0
ap
2n−p
(p+ 1)2n−2p
(n−p)!(1−λ−p)n−p
C2λn−−np+2p(x)tp.
Proof:
R.H.S =
∞
∑
n=0
wnσn(x, t)
=
∞
∑
n=0
wn
n
∑
p=0
ap
2n−p
(p+ 1)2n−2p
(n−p)!(1−λ−p)n−p
C2λn−−np+2p(x)tp
=
∞
∑
p=0
ap(wt)p ∞
∑
n=0
(p+ 1)2n
n!22n(1−λ−p) n
C2λ−nn+p+p(x) (2w)n
= (1−2w)
λ−1 2
{1−2w+ 2wx2}λ ∞
∑
p=0
apCpλ+p
( x
{1−2w+ 2wx2}12
){ wt(1−2w)
{1−2w+ 2wx2}32
}p
;
[using(4.2)]
= (1−2w)
λ−1 2
{1−2w+ 2wx2}λ G
( x
{1−2w+ 2wx2}12,
wt(1−2w)
{1−2w+ 2wx2}32
)
; [using(6.1)]
=L.H.S.
(6.2). So one can get a large number of bilateral generating functions by attributing different suitable values toanin (6.1).
REFERENCES
[1] Viswanathan, B., Generating functions for ultraspherical functions, Canad. J. Math., 20(1968), 120-134.
[2] Chongdar A.K., Group-theoretic origins of certain generating functions of Gegenbauer polynomi-als, Jour. Orissa math. Soc. 2(1983),.71-85.
[3] Ghosh B., Some generating functions of modified Gegenbauer polynomials, Proc. Indian Acad. Sci.(Math. Sci.), 2(96)(1987), 119-124.
[4] McBride,E.B., Obtaining generating functions, Springer verlag, New York, 1971.
[5] Weisner L., Group-theoretic origin of certain generating function,Pacific Jour. Math., 5(1955), 1033-1039.
[6] Rainville E.D., Special functions, Macmillan Company, New York, 1960.
[7] S.Das and Chatterjea S.K., On a partial differential operator for Gegenbauer polynomials, Bull. Cal. Math. Soc., 76(1984),351-361.
[8] Sultan T.I.,Weisner’smethodic survey for Gegenbauer polynomials, Jour.Pure Math.,4(1984),79-101.
