J ournal of
Classical
A nalysis
Volume 4, Number 2 (2014), 167–180
doi:10.7153/jca-04-11
ON A FAMILY OF MULTIVARIABLE POLYNOMIALS
DEFINED THROUGH RODRIGUES TYPE FORMULA
Ö VG Ü G ÜREL Y ILMAZ , R ABIA A KTAŞ AND A BDULLAH A LTIN
Abstract. In this paper, we present a family of multivariable polynomials defined by Rodrigues
formula and we discuss their some miscellaneous properties such as generating function and
recurrence relation. We also derive various classes of multilateral generating functions for these
multivariable polynomials and give some special cases of the results. Furthermore, we also show
that some particular cases of the polynomials reduce to the products of Hermite and Laguerre
orthogonal polynomials with one variable.
1. Introduction
The concept of generating function is one of the most surprising and useful inventions in Mathematics and other disciplines. It is a powerful tool for recurrence relations
which come up a lot in electrical engineering, circuit analysis, statistics, mathematics,
physics, geology and any other disciplines that use differential equations. Recently, several families of multivariable polynomials defined via generating functions have been
introduced and their some properties have been studied in [2, 4, 5, 6, 9, 11, 14]. This
paper deals with finding a generating function for a family of multivariable polynomials defined by Rodrigues formula and obtaining some recurrence relations for these
polynomials.
We recall that the classical Hermite polynomials Hn (x) of degree n are defined
by the Rodrigues formula
n 2 d
−x2
e
(1.1)
Hn (x) = (−1)n ex
dxn
and the familiar orthogonality property is as follows
∞
√
2
e−x Hn (x) Hm (x) dx = 2n n! πδm,n
(1.2)
−∞
∪∪ {0})
(m, n ∈ N0 := N∪
where δm,n being the Kronecker delta. Laguerre polynomials Ln (x) of degree n are
defined by the Rodrigues formula
Ln (x) =
ex d n −x n e x ,
n! dxn
(1.3)
Mathematics subject classification (2010): Primary: 33C45.
Keywords and phrases: Rodrigues formula, recurrence relation, generating function, bilateral generating function, Hermite polynomial, Laguerre polynomial.
c
, Zagreb
Paper JCA-04-11
167
168
Ö. G ÜREL Y ILMAZ , R. A KTAŞ AND A. A LTIN
and these polynomials hold the following orthogonality relation
∞
e−x Ln (x) Lm (x) dx = δm,n ;
m, n ∈ N0 .
(1.4)
0
As a generalization of Rodrigues formulas given by (1.1) and (1.3), in [1], the
authors defined a family of polynomials defined through Rodrigues formula:
φk+n(m−1) (x) = eϕm (x)
dn −ϕm (x)
ψ
(x)
e
k
dxn
(1.5)
where φ k+n(m−1) (x) is a polynomial of degree k + n (m − 1), n = 0, 1, 2, ... and, ψk (x)
and ϕm (x) are polynomials respectively of degree k and m; k , m = 0, 1, 2, ... In that
work, for these polynomials whose special cases reduce to Hermite polynomials, some
recurrence relations and a generating function were given. In [2], as a generalization
of polynomials (1.5), a family of polynomials in two variables was presented and some
properties of them were obtained.
Motivated essentially by these works in [1, 2], we first consider the following
Rodrigues formula in order to develop a general class of polynomials with r -variables
φn (x1 , ..., xr ) := φn1 ,...,nr (x1 , ..., xr )
= eϕm (x1 ,...,xr )
∂ |n|
∂ xn11 ...∂ xnr r
ψk (x1 , ..., xr ) e−ϕm (x1 ,...,xr )
(1.6)
( ni , k, m = 0, 1, 2, ...)
where ψk (x1 , ..., xr ) and ϕm (x1 , ..., xr ) are polynomials of total degree k and m with
respect to the variables x1 , ..., xr , respectively. φn (x1 , ..., xr ) is a polynomial of total degree N = (m − 1)|n| + k with respect to the variables x1 , ..., xr and |n| = n1 + ... + nr .
We then give a generating function for these polynomials by applying Cauchy’s integral
formula and derive various families of bilateral generating functions for them. Under
some special cases, we give some recurrence relations satisfied by φn (x1 , ..., xr ) . We
also show that some particular cases of these polynomials reduce to known multivariable polynomials which are products of Hermite or Laguerre orthogonal polynomials
with one variable.
