Available at
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Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 9, Issue 2 (December 2014), pp. 449 – 466
Certain Results for the
Laguerre-Gould Hopper Polynomials
Subuhi Khan1 and Ahmed Ali Al-Gonah2
1
Department of Mathematics
Aligarh Muslim University, Aligarh-202002, India
subuhi2006@gmail.com
&
2
Department of Mathematics
Aden University, Aden, Yemen
gonah1977@yahoo.com
Received: February 12, 2014; Accepted: August 6, 2014
Abstract
In this paper, we derive generating functions for the Laguerre-Gould Hopper polynomials in
terms of the generalized Lauricella function by using series rearrangement techniques. Further,
we derive the summation formulae for that polynomials by using different analytical means on
its generating function or by using certain operational techniques. Also, generating functions
and summation formulae for the polynomials related to Laguerre-Gould Hopper polynomials are
obtained as applications of main results.
Keywords: Laguerre-Gould Hopper polynomials; Lauricella function; Generating function; Summation formulae
MSC 2010 No.: 33B10, 33C45, 33C65
1.
Introduction and preliminaries
Very recently (Khan and Al-Gonah, 2012) introduced the Laguerre-Gould Hopper polynomials
(m,s)
(LGHP in the following) L Hn (x, y, z) and studied their properties. These polynomials are
449
450
S. Khan and A. A. Al-Gonah
defined as:
n
(m,s)
(x, y, z)
L Hn
= n!
[s]
X
z k m Ln−sk (x, y)
k!(n − sk)!
k=0
(1.1)
,
where the symbol ns denotes the greatest integer less than or equal ns and m Ln (x, y) are the
2-variable generalized Laguerre polynomials (2VgLP) defined by the series definition (Dattoli
et al., 1999):
n
]
[m
X
xr y n−mr
(1.2)
m Ln (x, y) = n!
(r!)2(n − mr)!
r=0
and by the operational definition
m Ln (x, y)
= exp
Dx−1
∂m
∂y m
n o
yn ,
(1.3)
∂
and is defined in such a way
where Dx−1 denotes the inverse of the derivative operator Dx := ∂x
that
Z x
1
−n
Dx f(x) =
(x − ξ)n−1 f(ξ)dξ,
(1.4)
(n − 1)! 0
so that for f(x) = 1, we have
(m,s)
The polynomials L Hn
xn
Dx−n 1 =
.
n!
(1.5)
(x, y, z) are defined by the following generating function:
exp(yt + zts )C0(−xtm ) =
∞
X
(m,s)
(x, y, z)
L Hn
n=0
tn
,
n!
(1.6)
where C0 (x) denotes the 0th order Tricomi function. The nth order Tricomi functions Cn (x) are
defined as (Andrews, 1985):
∞
X
(−1)r xr
.
(1.7)
Cn (x) =
r!(n + r)!
r=0
Also, we recall that the generalized Bessel function or the Bessel-Wright function is defined by
(Srivastava and Manocha, 1984)
Jn(m)(x)
=
∞
X
k=0
(−1)k xk
,
k!(n + mk)!
(1.8)
which for x → −x yields (Dattoli et al., 2006, p.33)
Wn (x; m) =
∞
X
k=0
xk
k!(n + mk)!
(n ∈ N0 := N ∪ {0}; m ∈ N)
(1.9)
and for m = 1, reduces to the nth order Tricomi functions (1.7). The associated Laguerre
polynomials Lαn (x) are defined by the series (Abramowitz and Stegun, 1992)
Lαn (x) =
n
X
(−1)k (n + α)! xk
,
(n − k)!(α + k)!k!
k=0
α = 0, 1, 2, ...
.
(1.10)
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
451
(m,s)
Also, the LGHP L Hn
(x, y, z) are defined by the following operational rules:
s n
o
∂
(m,s)
H
(x,
y,
z)
=
exp
z
L
(x,
y)
,
m n
L n
∂y s
n
o
m
(m,s)
(s)
−1 ∂
(x, y, z) = exp Dx
H
(y,
z)
,
L Hn
n
∂y m
(1.11a)
(1.11b)
(s)
where Hn (x, y) are the higher-order Hermite polynomials, some times called the Kampé de
Fériet or the Gould-Hopper polynomials (GHP) defined by (Gould and Hopper, 1962, p.58), see
also (Dattoli et al., 2000a)
n
[s]
X
y k xn−sk
s
(s)
gn (x, y) = Hn (x, y) = n!
.
k!(n − sk)!
(1.12)
k=0
(s)
The polynomials Hn (x, y) are specified by the generating function
exp(xt + yts ) =
∞
X
tn
n!
(1.13)
n o
xn .
