Proyecciones Journal of Mathematics
Vol. 29, No 3, pp. 201-207, December 2010.
Universidad Católica del Norte
Antofagasta - Chile
POLYNOMIAL SETS GENERATED BY
et φ(xt)ψ(yt)
MUMTAZ AHMAD KHAN
ALIGARH MUSLIM UNIVERSITY, INDIA
and
BAHMAN ALIDAD
GOLESTAN UNIVERSITY, IRAN
Received : August 2009. Accepted : October 2010
Abstract
The present paper deals with two variables polynomial sets generated by functions of the form et φ(xt)ψ(yt). Its special case analogous
to Laguerre polynomials have been discussed.
202
Mumtaz Ahmad Khan and Bahman Alidad
1. INTRODUCTION
(α)
Laguerre polynomials Ln (x) possess the generating relation (see Rainville
[8], pp. 130)
et 0 F1 (−; 1 + α; −xt) =
(α)
∞
X
Ln (x)tn
n=0
(1 + α)n
(1.1)
By studying generating relation
et ψ(xt) =
∞
X
σn (x) tn
(1.2)
n=0
(α)
One arrives at properties held by Ln (x) (see Rainville [8], p. 132-133)
Motivated by (1.2) an attempt has been made to study two variable
polynomials similar to one given in (1.2) and generated by functions of the
form et φ(xt)ψ(yt).
2. TWO VARIABLE POLYNOMIAL SETS ANALOGOUS
TO (1.2)
Let us consider the generating relation of the type
et φ(xt) ψ(yt) =
∞
X
σn (x, y) tn
(2.1)
n=0
Let
F = et φ(xt) ψ(yt)
Then
(2.2)
∂F
= t et φ0 ψ
∂x
(2.3)
∂F
= t et φ ψ 0
∂y
(2.4)
∂F
= et φ ψ + x et φ0 ψ + y et φ ψ 0
(2.5)
∂t
Eliminating φ, φ0 , ψ and ψ 0 from the four equations (2.2), (2.3), (2.4)
and (2.5), we obtain
µ
¶
∂
∂F
∂
x
F −t
+
= −t F
∂x ∂y
∂t
(2.6)
Polynomial sets generated by et φ(xt)ψ(yt)
Since
∞
X
F = et φ(xt) ψ(yt) =
203
σn (x, y) tn
n=0
Equation (2.6) yields
∞ µ
X
x
n=0
¶
∞
∞
X
X
∂
∂
+y
σn (x, y)tn −
n σn (x, y) tn = −
σn (x, y)tn+1
∂x
∂y
n=0
n=0
=−
∞
X
σn−1 (x, y)tn
n=1
from which the next theorem follows.
Theorem 1 :
From
et φ(xt) ψ(yt) =
∞
X
σn (x, y) tn
n=0
it follows that
∂
∂x σ0 (x, y)
µ
=
∂
∂y σ0 (x, y)
¶
= 0 and for n ≥ 1,
∂
∂
+y
x
σn (x, y) − n σn (x, y) = −σn−1 (x, y)
∂x
∂y
(2.7)
Next, let us assume that the functions φ and ψ in (2.1) have the formal
power - series expansion
φ(u) =
∞
X
γn un
(2.8)
δn v n
(2.9)
n=0
and
ψ(v) =
∞
X
n=0
Then (2.1) yields
∞
X
n=0
σn (x, y) tn =
̰
!Ã ∞
X tn
X
n!
n=0
n=0
γn xn tn
!Ã ∞
X
n=0
δn yn tn
!
204
Mumtaz Ahmad Khan and Bahman Alidad
=
∞ X
n n−r
X
X γr δs xr y s tn
(n − r − s)!
n=0 r=0 s=0
so that
σn (x, y) =
n n−r
X
X γr δs xr y s
r=0 s=0
Now consider the sum
∞
X
(c)n σn (x, y) tn =
n=0
∞ X
n n−r
X
X (c)n γr δs xr y s tn
∞ X
∞ X
∞
X
(c)n+r+s γr δs xr ys tn+r+s
n!
n=0 r=0 s=0
=
(n − r − s)!
n=0 r=0 s=0
=
(2.10)
(n − r − s)!
∞ X
∞
X
r
s
(c)r+s γr δs (xt) (yt)
r=0 s=0
∞
X
(c + r + s)n tn
n=0
=
n!
∞
∞ X
X
(c)r+s γr δs (xt)r (yt)s
r=0 s=0
(2.10)
(1 + t)c+r+s
We thus arrive at the following theorem:
Theorem 2 :
From
∞
∞
∞
P
P
P
σn (x, y) tn , φ(u) =
γn un , ψ(v) =
δn v n it
et φ(xt) ψ(yt) =
n=0
n=0
n=0
follows that for arbitrary c
−c
(1 − t)
F
µ
yt
xt
,
1−t 1−t
in which
F (u, v) =
∞ X
∞
X
¶
=
∞
X
(c)n σn (x, y) tn
(2.11)
n=0
(c)n+k γn δk un v k
(2.12)
n=0 k=0
The role of Theorem 2 is as follows: If a set σn (x, y) has a generating
function of the form et φ(xt) ψ(yt), Theorem 2 yields for σn (x, y) another
generating function of the form exhibited in (2.11). For instance, if φ(u) and
ψ(v) are specified p Fq , the theorem gives for σn (x, y) a class (c arbitrary)
of generating functions involving two variables hypergeometric functions.