2. A family of generating functions for the multivariable polynomials
φn1 ,...,nr (x1 , ..., xr )
With the help of the Cauchy’s integral formula, we give a family of generating
function for the polynomials φn1 ,...,nr (x1 , ..., xr ) as follows:
T HEOREM 2.1. Let the polynomials ψk (x1 , ..., xr ) and ϕm (x1 , ..., xr ) be independent of n1 , n2 , ..., nr . A generating function for the polynomials φn1 ,...,nr (x1 , ..., xr ) is
169
O N A FAMILY OF MULTIVARIABLE POLYNOMIALS
given by
∞
∑
n1 ,...,nr =0
φn1 ,...,nr (x1 , ..., xr )
t1n1 trnr
...
n1 ! nr !
= ψk (x1 + t1 , ..., xr + tr ) eϕm (x1 ,...,xr )−ϕm (x1 +t1 ,...,xr +tr ) .
(2.1)
Proof. We begin by considering the series
An1 ,...,nr (x1 , ..., xr ,t1 ) :=
t n1
∞
∑ φn1 ,...,nr (x1 , ..., xr ) n11 ! .
(2.2)
n1 =0
From Cauchy’s integral formula we have
n ! ψ (z , x , ..., x ) e−ϕm (z1 ,x2 ,...,xr )
∂ n1 r
1
k 1 2
−ϕm (x1 ,...,xr )
=
ψ
(x
,
...,
x
)
e
dz1
r
k 1
2π i
∂ xn11
(z1 − x1)n1 +1
∗
C1
where the closed contour C1∗ in the complex z-plane is a circle (centered at z1 =
x1 ) of sufficiently small radius, which is described in the positive direction (counterclockwise), so that we find from (1.6) and (2.2) that
An1 ,...,nr (x1 , ..., xr ,t1 )
⎧
⎫
⎪
⎪
⎨ 1 ψ (z , x , ..., x ) e−ϕm (z1 ,x2 ,...,xr )
⎬
r
k 1 2
ϕm (x1 ,...,xr )
t1n1
=e
dz
1
∑ ∂ xn2 ...∂ xnr r ⎪ 2π i
n1 +1
⎪
(z
−
x
)
⎩
⎭
2
1
1
n1 =0
∗
∞
∂ n2 +...+nr
C1
=
ψk (z1 , x2 , ..., xr ) e−ϕm (z1 ,x2 ,...,xr ) ∞
eϕm (x1 ,...,xr ) ∂ n2 +...+nr
∂ xn22 ...∂ xnr r
2π i
eϕm (x1 ,...,xr ) ∂ n2 +...+nr
=
2π i
∂ xn22 ...∂ xnr r
= eϕm (x1 ,...,xr )
∂ n2 +...+nr
∂ xn22 ...∂ xnr r
C1∗
C1
z1 − x1
∑
n1 =0
t1
z1 − x1
n1
dz1
t1 ψk (z1 , x2 , ..., xr ) e−ϕm (z1 ,x2 ,...,xr )
<1
dz1 , z1 − (x1 + t1 )
z1 − x1 ψk (x1 + t1 , x2 , ..., xr ) e−ϕm (x1 +t1 ,x2 ,...,xr )
where C1 is a circle (centered at z1 = x1 +t1 ) with radius ε > 0 in the complex z-plane.
Thus, we get
∞
t n1
∑ φn1 ,...,nr (x1 , ..., xr ) n11!
n1 =0
= eϕm (x1 ,...,xr )
∂ n2 +...+nr −ϕm (x1 +t1 ,x2 ,...,xr )
.
n2
nr ψk (x1 + t1 , x2 , ..., xr ) e
∂ x2 ...∂ xr
(2.3)
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Ö. G ÜREL Y ILMAZ , R. A KTAŞ AND A. A LTIN
When multiplying both sides of (2.3) by
t2n2
and then summing both sides, we obtain
n2 !
t1n1 t2n2
n1 ! n2 !
n1 ,n2 =0
n2
∞
∂ n2 +...+nr −ϕm (x1 +t1 ,x2 ,...,xr ) t2
.
= ∑ eϕm (x1 ,...,xr ) n2
nr ψk (x1 + t1 , x2 , ..., xr ) e
n2 !
∂ x2 ...∂ xr
n2 =0
∞
∑
φn1 ,...,nr (x1 , ..., xr )
(2.4)
Applying the Cauchy’s integral formula again to the right hand-side of (2.4), for suitable
contour C2 , we have
∞
∑
φn1 ,...,nr (x1 , ..., xr )
n1 ,n2 =0
eϕm (x1 ,...,xr ) ∂ n3 +...+nr
=
2π i
∂ xn33 ...∂ xnr r
= eϕm (x1 ,...,xr )
∂ n3 +...+nr
∂ xn33 ...∂ xnr r
C2
t1n1 t2n2
n1 ! n2 !