(1.14)
Hn(s) (x, y)
n=0
and by the operational definition
Hn(s) (x, y)
∂s
= exp y s
∂x
(Gould and Hopper, 1962) used the notation gns (x, y) for these polynomials, but due to their link
(s)
with the Hermite polynomials the notation Hn (x, y) was used in most of the works. We also
(s)
follow the notation Hn (x, y) in this paper. We note the following special cases:
1 Ln (−x, y)
= Ln (x, y),
(1.15)
Hn(2)(x, y) = Hn (x, y),
(1.16a)
Hn(s) (x, 0) = Hn (x, 0) = xn ,
(1.16b)
where Ln (x, y) denotes the 2-variable Laguerre polynomials (2VLP) (Dattoli and Torre, 1998)
and Hn (x, y) denotes the 2-variable Hermite-Kampé de Fériet polynomials
(2VHKdFP) (Appell and de Fériet, 1926) respectively. Also, we note that
1
= Hen (x),
(1.17)
Hn x, −
2
where Hen (x) denotes the classical Hermite polynomials (Andrews, 1985). Further, we have
(1,2)
(−x, y, z)
L Hn
= L Hn (x, y, z),
1
1
(1,2)
−x, y, −
= L Hn x, y, −
= L Hn∗ (x, y)
L Hn
2
2
(1.18)
(1.19)
and
(1,1)
(−x, y, z)
L Hn
= Ln (x, y + z),
(1.20)
452
S. Khan and A. A. Al-Gonah
where L Hn (x, y, z) denotes the 3-variable Laguerre-Hermite polynomials (3VLHP) (Dattoli et al.,
2000a) and L Hn∗ (x, y) denotes the 2-variable Laguerre-Hermite polynomials (2VLHP) (Dattoli
et al., 2000b), respectively. The study of the properties of multi-variable generalized special
functions has provided new means of analysis for the solution of large classes of partial differential
equations often encountered in physical problems. The relevance of the special functions in
physics is well established. Most of the special functions of mathematical physics as well as
their generalizations have been suggested by physical problems. We know that the 2VHKdFP
Hn (x, y) are solutions of the heat equation
∂2
∂
Hn (x, y) =
Hn (x, y),
∂y
∂x2
(1.21a)
Hn (x, 0) = xn .
(1.21b)
with the initial condition
Also, the 2VLP Ln (x, y) are natural solutions of the equation
∂ ∂
∂
Ln (x, y) = −
x
Ln (x, y),
(1.22a)
∂y
∂x ∂x
with the initial condition
(−x)n
,
(1.22b)
Ln (x, 0) =
n!
which is a kind of heat diffusion equation of Fokker-Plank type and is used to study the beam
life-time due to quantum fluctuation in storage rings (Wrüullich, 1992). In reference (Dattoli,
2004), it has been shown that the summation formulae of special functions, often encountered in
applications ranging from electromagnetic processes to combinatorics, can be written in terms of
Hermite polynomials of more than one variable. The work of this paper is motivated by the results
on generating functions for the extended generalized Hermite polynomials due to (Greubel, 2006)
and the recently derived summation formulae for Hermite and the Gould-Hopper polynomials by
(Khan et al., 2008) and (Khan and Al-Saad, 2011). Throughout this work, we use the Pochhammer
symbol (λ)n , defined by (Srivastava and Manocha, 1984, p.21)
1, if n = 0
(λ)n =
(1.23)
λ(λ + 1) · · · (λ + n − 1), if n ∈ 1, 2, 3, · · · ,
also, we note that
(λ)m+n = (λ)m (λ + m)n ,
(n − mk)! =
hni
(−1)mk n!
, 0≤k≤
(−n)mk
m
and
(1.24)
(1.25)
(−1)M n!
, 0 ≤ M ≤ n,
(1.26)
(−n)M
where M is defined by M = m1k1 + m2k2 + · · · + mj kj , m1 , m2, ..., mj ∈ N ; k1 , k2 , ..., kj ∈ N0 .
We recall that, the Kampé de Fériet function of two variables is defined by (Srivastava and
Manocha, 1984, p.63)
Qq
Qk
∞ Qp
h
i X
r s
(a
)
(b
)
j
r+s
j
r
(a
):
(b
);(c
);
j=1
j=1
j=1 (cj )s x y
p:q;k
p
q
k
Fl:m;n
x,
y
=
.
(1.27)
Q
Q
Q
(αl):(βm );(γn);
l
m
n
r!
s!
(α
)
(β
)
(γ
)
j
r+s
j
r
j
s
j=1
j=1
j=1
r,s=0
(n − M)! =
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
453
A further generalization of the Kampé de Fériet function (1.27) is the generalized Lauricella
function of several variables, which is defined as (Srivastava and Manocha, 1984, p.64):
0
(n)
00
A:B ;B ;...;B
FC:D
0 ;D 00 ;...;D (n) (z1 , z2 , ..., zn)
[(a):θ0 ,θ00 ,...,θ(n) ]:[(b0):φ0 ];[(b00 ):φ00];...;[(b(n) ):φ(n) ];
A:B 0;B 00;...;B (n)
≡ FC:D
z
,
z
,
...,
z
n
0 ;D 00 ;...;D (n)
[(c):ψ0 ,ψ00 ,...,ψ(n) ]:[(d0 ):δ 0 ];[(d00):δ 00 ];...;[(d(n)):δ (n) ]; 1 2
∞
X
=
Ω(m1 , m2, ..., mn)
m1 ,m2 ,...,mn =0
z mn
z1m1 z2m2
··· n ,
m1 ! m2 !
mn !