Polynomial sets generated by et φ(xt)ψ(yt)
205
Let us now apply Theorems 1 and 2 to Laguerre polynomials of two
(α,β)
variables Ln (x, y) due to S. F. Ragab [7] defined by
n
(−y)r Lαn−r (x)
Γ(n + α + 1)Γ(n + β + 1) X
=
n!
r! Γ(α + n − r + 1)Γ(β + r + 1)
r=0
(2.13)
(α)
Where Ln (x) is the well - known Laguerre polynomials of one variable.
The definition (2.13) is equivalent to the following explicit representa(α,β)
tion of Ln (x, y), given by Ragab:
(x, y)
L(α,β)
n
n n−r
X
(−n)r+s xs y r
(α + 1)n (β + 1)n X
(n!)2
(α + 1)s (β + 1)r r! s!
r=0 s=0
L(α,β)
(x, y) =
n
(2.14)
Later, the same year Chatterjea [1] gave the following generating func(α,β)
tion for Ln (x, y):
et 0 F1 (−; α + 1; −xt) 0 F1 (−; β + 1; −yt) =
We use Theorem 1 to conclude that
for n ≥ 1.
µ
(α,β)
∞
X
n! Ln (x, y) tn
n=0
(α + 1)n (β + 1)n
(α,β)
L0 (x, y)
(2.15)
is a constant and, and
¶
∂
∂
(α + n)(β + n) (α,β)
+y
Ln−1 (x, y)
x
Ln(α,β) (x, y)−n Ln(α,β) (x, y) = −
∂x
∂y
n
(2.16)
(α,β)
In applying theorem 2 to Laguerre polynomials of two variables Ln (x, y),
(α,β)
note that σn (x, y) =
n! Ln
(x,y)
(α+1)n (β+1)n
φ(u) =
ψ(v) =
Then γn =
0 F1 (−;
and that
1 + α; −u) =
0 F1 (−; 1 + β; −v) =
(−1)n
n! (1+α)n ,
δn =
(−1)n
n! (1+β)n
and
∞
X
(−1)n un
n=0
n! (1 + α)n
∞
X
(−1)n v n
n=0
n! (1 + β)n
206
Mumtaz Ahmad Khan and Bahman Alidad
F (u, v) =
∞ X
∞
X
(c)n+k γn δk un vn =
n=0 k=0
∞ X
∞
X
(c)n+k (−1)n+k un v k
n=0 k=0
n! k! (1 + α)n (1 + β)k
= ψ2 [c; 1 + α, 1 + β; −u, −v]
Therefore Theorem 2, yields
∙
¸
(α,β)
∞
X
yt
n! (c)n Ln (x, y)tn
xt
,−
=
(1 − t) ψ2 c; 1 + α, 1 + β; −
1−t 1−t
(1 + α)n (1 + β)n
n=0
(2.17)
(α,β)
a class of generating relations for Ln (x, y) due to M.A. Khan and
A.K. Shukla [2].
−c
Concluding Remark
Application of the theorems given in this paper have already been shown
in case of Laguerre polynomials of two variables. Thus, this class of product
may be used whenever Laguerre polynomials of two variables occur.
References
[1] CHATTERJEA, S. K. A note on Laguerre polynomials of two variables, Bull. Cal. Math. Soc., Vol. 83, pp. 263, (1991).
[2] KHAN, M. A. AND SHUKLA, A. K. On Laguerre polynomials of
several variables, Bull. Cal. Math. Soc., Vol. 89, pp. 155-164, (1997).
[3] KHAN, M. A. AND SHUKLA, A. K. A note on Laguerre polynomials
of m variables, Bulletin of the Greek Mathematical Society., Vol. 40,
pp. 113-117, (1998).
[4] KHAN, M. A. AND ABUKHAMMASH, G. S. On Hermite polynomials
of two variables suggested by S. F. Ragab’s Laguerre polynomials of two
variables, Bull. Cal. Math. Soc., Vol. 90, pp. 61-76, (1998).
Polynomial sets generated by et φ(xt)ψ(yt)
207
[5] KHAN, M. A. AND AHMAD, K. On a general class of polynomials
(α,β;γ,δ)
(α,β)
(x, y) of two variables suggested by the polynomials Ln (x, y)
Ln
(α,β)
of Ragab and Ln (x) of Prabhakar and Rekha, Pro Mathematica, Vol.
xix/Nos. 37-38, pp. 21-38, (2005).
[6] KHAN, M. A. AND ALIDAD, B. Polynomial sets generated by functions of the form G(2xt − t2 )K(2yt − t2 ) , Communicated for publication.
(α,β)
[7] RAGAB, S. F. On Laguerse polynomials of two variables Ln
Bull. Cal. Math. Soc., Vol. 83, pp. 253, (1991).
(x, y),
[8] RAINVILLE, E. D. Special Functions, Macmillan, New York;
Reprinted by Chelsea Publ. Co., Bronx., New York, (1971).
[9] SRIVASTAVA, H. M. AND MANOCHA, H. L. A Treatise on Generating Functions, Ellis Horwood Limited Publishers, Chichester, Halsted
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MUMTAZ AHMAD KHAN
Department of Applied Mathematics,
Faculty of Engineering and Technology,
Aligarh Muslim University,
Aligarh - 202002, U. P.,
India
e-mail : mumtaz ahmad khan 2008@yahoo.com
AND
BAHMAN ALIDAD
Department of Mathematics
Faculty of Science,
Golestan University,
Gorgan,
Iran
e-mail: bahman alidad@yahoo.com