ψk (x1 + t1, z2 , x3 , ..., xr ) e−ϕm (x1 +t1 ,z2, ,x3 ,....,xr )
dz2
z2 − (x2 + t2 )
ψk (x1 + t1 , x2 + t2 , x3, ..., xr ) e−ϕm (x1 +t1 ,x2 +t2 ,x3 ,...,xr )
t2 for z2 −x
< 1 . When the above method is applied (r − 3) times repeatedly by means
2
of Cauchy’s integral formula, we obtain
n
∞
∑
n1 ,...,nr−1 =0
φn1 ,...,nr (x1 , ..., xr )
= eϕm (x1 ,...,xr )
r−1
t1n1 t2n2 tr−1
...
n1 ! n2 ! nr−1 !
∂ nr −ϕm (x1 +t1 ,...,xr−1 +tr−1 ,xr )
.
nr ψk (x1 + t1 , ..., xr−1+ tr−1, xr ) e
∂ xr
By multiplying both sides of (2.5) by
∞
∑
n1 ,...,nr =0
φn1 ,...,nr (x1 , ..., xr )
= eϕm (x1 ,...,xr )
(2.5)
trnr
and then summing both sides, we have
nr !
t1n1 t2n2 trnr
...
n1 ! n2 ! nr !
nr
∂ nr −ϕm (x1 +t1 ,...,xr−1 +tr−1 ,xr ) tr
,
ψ
(x
+
t
,
...,
x
t
x
)
e
∑ nr k 1 1 r−1+ r−1, r
nr !
nr =0 ∂ xr
∞
in which considering Cauchy’s integral formula in the right hand for suitable contour
Cr (centered at zr = xr + tr ), we conclude that
∞
∑
n1 ,...,nr =0
φn1 ,...,nr (x1 , ..., xr )
eϕm (x1 ,...,xr )
=
2π i
Cr
t1n1 trnr
...
n1 ! nr !
tr ψk (x1 +t1 , ..., xr−1 +tr−1, zr ) e−ϕm (x1 +t1 ,...,xr−1 +tr−1 ,zr )
<1
dzr , zr − (xr +tr )
zr − xr = ψk (x1 + t1 , x2 + t2 , ..., xr + tr ) eϕm (x1 ,...,xr )−ϕm (x1 +t1 ,x2 +t2 ,...,xr +tr )
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O N A FAMILY OF MULTIVARIABLE POLYNOMIALS
which completes the proof.
Now, we consider many general families of bilateral generating relations for the
polynomials φn1 ,...,nr (x1 , ..., xr ) which are generated by (2.1) without using grouptheoretic (or Lie algebraic) technique but, with the help of the similar method as given
in [2, 3, 4, 5, 6, 8, 9, 10, 14].
We begin by stating the following theorem.
T HEOREM 2.2. Corresponding to an identically non-vanishing function Ωμ (y1 ,
..., ys ) of s complex variables y1 , ..., ys (s ∈ N) and of complex order μ , suppose that
∞
Λμ ,ν (y1 , ..., ys ; z) := ∑ al Ωμ +ν l (y1 , ..., ys )zl
(2.6)
l=0
(al = 0,
μ , ν ∈ C).
and
Θn,p,μ ,ν (x1 , ..., xr ; y1 , ..., ys ; ς )
:=
[n1 /p]
∑
l=0
al
φn −pl,n2 ,...,nr (x1 , ..., xr ) Ωμ +ν l (y1 , ..., ys )ς l
(n1 − pl)! 1
(2.7)
(n1 , p ∈ N) .
Then we have
1
η n1 n2 nr
Θ
,
...,
x
;
y
,
...,
y
;
x
n,p, μ ,ν
r 1
s p t1 t2 ...tr
1
∑
t1
n1 ,...nr =0 n2 !...nr !
∞
(2.8)
= Λμ ,ν (y1 , ..., ys ; η )ψk (x1 + t1 , ..., xr + tr ) eϕm (x1 ,...,xr )−ϕm (x1 +t1 ,...,xr +tr )
provided that each member of (2.8) exists.
Proof. For the proof of Theorem 2.2, we find it to be convenient to denote the first
member of the assertion (2.8) by S . Then, upon substituting for the polynomials
η
Θn,p,μ ,ν x1 , ..., xr ; y1 , ..., ys ; p
t1
from the definition (2.7) into the left-hand side of (2.8), we obtain
S=
∞
∑
[n1 /p]
∑
n1 ,...nr =0 l=0
al
φn −pl,n2 ,....,nr (x1 , ..., xr )
(n1 − pl)!n2 !...nr ! 1
× Ωμ +ν l (y1 , ..., ys )t1n1 −pl t2n2 ...trnr η l
=
∞
∑ al Ωμ +ν l (y1 , ..., ys )η l
l=0
∞
1
φn1 ,...,nr (x1 , ..., xr )t1n1 t2n2 ...trnr
n
!n
!...n
!
r
1
2
n1 ,...nr =0
∑
= Λμ ,ν (y1 , ..., ys ; η )ψk (x1 + t1 , ..., xr + tr ) eϕm (x1 ,...,xr )−ϕm (x1 +t1 ,...,xr +tr )
which completes the proof of Theorem 2.2.