(1.28)
where
Ω(m1, m2 , ..., mn)
QA
QB 0 0
QB 00 00
QB (n) (n)
0
00
j=1 (aj )m1 θj0 +m2 θj00 +···+mn θj(n)
j=1 (bj )m1 φj
j=1 (bj )m2 φj ...
j=1 (bj )mn φ(n)
j
:= QC
QD0 0
QD00 00
QD(n) (n)
0
00
j=1 (cj )m1 ψ0 +m2 ψ00 +···+mn ψ(n)
j=1 (dj )m1 δj
j=1 (dj )m2 δj ...
j=1 (dj )mn δ (n)
j
j
j
j
and the coefficients
(k)
(k)
(k)
(k)
θj , j = 1, 2, ..., A; φj , j = 1, 2, ..., B (k); ψj , j = 1, 2, ..., C; δj , j = 1, 2, ..., D(k) ;
for all k ∈ 1, 2, ..., n are real and positive, (a) abbreviates the array of A parameters a1, a2, ..., aA,
(k)
(b(k)) abbreviates the array of B (k) parameters bj , j = 1, 2, ..., B (k); for all k ∈ 1, 2, ..., n
with similar interpretations for (c) and (d(k) ), k = 1, 2, ..., n; et cetera. Note that, when the
coefficients in equation (1.28) equal to 1, the generalized Lauricella function (1.28) reduces to a
direct multivariable extension of the Kampé de Fériet function (1.27). Taking coefficients equal
to 1 in definition (1.28) and for n = 3, we have the Kampé de Fériet function of three variables
0
00
000
p:q1;q2;q3
p:q1;q2 ;q3 (ap ):(bq1 );(bq2 );[(bq3 );
Fl:s
(z
,
z
,
z
)
≡
F
z
,
z
,
z
000
1 2 3
1 2 3
l:s1 ;s2 ;s3
(cl ):(d0s1 );(d00
1 ;s2 ;s3
s2 );(ds3 );
Q
Q
Q
Qq3 000
p
q1
q2
∞
00
0
X
j=1 (bj )m3
j=1 (aj )m1 +m2 +m3
j=1 (bj )m1
j=1 (bj )m2
=
Ql
Qs1
Q
Q
s
s
2
3
0
00
000
j=1 (cj )m1 +m2 +m3
j=1 (dj )m1
j=1 (dj )m2
j=1 (dj )m3
m1 ,m2 ,m3 =0
z1m1 z2m2 z3m3
.
m1 ! m2 ! m3 !
As an illustration, we derive the following explicit representation of the LGHP
(m,s)
(x, y, z) in terms of the generalized Lauricella function of two variables:
L Hn
s
m −1
−1
(m,s)
n 1:0;0 [−n:s,m]:−; −;
(x, y, z) = y F0:0;1
, x
.
L Hn
−:−;[1:1]; z
y
y
×
In order to derive representation (1.30), we use series definition (1.2) of the 2VgLP
in the r.h.s. of definition (1.1), so that we have
(m,s)
(x, y, z)
L Hn
= n!
sk+mr≤n
X
k,r=0
z k xr y n−sk−mr
.
k!(r!)2 (n − sk − mr)!
(1.29)
(1.30)
m Ln (x, y)
(1.31)
Now, using equation (1.26) and the elementary factorial property n! = (1)n in the r.h.s. of equation
(1.31) and in view of definition (1.28) (for n = 2), we get representation (1.30). In this paper,
454
S. Khan and A. A. Al-Gonah
(m,s)
we derive the generating functions for the LGHP L Hn (x, y, z) in terms of the generalized
A:B 0;B 00 ;B 000
Lauricella function of three variables FC:D
0 ;D 00 ;D 000 [.] by using series rearrangement techniques.
Further, we derive the summation formulae for the LGHP by using different analytical means on
(m,s)
the generating function of the LGHP L Hn (x, y, z) or by using certain operational techniques.
2.
Generating functions for the Laguerre-Gould Hopper polynomials
(m,s)
First, we prove the following generating function for the LGHP L Hn
(x, y, z).
Theorem 2.1. For a suitable bounded sequence {f(n)}∞
n=0 , the following generating function
(m,s)
for the LGHP L Hn (x, y, z) holds true:
∞
X
f(n) L Hn(m,s) (x, y, z) tn
n=0
=
∞
X
f(n + sk + mr)
n,k,r=0
(1)n+sk+mr (yt)n (zts )k (xtm )r
.
(1)r
n!
k!
r!
(2.1)
Proof. Denoting the l.h.s. of equation (2.1) by ∆1 and using equation (1.31), we find
∆1 = n!
∞ sk+mr≤n
X
X
n=0
f(n)
k,r=0
z k xr y n−sk−mr tn
.
k!(r!)2(n − sk − mr)!
Replacing n by n+sk +mr in the above equation and using the lemma (Srivastava and Manocha,
1984, p.102):
∞
X
M
≤n
X
φ(k1 , k2 , · · · , kr ; n) =
n=0 k1 ,k2 ,...,kr =0
∞
X
∞
X
φ(k1 , k2 , · · · , kr ; n + M),
(2.2)
n=0 k1 ,k2 ,...,kr =0
where M is defined by M = m1k1 + m2 k2 + · · · + mj kj , m1, m2 , ..., mj ∈ N ; k1 , k2 , ..., kj ∈ N0 ,
we find
∞
X
(n + sk + mr)! (yt)n (zts )k (xtm )r
∆1 =
f(n + sk + mr)
.
n!k!(r!)2
n,k,r=0
Now, using the elementary factorial property n! = (1)n , we get the r.h.s. of assertion (2.1) of
Theorem 2.1.
Qp
(aj )n
Ql
Remark 2.1. Taking f(n) = Qq (b j=1
in assertion (2.1) of Theorem 2.1 and using
j=1 j )n
j=1 (cj )n
definition (1.28) (for n = 3), we deduce the following consequence of Theorem 2.1.
(m,s)
Corollary 2.1. The following generating function for the LGHP L Hn (x, y, z) holds true:
Qp
∞
X
j=1 (aj )n
(m,s)
(x, y, z) tn
Qq
Ql
L Hn
j=1 (bj )n
j=1 (cj )n
n=0
[(a)p1 :1,s,m],[1:1,s,m]:−;−; −;
s
m
= F p+1:0;0;0
yt,
zt
,
xt
,
(2.3)
q
q+l:0;0;1 [(b) :1,s,m],[(c)l :1,s,m]:−;−;[1:1];
1
1
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
455
where the notation (a)p1 is used to represent the product
Qp
Qp
j=1
aj .
(aj )n
in assertion (2.1) of Theorem 2.1 and using definition
Remark 2.2. Taking f(n) = Qj=1
q
j=1 (bj )n
(1.28) (for n = 3), we deduce the following consequence of Theorem 2.1.
(m,s)
Corollary 2.2. The following generating function for the LGHP L Hn (x, y, z) holds true:
∞ Qp
X
(aj )n
(m,s)
Qj=1
(x, y, z) tn
L Hn
q
j=1 (bj )n
n=0
p
[(a)1 :1,s,m],[1:1,s,m]:−;−; −;
s
m
= F p+1:0;0;0
yt,
zt
,
xt
.
(2.4)
q
q:0;0;1
[(b) :1,s,m]:−;−;[1:1];
1
(m,s)
Next, we prove the following bilateral generating function for the LGHP L Hn
(x, y, z).
(m,s)
Theorem 2.2. The following bilateral generating function for the LGHP L Hn (x, y, z) holds
true:
∞
X
tn
0:0;0;0
−:−;−; −;
s
m
Jn(s) (w) L Hn(m,s) (x, y, z) = F1:0;0;1
yt,
zt
−
w,
xt
.
(2.5)
[1:1,s,m]:−;−;[1:1];
n!
n=0
Proof. Denoting the l.h.s. of equation (2.5) by ∆2 and using definitions (1.8) and (1.31), we find
∆2 =
∞
X
n,p=0
sk+mr≤n
X
k,r=0
(−1)pwp z k xr y n−sk−mr tn
.
p!(n + sp)!k!(r!)2(n − sk − mr)!
Replacing n by n + sk + mr and using equation (2.2) in the resultant equation, we find
∆2 =
∞
X
n,p,k,r=0
(−1)p wp y n z k xr tn+sk+mr
.
(n + sk + sp + mr)! p!n! k! (r!)2
Now, replacing k by k − p in the above equation and using equation (1.25) (for m = 1) in the
resultant equation, we find
∆2 =
∞
X
n,k,r=0
k
(yt)n (zts)k (xtm)r X (−k)p w p
.
(1)n+sk+mr (1)r n!k!r! p=0 p!
zts
(2.6)
Finally, using the expansion (Srivastava and Manocha, 1984)
(1 − x)
−λ
=
∞
X
n=0
(λ)n
xn
,
n!
(2.7)
and definition (1.28) (for n = 3) in equation (2.6), we get the r.h.s. of assertion (2.5) of Theorem
2.2.
Remark 2.3. Taking w = zts in assertion (2.5) of Theorem 2.2, we deduce the following
consequence of Theorem 2.2.
456
S. Khan and A. A. Al-Gonah
(m,s)
Corollary 2.3. The following bilateral generating function for the LGHP L Hn (x, y, z) holds
true:
∞
X
tn
0:0;0
−:−; −;
m
(s)
s
(m,s)
Jn (zt ) L Hn (x, y, z) = F1:0;1 [1:1,m]:−;[1:1]; yt, xt .
(2.8)
n!
n=0
(m,s)
In the forthcoming section, we establish summation formulae for the LGHP L Hn (x, y, z) by
using series rearrangement techniques and also by making use of the operational techniques.
3.
Summation formulae for the Laguerre-Gould Hopper polynomials
(m,s)
First, we prove the following result involving the LGHP L Hn
rangement techniques:
(x, y, z) by using series rear-
(m,s)
Theorem 3.1. The following summation formula for the LGHP L Hn (x, y, z) holds true:
k,l X
k
l
(s)
(m,s)
(m,s)
Hn+r (w − y, v − z) L Hk+l−n−r (x, y, z).
(3.1)
L Hk+l (x, w, v) =
n
r
n,r=0
Proof. Replacing t by t + u in equation (1.6) and using the formula (Srivastava and Manocha,
1984, p.52):
∞
∞
X
X
(x + y)n
xn y m
f(n)
=
f(n + m)
,
(3.2)
n!
n! m!
n=0
n,m=0
(m,s)
in the resultant equation, we find the following generating function for the LGHP L Hn
∞
X
t k ul
exp(y(t + u) + z(t + u) )C0 (−x(t + u) ) =
k! l!
k,l=0
s
m
(x, y, z):
(m,s)
L Hk+l (x, y, z),
which can be written as
∞
X
t k ul
C0 (−x(t + u) ) = exp(−y(t + u) − z(t + u) )
k! l!
m
s
(m,s)
L Hk+l (x, y, z).