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Ö. G ÜREL Y ILMAZ , R. A KTAŞ AND A. A LTIN
3. Further Remarks and Observations
If we express the multivariable function
Ωμ +ν l (y1 , ..., ys ) (l ∈ N0 , s ∈ N)
in terms of several relatively simpler functions of one and more variables, Theorem 2.2
can be applied to yield various families of bilateral generating functions. For instance,
if we set
(γ ,...,γ )
s = r and Ωμ +ν l (y1 , ..., yr ) = hμ +1 ν l r (y1 , ..., yr )
(α1 ,...,αr )
in Theorem 2.2, where the multivariable Lagrange-Hermite polynomials hn
..., xr ) [4] are generated by
∞
r −α (α ,...,α )
∏ 1 − x jt j j = ∑ hn 1 r (x1, ..., xr )t n
j=1
(x1 ,
(3.1)
n=0
where |t| < min |x1 |−1 , |x2 |−1/2 , ..., |xr |−1/r , then we obtain the following result
which provides a class of bilateral generating functions for the Lagrange-Hermite multivariable polynomials and the polynomials φn1 ,....,nr (x1 , ..., xr ) generated by (2.1).
∞
(γ ,...,γr )
C OROLLARY 3.1. If Λμ ,ν (y1 , ..., yr ; z) := ∑ al hμ +1 ν l
l=0
0 , ν , μ ∈ N0 ; and
(y1 , ..., yr )zl where al =
Θn,p,μ ,ν (x1 , ..., xr ; y1 , ..., yr ; ς )
:=
[n1 /p]
∑
l=0
al
(γ ,...,γ )
φn −pl,n2 ,...,nr (x1 , ..., xr ) hμ +1 ν l r (y1 , ..., yr )ς l
(n1 − pl)! 1
where n1 , p ∈ N. Then we have
1
η n1 nr
Θ
,
...,
x
;
y
,
...,
y
;
x
n,p, μ ,ν
r 1
r p t1 ...tr
1
∑
t1
n1 ,...nr =0 n2 !...nr !
∞
(3.2)
= Λμ ,ν (y1 , ..., yr ; η )ψk (x1 + t1 , ..., xr + tr ) eϕm (x1 ,....,xr )−ϕm (x1 +t1 ,...,xr +tr )
provided that each member of (3.2) exists.
R EMARK 3.2. Using the generating function (3.1) and taking al = 1, μ = 0, ν =
1, we have
[n1 /p]
∞
∑
∑
n1 ,...nr =0 l=0
(γ1 ,...,γr )
× hl
1
φn −pl,n2 ,...,nr (x1 , ..., xr )t2n2 ...trnr
(n1 − pl)!n2 !...nr ! 1
(y1 , ..., yr )η l t1n1 −pl
r
= ψk (x1 + t1 , ..., xr + tr ) eϕm (x1 ,...,xr )−ϕm (x1 +t1 ,...,xr +tr ) ∏
j=1
−γ 1 − y jη j j
O N A FAMILY OF MULTIVARIABLE POLYNOMIALS
where
173
|η | < min |y1 |−1 , |y2 |−1/2 , ..., |yr |−1/r .
Set
s = 1 and Ωμ +ν l (y ) = Hμ +ν l (y)
in Theorem 2.2, where the n th Hermite polynomial Hn (x) is generated by
∞
tn
∑ Hn (x) n! = exp
2xt − t 2 .
(3.3)
n=0
Then, we get the following result which provides a class of bilateral generating functions for Hermite polynomials and the polynomials φn1 ,....,nr (x1 , ..., xr ) .
∞
C OROLLARY 3.3. Let Λμ ,ν (y; z) := ∑ al Hμ +ν l (y)zl where al = 0 , ν , μ ∈ N0 ;
l=0
and
Θn,p,μ ,ν (x1 , ..., xr ; y; ς )
:=
[n1 /p]
∑
l=0
al
φn −pl,n2 ,...,nr (x1 , ..., xr ) Hμ +ν l (y)ς l
(n1 − pl)! 1
where n1 , p ∈ N. Then we get
1
η n1 nr
Θn,p,μ ,ν x1 , ..., xr ; y; p t1 ...tr
∑
t1
n1 ,...nr =0 n2 !...nr !