(3.3)
k,l=0
Replacing y by w and z by v in equation (3.3) and equating the resultant equation to itself, we
find
∞
X
t k ul
(m,s)
L Hk+l (x, w, v)
k! l!
k,l=0
∞
X
t k ul
= exp((w − y)(t + u) + (v − z)(t + u) )
k! l!
s
(m,s)
L Hk+l (x, y, z),
k,l=0
which on using the generating function (1.13) in the exponential on the r.h.s., becomes
∞
X
t k ul
k! l!
k,l=0
(m,s)
L Hk+l (x, w, v)
(3.4)
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
=
∞
X
(t + u)n
n=0
n!
457
Hn(s) (w − y, v − z)
∞
X
t k ul
k! l!
(m,s)
L Hk+l (x, y, z).
(3.5)
k,l=0
Again, using formula (3.2) in the first summation on the r.h.s. of equation (3.5), we have
∞
X
t k ul
k! l!
k,l=0
(m,s)
L Hk+l (x, w, v)
=
∞
∞
X
X
tn ur (s)
t k ul
Hn+r (w − y, v − z)
n!r!
k! l!
n,r=0
k,l=0
(m,s)
L Hk+l (x, y, z).
(3.6)
Now, replacing k by k − n and l by l − r in the r.h.s. of equation (3.6) and using the lemma
(Srivastava and Manocha, 1984, p.100):
∞ X
∞
X
k=0 n=0
A(n, k) =
∞ X
k
X
A(n, k − n),
(3.7)
k=0 n=0
in the resultant equation, we find
∞
X
t k ul
k! l!
k,l=0
(m,s)
L Hk+l (x, w, v)
k,l
∞ X
(s)
X
tk ul Hn+r (w − y, v − z)
(m,s)
=
L Hk+l−n−r (x, y, z).
n!r!(k
−
n)!(l
−
r)!
k,l=0 n,r=0
(3.8)
Finally, on equating the coefficients of like powers of t and u in equation (3.8), we get assertion
(3.1) of Theorem 3.1.
Remark 3.1. Replacing v by z in assertion (3.1) of Theorem 3.1 and using relation (1.16b), we
deduce the following consequence of Theorem 3.1.
(m,s)
Corollary 3.1. The following summation formula for the LGHP L Hn (x, y, z) holds true:
k,l X
k
l
(m,s)
(m,s)
(w − y)n+r L Hk+l−n−r (x, y, z).
(3.9)
L Hk+l (x, w, z) =
n
r
n,r=0
(m,s)
Next, we prove the following result involving the product of the LGHP L Hn
series rearrangement techniques:
(x, y, z) by using
(m,s)
Theorem 3.2. The following summation formula involving product of the LGHP L Hn
holds true:
n,r X
n r
(s)
(m,s)
(m,s)
(x, w, u) L Hr
(X, W, U) =
Hk (w − y, u − z)
L Hn
k
p
(x, y, z)
k,p=0
(m,s)
(m,s)
×Hp(s) (W − Y, U − Z) L Hn−k (x, y, z) L Hr−p (X, Y, Z).
(3.10)
458
S. Khan and A. A. Al-Gonah
Proof. Consider the product of the LGHP generating function (1.6) in the following form:
exp(yt + Y T + zts + ZT s) C0(−xtm ) C0 (−XT m )
=
∞ X
∞
X
Tr
.
n! r!
t
(m,s)
(x, y, z) L Hr(m,s) (X, Y, Z)
L Hn
n=0 r=0
n
(3.11)
Replacing y by w, z by u, Y by W and Z by U in equation (3.11) and equating the resultant
equation to itself, we find
∞ X
∞
X
Tr
= exp((w − y)t + (u − z)ts)
n! r!
t
(m,s)
(x, w, u) L Hr(m,s) (X, W, U)
L Hn
n=0 r=0
× exp((W − Y )T + (U − Z)T s)
∞ X
∞
X
n
(m,s)
(x, y, z) L Hr(m,s) (X, Y, Z)
L Hn
n=0 r=0
tn T r
,
n! r!
(3.12)
which on using the generating function (1.13) in the exponentials on the r.h.s., becomes
∞ X
∞
X
(m,s)
(x, w, u) L Hr(m,s) (X, W, U)
L Hn
n=0 r=0
=
∞ X
∞
X
tn T r
n! r!
(s)
Hk (w − y, u − z) L Hn(m,s) (x, y, z)
n,k=0 r,p=0
× L Hr(m,s) (X, Y, Z)
tn+k (s)
H (W − Y, U − Z)
n!k! p
T r+p
.
r!p!
(3.13)
Finally, replacing n by n − k, r by r − p and using equation (3.7) in the r.h.s. of the above
equation and then equating the coefficients of like powers t and T , we get assertion (3.10) of
Theorem 3.2.
Remark 3.2. Replacing u by z and U by Z in assertion (3.10) of Theorem 3.2 and using relation
(1.16b), we deduce the following consequence of Theorem 3.2.
(m,s)
Corollary 3.2. The following summation formula involving product of the LGHP L Hn
holds true:
n,r X
n r
(m,s)
(m,s)
(x, w, z) L Hr
(X, W, Z) =
(w − y)k (W − Y )p
L Hn
k
p
(x, y, z)
k,p=0
(m,s)
(m,s)
(3.14)
(m,s)
(x, y, z) with the 2VgLP
× L Hn−k (x, y, z) L Hr−p (X, Y, Z).
Further, we prove the following results connecting the LGHP L Hn
(s)
m Ln (x, y) and the GHP Hn (x, y) by using operational techniques:
(m,s)
Theorem 3.3. The following summation formula for the LGHP L Hn (x, y, z) holds true:
k,l X
k
l
(s)
(m,s)
(3.15)
L Hk+l (z, w, y) =
m Ln+r (z, w − x) Hk+l−n−r (x, y).
n
r
n,r=0
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
459
(s)
Proof. We start by a recently derived summation formula for the GHP Hn (x, y) (Khan and
Al-Saad, 2011, p.1538)
k,l X
k
l
(s)
(s)
Hk+l (w, y) =
(w − x)n+r Hk+l−n−r (x, y).
(3.16)
n r
n,r=0
∂m
on both sides of equation (3.16), we have
Operating exp Dz−1 ∂w
m
∂m
(s)
exp Dz−1 m Hk+l (w, y)
∂w
k,l m
X
k
l
(s)
−1 ∂
=
Hk+l−n−r (x, y) exp Dz
(w − x)n+r .
(3.17)
m
n
r
∂w
n,r=0
Using the operational definitions (1.11b) and (1.3) in the l.h.s. and r.h.s., respectively of equation
(3.17), we get assertion (3.15) of Theorem 3.3.
(m,s)
Theorem 3.4. The following summation formula for the LGHP L Hn (x, y, z) holds true:
k,l X
k
l
(m,s)
(s)
Hn+r (w − x, z) m Lk+l−n−r (y, x).
(3.18)
L Hk+l (y, w, z) =
n
r
n,r=0
Proof. Replacing s by m and y by Dy−1 in equation (3.16) and then using the following link
(s)
between the GHP Hn (x, y) and the 2VgLP m Ln (x, y) (Dattoli et al., 1999, p.213):
Hn(m) (y, Dx−1 ) = m Ln (x, y),
(3.19)
in the resultant equation, we find the following summation formula for the 2VgLP m Ln (x, y):
k,l X
k
l
(w − x)n+r m Lk+l−n−r (y, x).
(3.20)
m Lk+l (y, w) =
r
n
n,r=0
s
∂
Now, operating exp z ∂w
on equation (3.20) and using operational definitions (1.11a) and (1.14)
s
in the l.h.s. and r.h.s., respectively of the resultant equation, we get assertion (3.18) of Theorem
3.4.
4.
Applications
I. Taking p = q = l = 1 in equation (2.3), we get
∞
X
(a1)n
2:0;0;0 [a1 :1,s,m],[1:1,s,m]:−;−; −;
(m,s)
n
s
m
(x, y, z) t = F2:0;0;1 [b1 :1,s,m],[c1 :1,s,m]:−;−;[1:1]; yt, zt , xt .
LH
(b1 )n (c1 )n n
n=0
Further, taking b1 = c1 = 1 and replacing a1 by a + 1 in equation (4.1), we get
∞ X
a+n
tn
1:0;0;0 [a+1:1,s,m]:−;−; −;
(m,s)
s
m
H
(x,
y,
z)
=
F
yt,
zt
,
xt
.
L n
1:0;0;1
[1:1,s,m]:−;−;[1:1];
n
n!
n=0
(4.1)
(4.2)
460
S. Khan and A. A. Al-Gonah
II. Taking p = q = 1 in equation (2.4), we get
∞
X
(a1)n
2:0;0;0 [a1 :1,s,m],[1:1,s,m]:−;−; −;
(m,s)
n
s
m
H
(x,
y,
z)
t
=
F
yt,
zt
,
xt
.
L n
1:0;0;1
[b1 :1,s,m]:−;−;[1:1];
(b
)
1
n
n=0
(4.3)
Next, taking b1 = 1 and replacing a1 by a in equation (4.3) and using equations (1.24) and (2.7)
in the r.h.s. of the resultant equation, we get
∞
X
tn
n!
(a)n L Hn(m,s) (x, y, z)
n=0
= (1 − yt)
1:0;0
F0:0;1
−a
[a:s,m]:−; −;
−:−;[1:1];
z
t
1 − yt
s
t
,x
1 − yt
m ,
(4.4)
which for a = 1, becomes
∞
X
(m,s)
(x, y, z) tn
L Hn
n=0
= (1 − yt)
−1
1:0;0
F0:0;1
[1:s,m]:−; −;
−:−;[1:1];
z
t
1 − yt
s
t
,x
1 − yt
m .
(4.5)
Again, taking s = m in equation (4.4), we get the following generating function for the LGHP
(m,s)
p:q;k
(x, y, z) in terms of the Kampé de Fériet function of two variables Fl:m;n
[.] :
L Hn
∞
X
(a)n L Hn(m,m) (x, y, z)
n=0
= (1 − yt)
−a
F m:0;0
0:0;1
tn
n!
∆(m,a):−;−;
−:−; 1;
z
m
1 − yt
where ∆(m; a) abbreviates the array of m parameters
m
m
,x
1 − yt
a a−1
,
, . . . , a−m+1
m m
m
m ,
(4.6)
m > 1.
III. Replacing w by −w in equation (2.5) and using definition (1.9), we get
∞
X
tn
0:0;0;0
−:−;−; −;
s
m
Wn (w; s) L Hn(m,s) (x, y, z) = F1:0;0;1
yt,
zt
+
w,
xt
.