∞
= Λμ ,ν (y; η )ψk (x1 + t1 , ..., xr + tr ) eϕm (x1 ,....,xr )−ϕm (x1 +t1 ,...,xr +tr )
(3.4)
provided that each member of (3.4) exists.
R EMARK 3.4. If we take al = l!1 , μ = 0, ν = 1 and then use the generating
relation (3.3) for Hermite polynomials, we have
∞
∑
[n1 /p]
∑
n1 ,...nr =0 l=0
φn1 −pl,n2 ,...,nr (x1 , ..., xr ) Hl (y)
1
η l t n1 −pl ...trnr
l! (n1 − pl)!n2 !...nr ! 1
= exp 2yη − η 2 ψk (x1 + t1, ..., xr + tr ) eϕm (x1 ,....,xr )−ϕm (x1 +t1 ,...,xr +tr ) .
We conclude this section by remarking that for each suitable choice of the coefficients al (l ∈ N0 ), if the multivariable function Ωμ +ν l (y1 , ..., ys ), (s ∈ N), is expressed
as an appropriate product of several simpler relatively functions, the assertion of Theorem 2.2 can be applied to yield many different families of multilateral generating functions for the polynomials φn1 ,...,nr .
174
Ö. G ÜREL Y ILMAZ , R. A KTAŞ AND A. A LTIN
4. Some recurrence relations of the polynomials φn1 ,...,nr (x1 , ..., xr )
In this section, we give some recurrence relations satisfied by the polynomial set
φn1 ,...,nr (x1 , ..., xr ) for some special choices of the polynomials ψk (x1 , ...xr ) .
Setting ψk (x1 , ...xr ) = (a1 x1 + a2 x2 + ... + ar xr + ar+1)k (a1 , ..., ar+1 ∈ R, k =
0, 1, ...) in (1.6), we get
φn1 ,...,nr (x1 , ..., xr ) = e
ϕm (x1 ,...,xr )
|n|
∂
k −ϕm (x1 ,...,xr )
.
n1
nr (a1 x1 + ... + ar xr + ar+1 ) e
∂ x1 ...∂ xr
(4.1)
which are generated by
∞
∑
n1 ,...,nr =0
φn1 ,...,nr (x1 , ..., xr )
t1n1 trnr
...
n1 ! nr !
(4.2)
= eϕm (x1 ,...,xr )−ϕm (x1 +t1 ,x2 +t2 ,...,xr +tr ) [a1 (x1 + t1) + ... + ar (xr + tr ) + ar+1 ]k
from Theorem 2.1.
For brevity, we need the following notations: for x = (x1 , ..., xr ) and n = (n1 , ..., nr ) ,
let denote
∂ |n|
∂ |n|
f (x1 , ..., xr ) .
n1
nr f (x1 , ..., xr ) =
∂ xn
∂ x1 ...∂ xr
Let e1 , ..., er be the standard basis of Rr , that is, the ith coordinate of e j is 1 if i = j ,
0 if i = j .
Now, we can give the following results for these polynomials.
T HEOREM 4.1. Let
Ω(l1 ,...,lr ) (x1 , ..., xr ; n1 , ..., nr ) = (a1 x1 + ... + ar xr + ar+1) φn−l1 e1 −...−lr er (x1 , ..., xr )
r
+ ∑ a j (n j − l j ) φn−l1 e1 −...−lr er −e j (x1 , ..., xr )
j=1
(l1 , l2 , ..., lr ∈ N0 ).
Then for the polynomials φn1 ,...,nr (x1 , ..., xr ) , we have the following recurrence
relations for 1 i r
m−1
n1
nr
∂ p+1
−∑
...
Ω(l1 ,...,lr ) (x1 , ..., xr ; n1 , ..., nr ) l+e ϕm (x1 , ..., xr )
∑
lr
∂x i
p=0 l +...+lr =p l1
1
= (a1 x1 + ... + ar xr + ar+1 ) φn+ei (x1 , ..., xr )
r
+ ∑ a j n j φn+ei −e j (x1 , ..., xr ) − ai kφn (x1 , ..., xr )
j=1
where l = (l1 , l2 , ..., lr ) and
m 1,
ni li + 1;
i = 1, 2, ..., r.
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O N A FAMILY OF MULTIVARIABLE POLYNOMIALS
Proof. Without loss generality, we can assume i = 1 . Differentiating each member
of the generating function (4.2) with respect to t1 and then using (4.2) again, we find
that
∞
∑
n1 ,...,nr =0
φn1 +1,n2 ,...,nr (x1 , ..., xr )
t1n1 t2n2 ...trnr
n1 !...nr !