[1:1,s,m]:−;−;[1:1];
n!
n=0
(4.7)
Again, taking s = m = 1 and replacing x by −x in equation (2.5) and using equations (1.7) and
(1.20) in the l.h.s. of the resultant equation, we get the following bilateral generating function
for the 2VLP Ln (x, y) in terms of the Kampé de Fériet function of three variables:
∞
X
n=0
Cn (w) Ln (x, y + z)
tn
0:0;0;0
= F1:0;0;1
n!
−:−;−;−;
1:−;−; 1;
yt, zt − w, −xt ,
(4.8)
IV. Taking l = 0 in equation (3.1) and replacing w by w + y, v by v + z in the resultant equation,
we get
k X
k
(m,s)
(m,s)
(x, w + y, v + z) =
Hn(s) (w, v) L Hk−n (x, y, z),
(4.9)
L Hk
n
n=0
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
461
which on taking v = 0 and using relation (1.16b), yields
k X
k
(m,s)
(m,s)
(x, w + y, z) =
wn L Hk−n (x, y, z).
L Hk
n
n=0
(4.10)
Next, taking m = 1, s = 2 and replacing x by −x in equations (3.1), (3.9), (4.9) and (4,10) and
using equation (1.18), we get the following summation formulae for the 3VLHP L Hn (x, y, z):
k,l X
k
l
Hn+r (w − y, v − z) L Hk+l−n−r (x, y, z),
(4.11)
L Hk+l (x, w, v) =
n r
n,r=0
k,l X
k
l
(w − y)n+r L Hk+l−n−r (x, y, z),
L Hk+l (x, w, z) =
n r
(4.12)
k X
k
Hn (w, v) L Hk−n (x, y, z),
L Hk (x, w + y, v + z) =
n
n=0
(4.13)
n,k=0
L Hk (x, w
+ y, z) =
k X
k
n=0
n
wn L Hk−n (x, y, z).
(4.14)
− 12
Again, taking z =
in equations (4.12) and (4.14) and using relation (1.19), we get the
following summation formulae for the 2VLHP L Hn∗ (x, y):
k,l X
k
l
∗
∗
(w − y)n+r L Hk+l−n−r
(x, y),
(4.15)
L Hk+l (x, w) =
n
r
n,r=0
∗
L Hk (x, w
+ y) =
k X
k
n=0
n
wk−n L Hn∗ (x, y).
(4.16)
Further, taking v = z = − 12 in equation (4.13) and using relations (1.17) and (1.19), we get
k X
k
Hek−n (w) L Hn∗ (x, y).
(4.17)
L Hk (x, w + y, −1) =
n
n=0
V. Taking m = 1, s = 2 and replacing x and X by −x and −X, respectively in equations (3.10)
and (3.14) and using relations (1.18) and (1.16a), we get the following summation formulae
involving product of the 3VLHP L Hn (x, y, z):
n,r X
n r
Hk (w − y, u − z)
L Hn (x, w, u) L Hr (X, W, U) =
k
p
k,p=0
×Hp (W − Y, U − Z) L Hn−k (x, y, z) L Hr−p (X, Y, Z),
n,r X
n r
(w − y)k (W − Y )p
L Hn (x, w, z) L Hr (X, W, Z) =
k
p
(4.18)
k,p=0
× L Hn−k (x, y, z) L Hr−p (X, Y, Z).
(4.19)
462
S. Khan and A. A. Al-Gonah
Further, taking z = Z = − 12 in equation (4.19) and using relation (1.19), we get the following
summation formula involving product of the 2VLHP L Hn∗ (x, y):
n,r X
n r
∗
∗
(w − y)k (W − Y )p
L Hn (x, w) L Hr (X, W ) =
k
p
k,p=0
∗
∗
× L Hn−k
(x, y) L Hr−p
(X, Y ).
VI. Taking l = 0 in equations (3.15) and (3.18), we get
k X
k
(m,s)
(s)
(z, w, y) =
L Hk
m Ln (z, w − x)Hk−n (x, y)
n
n=0
and
(m,s)
(y, w, z)
L Hk
=
k X
k
n=0
n
Hn(s) (w − x, z)
m Lk−n (y, x).
(4.20)
(4.21)
(4.22)
Next, taking m = 1, s = 2 and replacing z by −z in equations (3.15) and (4.21) and then using
equations (1.18), (1.15) and (1.16a), we get the following summation formulae for the 3VLHP
L Hn (x, y, z):
k,l X
k
l
Ln+r (z, w − x)Hk+l−n−r (x, y),
(4.23)
L Hk+l (z, w, y) =
n r
n,r=0
L Hk (z, w, y)
=
k X
k
n=0
n
Ln (z, w − x)Hk−n (x, y).
(4.24)
Again, taking m = 1, s = 2 and replacing y by −y in equations (3.18) and (4.22) and then using
equations (1.18), (1.15) and (1.16a), we get the following summation formulae for the 3VLHP
L Hn (x, y, z):
k,l X
k
l
Hn+r (w − x, z) Lk+l−n−r (y, x),
(4.25)
L Hk+l (y, w, z) =
n r
n,r=0
L Hk (y, w, z)
=
k X
k
n=0
n
Hn (w − x, z) Lk−n (y, x).