(4.3)
=
∞
t1n1 trnr
a1 k
φ
(x
,
...,
x
)
∑ n1 ,...,nr 1 r n1! ... nr !
[a1 (x1 + t1 ) + ... + ar (xr + tr ) + ar+1 ] n1 ,...,n
r =0
−
∞
t n1 t nr
∂
ϕm (x1 + t1 , ..., xr + tr ) ∑ φn1 ,...,nr (x1 , ..., xr ) 1 ... r .
∂ t1
n1 ! nr !
n1 ,...,nr =0
Since the Taylor series of the polynomial
(0, ..., 0) is
∂
ϕm (x1 + t1 , ..., xr + tr ) at (t1 , ...,tr ) =
∂ t1
∂
ϕm (x1 + t1 , ..., xr + tr )
∂ t1
=
m−1
∑
p=0 l1
1
∂ p+1
ϕ (x , ..., xr )t1l1 ...trlr ,
l
+1
l2
lr m 1
1
l
!...l
!
r
∂x
∂ x ...∂ xr
+...+l =p 1
∑
2
1
r
the equality (4.3) can be written in the form
(a1 x1 + ... + ar xr + ar+1)
+ (a1t1 + ... + artr )
= a1 k
∞
∑
n1 ,...,nr =0
∞
∑
n1 ,...,nr =0
∞
∑
n1 ,...,nr =0
φn1 +1,n2 ,...,nr (x1 , ..., xr )
φn1 ,...,nr (x1 , ..., xr )
− (a1 x1 + ... + ar xr + ar+1)
× φn1 ,...,nr (x1 , ..., xr )
− (a1t1 + ... + artr )
φn1 +1,n2 ,...,nr (x1 , ..., xr )
t1n1 trnr
...
n1 ! nr !
∞
∑
∑
∞
∑
m−1
∑
∑
n1 ,...,nr =0 p=0 l1 +...+lr =p
× φn1 ,...,nr (x1 , ..., xr )
∑
n1 ,...,nr =0 p=0 l1 +...+lr =p
t1n1 +l1 trnr +lr
...
n1 !l1 ! nr !lr !
t1n1 ...trnr
n1 !...nr !
m−1
t1n1 ...trnr
n1 !...nr !
∂ p+1
∂ x1l1 +1 ∂ xl22 ...∂ xlrr
∂ p+1
∂ x1l1 +1 ∂ xl22 ...∂ xlrr
ϕm (x1 , ..., xr )
ϕm (x1 , ..., xr )
t1n1 +l1 trnr +lr
...
.
n1 !l1 ! nr !lr !
If we make necessary calculations, we find the desired recurrence relation for i = 1 .
Similarly, it can be easily obtained for i = 2, 3, ...r. 176
Ö. G ÜREL Y ILMAZ , R. A KTAŞ AND A. A LTIN
Other recurrence relations for the polynomials φn1 ,...,nr (x1 , ..., xr ) can be obtained
by differentiating the generating function (4.2) with respect to xi , i = 1, 2, ..., r as follows:
T HEOREM 4.2. Let Ω(l1 ,...,lr ) (x1 , ..., xr ; n1 , ..., nr ) be given as in Theorem 4.1. The
polynomials φn1 ,...,nr (x1 , ..., xr ) satisfy the following recurrence relations
∂ p+1
n1
nr
...
Ω(l1 ,...,lr ) (x1 , ..., xr ; n1 , ..., nr ) l+e ϕm (x1 , ..., xr )
∑ ∑
lr
∂x i
p=0 l1 +...+lr =p l1
∂
∂
= (a1 x1 + ... + arxr + ar+1 )
φn (x1 , ..., xr ) − φn (x1 , ..., xr )
ϕm (x1 , ..., xr )
∂ xi
∂ xi
r
∂
∂
φn−e j (x1 , ..., xr ) − φn−e j (x1 , ..., xr )
ϕm (x1 , ..., xr )
+ ∑ a jn j
∂ xi
∂ xi
j=1
−
m−1
− ai kφn (x1 , ..., xr )
where
m 1,
ni li + 1;
i = 1, 2, ..., r.
The next results can be easily obtained from Theorem 4.1 and Theorem 4.2.
T HEOREM 4.3. The polynomials φn1 ,...,nr (x1 , ..., xr ) hold that
∂
(a1 x1 + ... + arxr + ar+1 ) φn+ei (x1 , ..., xr ) + φn (x1 , ..., xr )
ϕm (x1 , ..., xr )
∂ xi
r
∂
∂
= ∑ a jn j
φn−e j (x1 , ..., xr ) −φn−e j (x1 , ..., xr )
ϕm (x1 , ..., xr ) −φn+ei −e j (x1 , ..., xr )
∂ xi
∂ xi
j=1
+ (a1 x1 + ... + ar xr + ar+1)
∂
φn (x1 , ..., xr ) .
∂ xi
For the special case ψk (x1 , ..., xr ) = 1 in (1.6), we have the polynomials
φn1 ,...,nr (x1 , ..., xr ) = eϕm (x1 ,...,xr )
∂ |n|
∂ xn11 ...∂ xnr r
e−ϕm (x1 ,...,xr ) .
(4.4)
which are of degree N1 = (m − 1)|n| ; |n| = n1 + ... + nr . As a result of Theorems
4.1–4.3, we have following:
C OROLLARY 4.4. For the polynomials φn1 ,...,nr (x1 , ..., xr ) , we have the following
recurrence relations
m−1
n1
nr
φn+ei (x1 , ..., xr ) = − ∑
φn−l1 e1 −...−lr er (x1 , ..., xr )
...
∑
l
lr
1
p=0 l1 +...+lr =p
×
∂ p+1
ϕm (x1 , ..., xr ) ,
∂ xl+ei
177
O N A FAMILY OF MULTIVARIABLE POLYNOMIALS
∂
∂
φn (x1 , ..., xr ) − φn (x1 , ..., xr )
ϕm (x1 , ..., xr )
∂ xi
∂ xi
m−1
n1
nr
∂ p+1
=− ∑
φn−l1 e1 −...−lr er (x1 , ..., xr ) l+e ϕm (x1 , ..., xr ) ,
...
∑
lr
∂x i
p=0 l1 +...+lr =p l1
ni li + 1;
(m 1,
i = 1, 2, ..., r)
and
φn+ei (x1 , ..., xr ) + φn (x1 , ..., xr )
∂
∂
ϕm (x1 , ..., xr ) =
φn (x1 , ..., xr )
∂ xi
∂ xi
for 1 i r.
5. Some special cases of the polynomials φn1 ,...,nr (x1 , ..., xr )
Under suitable choices, the family of polynomials φn1 ,...,nr (x1 , ..., xr ) reduce to
several known multivariable polynomials which are products of Hermite or Laguerre
orthogonal polynomials with one variable. Now, let consider some special cases.
x2r ,
R EMARK 5.1. By getting ψk (x1 , ..., xr ) = x1 ...xr and ϕm (x1 , ..., xr ) = x21 + ... +
the family of polynomials given by (1.6) reduces to the following polynomials
2
2
∂ |n|
−x21 −...−x2r
(x
φn1 ,...,nr (x1 , ..., xr ) = ex1 +...+xr n1
...x
)e
r
1
∂ x1 ...∂ xnr r
2
2
1 2 d nr +1
1 2 d n1 +1
= − ex1 n +1 e−x1 ... − exr nr +1 e−xr
2 dx11
2 dxr
=
(−1)n1 +...+nr
Hn1 +1 (x1 ) ...Hnr +1 (xr ) ;
2r
n1 , ..., nr = 0, 1, ...,
which are generated by
∞
∑
n1 ,...,nr =0
φn1 ,...,nr (x1 , ..., xr )
t1n1 trnr
...
n1 ! nr !
= (x1 + t1) ... (xr + tr ) e−(t1 +...+tr +2x1t1 +...+2xr tr )
2
2
where Hn1 +1 (x1 ) ,...,Hnr +1 (xr ) are Hermite polynomials of degree n1 + 1, ..., nr + 1,
respectively. These polynomials satisfy the following orthogonality relation from (1.2)
(see also [7])
∞
−∞
...
∞
2
2
e−(x1 +...+xr ) φn1 ,...,nr (x1 , ..., xr ) φk1 ,...,kr (x1 , ..., xr ) dx1 ...dxr
−∞
=2
n1 +...+nr −r
r
(n1 + 1)!... (nr + 1)!π 2 δn1 ,k1 ...δnr ,kr
where δn1 ,k1 , ..., δnr ,kr are Kronecker delta.
178
Ö. G ÜREL Y ILMAZ , R. A KTAŞ AND A. A LTIN
R EMARK 5.2. If we take ϕm (x1 , ..., xr ) = x21 + ... + x2r + α1 x1 + ... + αr xr in (4.4),
we have
φn1 ,...,nr (x1 , ..., xr )
2
2
∂ |n|
−x21 −...−x2r −α1 x1 −...−αr xr
= ex1 +...+xr +α1 x1 +...+αr xr n1
nr e
∂ x1 ...∂ xr
n1
2
∂
2
2
∂ nr
2
= ex1 +α1 x1 n1 e−x1 −α1 x1 ... exr +αr xr nr e−xr −αr xr
∂ xr
∂ x1
nr
2
2
α1 2 ∂ n1
α1 2
−(x1 + 2 )
(xr + α2r ) ∂ e−(xr + α2r )
e
= e(x1 + 2 )
...
e
∂ xnr r
∂ xn11
n1 +...+nr
α1
αr
= (−1)
...Hnr xr +
; n1 , ..., nr = 0, 1, ...