(4.26)
Further, taking y = − 12 in equations (4.23), (4.24) and z = − 21 in equations (4.25), (4.26),
respectively and using relations (1.19) and (1.17), we get the following summation formulae for
the 2VLHP L Hn∗ (x, y):
k,l X
k
l
∗
Ln+r (z, w − x)Hek+l−n−r (x),
(4.27)
L Hk+l (z, w) =
n
r
n,r=0
∗
L Hk (z, w)
k X
k
=
Ln (w − x, z)Hek−n (x),
n
n=0
(4.28)
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
and
∗
L Hk+l (y, w)
463
k,l X
k
l
=
Hen+r (w − x) Lk+l−n−r (y, x),
n
r
n,r=0
∗
L Hk (y, w)
=
k X
k
n=0
n
Hen (w − x) Lk−n (y, x),
(4.29)
(4.30)
respectively.
5.
Concluding remarks
In Section 3, we have established the summation formula (3.10), containing the product of the
(m,s)
LGHP L Hn (x, y, z). The formula was proved by using series rearrangement techniques on the
(m,s)
product of the generating functions of the LGHP L Hn (x, y, z). To explore another possibility,
(m,s)
we consider the product of the LGHP L Hn (x, y, z) generating function (1.6) in the following
form:
exp((Y + y)t + (Z + z)ts ) C0 (−Xtm ) C0 (−xtm)
=
∞ X
∞
X
(m,s)
(X, Y, Z) L Hn(m,s) (x, y, z)
L Hr
n=0 r=0
tn+r
.
n!r!
(5.1)
Replacing n by n − r and using equation (3.7) in the r.h.s. of the above equation, we find
exp((Y + y)t + (Z + z)ts ) C0 (−Xtm ) C0 (−xtm)
∞ X
n X
n
tn
(m,s)
(m,s)
=
(X, Y, Z) L Hn−r (x, y, z) ,
L Hr
r
n!
n=0 r=0
(5.2)
which on shifting the exponential to the r.h.s. and using the generating function (1.12), becomes
∞ X
∞ X
n X
n
(s)
m
m
C0(−Xt ) C0(−xt ) =
Hk (−(Y + y), −(Z + z))
r
n=0 k=0 r=0
tn+k
.
(5.3)
n!k!
Again, replacing n by n − k and using equation (3.7) in the r.h.s. of the above equation, we get
∞ X
n X
n−k X
n n−k
m
m
C0 (−Xt ) C0 (−xt ) =
k
r
n=0 k=0 r=0
(m,s)
×LHr(m,s) (X, Y, Z) L Hn−r (x, y, z)
tn
.
(5.4)
n!
Taking m = 1, s = 2 and replacing X and x by −X and −x, respectively, in equation (5.4) and
using relations (1.16a) and (1.18), we find
∞ X
n X
n−k X
n n−k
C0 (Xt) C0 (xt) =
k
r
n=0
r=0
(s)
(m,s)
×Hk (−(Y + y), −(Z + z)) L Hr(m,s) (X, Y, Z) L Hn−k−r (x, y, z)
k=0
464
S. Khan and A. A. Al-Gonah
×Hk (−(Y + y), −(Z + z)) L Hr (X, Y, Z) L Hn−k−r (x, y, z)
tn
.
n!
(5.5)
Now, using the following generating function for the 2-variable Legendre polynomials Rn (x, y)
(Dattoli et al., 2001):
∞
X
tn
C0 (−yt) C0 (xt) =
Rn (x, y)
(5.6)
(n!)2
n=0
in the l.h.s. of equation (5.5) and then equating the coefficients of like powers of t in the resultant
equation, we get the following summation formula for Rn (x, y):
Rn (X, −x) = n!
n X
n−k X
n n−k
k=0 r=0
k
r
Hk (−(Y + y), −(Z + z))
× L Hr (X, Y, Z) L Hn−k−r (x, y, z),
(5.7)
which on using the following link between the 2-variable Legendre polynomials Rn (x, y) and
the classical Legendre polynomials Pn (x) (Andrews, 1985):
1−x 1+x
,
= Pn (x),
(5.8)
Rn
2
2
yields the following summation formula for Pn (x):
n X
n−k X
n n−k
Pn (x) = n!
Hk (−(Y + y), −(Z + z))
k
r
k=0 r=0
×L Hr
6.
1−x
, Y, Z
2
L Hn−k−r
−1 − x
, y, z .
2
(5.9)
Conclusion
In this paper, we have established some generating functions for the Laguerre-Gould Hopper
polynomials by using series rearrangement techniques. Also, some summation formulae for that
polynomials are derived by using certain operational techniques and by using different analytical
means on its generating function. Further, many generating functions and summation formulae
for the polynomials related to Laguerre-Gould Hopper polynomials are obtained as applications
of main results. The approach presented in this paper is general and can be extended to establish
other properties of special polynomials.
Acknowledgments
The authors are thankful to anonymous referees and the Editor-in-Chief Professor Aliakbar
Montazer Haghighi for useful comments and suggestions towards the improvement of this paper.
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
465
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Abramowitz, M. and Stegun, I. A. (1992). Handbook of mathematical functions with formulas,
graphs and mathematical tables, Reprint of the 1972 edition. Dover Publications, Inc. New
York.
Andrews, L. (1985). Special Functions for Engineers and Applied Mathematicians. Macmillan
Co., New York.
Appell, P. and de Fériet, J. K. (1926). Fonctions hypergéométriques et hypersphériques:
Polynômes d’ Hermite. Gauthier-Villars, Paris.
Dattoli, G. (2004).
Summation formulae of special functions and multivariable hermite polynomials.
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