Hn1 x1 +
(5.1)
2
2
where Hn1 x1 + α21 , ..., Hnr xr + α2r are Hermite polynomials of degree n1 , ..., nr , respectively.
From Theorem 2.1, these polynomials defined by (5.1) are generated by
∞
∑
n1 ,...,nr =0
φn1 ,...,nr (x1 , ..., xr )
t1n1 trnr
2
2
...
= e−(t1 +...+tr +(2x1 +α1 )t1 +...+(2xr +αr )tr )
n1 ! nr !
and they verify the following orthogonality relation
∞
−∞
...
∞
2
α1
e−(x1 + 2 )
−...−(xr + α2r )
2
φn1 ,...,nr (x1 , ..., xr ) φk1 ,...,kr (x1 , ..., xr ) dx1 ...dxr
−∞
= 2n1 +...+nr (n1 )!... (nr )!π r/2 δn1 ,k1 ...δnr ,kr .
From Corollary 4.4, they satisfy the following recurrence relations for 1 i r
φn+ei (x1 , ..., xr ) + (2xi + αi ) φn (x1 , ..., xr ) + 2niφn−ei (x1 , ..., xr ) = 0,
∂
φn (x1 , ..., xr ) + 2ni φn−ei (x1 , ..., xr ) = 0
∂ xi
and
∂
φn (x1 , ..., xr ) − (2xi + αi ) φn (x1 , ..., xr ) ,
∂ xi
which give the results presented by Altın et.al in [3].
φn+ei (x1 , ..., xr ) =
R EMARK 5.3. The case of ψk (x1 , ..., xr ) = xn11 ...xnr r and ϕm (x1 , ..., xr ) = x1 +...+
xr in (1.6) gives
n1 n −x −...−x ∂ |n|
r
x ...xr r e 1
φn1 ,...,nr (x1 , ..., xr ) = ex1 +...+xr n1
∂ x1 ...∂ xnr r 1
n1
nr
n1 −x1
x1 ∂
xr ∂
nr −xr
= e
x e
x e
... e
∂ xnr r r
∂ xn11 1
= n1 !...nr !Ln1 (x1 ) ...Lnr (xr ) ;
n1 , ..., nr = 0, 1, ...,
O N A FAMILY OF MULTIVARIABLE POLYNOMIALS
179
which verifies the following orthogonality relation from (1.4) (see also [7])
∞ ∞
... e−x1 −...−xr φn1 ,...,nr (x1 , ..., xr ) φk1 ,...,kr (x1 , ..., xr ) dx1 ...dxr
0
0
= (n1 !)2 ... (nr !)2 δn1 ,k1 ...δnr ,kr
where Ln1 (x1 ) , ..., Lnr (xr ) are Laguerre polynomials of degree n1 , ..., nr , respectively.
R EMARK 5.4. If we take ψk (x1 , ..., xr ) = xn11 ...xnr r and ϕm (x1 , ..., xr ) = p1 xk11 +
... + pr xkr r in (1.6), we have
k1
+...+p
φn1 ,...,nr (x1 , ..., xr ) = e p1 x1
(0,0,...,0)
= Ln
kr
r xr
∂ |n|
n1
∂ x1 ...∂ xnr r
(x; p; k)
(α1 ,...,αr )
where the family of polynomials Ln
(α1 ,...,αr )
Ln
k
− p x 1 +...+pr xkr r
xn11 ...xnr r e 1 1
(x; p; k) is defined by (see [10])
r
ki
(x; p; k) = ∏e pi xi
i=1
∂ ni ni −pi xki
i
x e
∂ xni i i
where p = (p1 , ..., pr ); k = (k1 , ..., kr ) and pi , ki ∈ N for i = 1, 2, ..., r.
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(Received August 15, 2013)
Övgü Gürel
Ankara University
Faculty of Science
Department of Mathematics
Tandoğan, 06100 Ankara
e-mail: [email protected]
Rabia Aktaş
Ankara University
Faculty of Science
Department of Mathematics
Tandoğan, 06100 Ankara
e-mail: [email protected]
Abdullah Altın
Ankara University
Faculty of Science
Department of Mathematics
Tandoğan, 06100 Ankara
e-mail: [email protected]
Journal of Classical Analysis
www.ele-math.com
[email protected]