Decomposition of Laguerre polynomials with respect
to the cyclic group of order
n
Youssef Ben Cheikh
Departement de Mathematiques, Faculte des Sciences, 5019 Monastir, Tunisia
Received 15 November 1997; received in revised form 20 April 1998
Abstract
Letnbe an arbitrary positive integer, We decompose the Laguerre polynomialsL()m as the sum ofnpolynomialsL (; n; k) m ; m∈N;k= 0;1; : : : ; n−1; dened by
L(; n; k)m (z) = 1 n
n−1
X
l=0 exp
−2ikl
n
L()m
zexp
2il
n
; z∈C:
In this paper, we establish the close relation between these components and the Brafman polynomials. The use of a technique described in an earlier work [2] leads us rstly to derive, from the basic identities and relations for L()m , other analogous forL(; n; k)m that turn out to be two integral representations, an operational representation, some generating functions dened by means of the generalized hyperbolic functions of ordernand the hyper-Bessel functions, some nite sums including multiplication and addition formulas, a non standard (2n+ 1)-term recurrence relation and a dierential equation of order 2n. Secondly, to express some identities ofL()m as functions of the polynomialsL(; n; k)m . Some particular properties of L(; n;0)m , the rst component, will be pointed out. c1998 Elsevier Science B.V. All rights reserved.
Keywords:Orthogonal polynomials; Laguerre polynomials; Brafman polynomials; Decomposition with respect to the cyclic group of order n.
1. Introduction
It is assumed that the reader is acquainted with the results reported in an earlier work [2]. The notations and terminologies used in this paper will be continued. In particular, the reader is reminded that (I)≡ denotes the space of complex functions admitting a Laurent expansion in an annulus
I with center in the origin and for an arbitrary positive integer n, every function f in can be
written as the sum of n functions f[n; k]; k= 0;1; : : : ; n−1; dened by (cf. [19, p. 44, Eq. (3.3)]):
f[n; k](z) =
1
n
n−1
X
l=0
!−nklf(!lnz); z∈I
with !n= exp(2i=n) the complex n-root of unity.
This paper deals with the case of Laguerre polynomials L()
m , we shall study in detail the
polyno-mials
L(m; n; k)(z) =1
n
n−1
X
l=0
!−nklL(m)(!lnz); z∈C:
With the two additional parameters n and k, these polynomials can be viewed as generalizations of the polynomials L()
m , so we begin by situating the components L(m; n; k) among the generalizations of
L()
m in the literature. More precisely, we shall state a relation between these components and the
Brafman polynomials. Thereafter we use some results established in [2] to derive, from the basic identities and relations for L()
m , other analogous for L(m; n; k). More precisely, we shall state for these
components two integral representations, an operational representation, some generating functions, a nonstandard (2n+ 1)-term recurrence relation, a dierential equation of order 2n, some nite sums including multiplication and addition formulae. The converse of L(; n;0)
m , the rst component, will be
classied according to some known families in the literature.
2. Representation as hypergeometric series
The action of the projection operator [n; k] on both sides of the following representation (see, for
instance, [26, p. 103, Eq. (5.3.3)]):
L(m)(z) =(+ 1)m
m! 1F1
−
m
; z
+ 1
considered as functions of the variable z and the use of the Osler–Srivastava identity (cf. [16, p. 890, Eq. (5)] or [24, p. 194, Eq. (12)]) give rise to the relation
L(; n; k)
m (z) =
(m++ 1)(−m)k
(k++ 1)m!k! z
k nF2n−1
(n;−m+k)
∗(n; k+ 1); (n; +k+ 1); (
z n)
n !
; (2.1)
where (n; ) is the set of n parameters:
(n; )≡
n;
+ 1
n ; : : : ;
+n−1
n
; n∈N∗
It is possible to express these components by the Brafman polynomials dened by (cf. [7, p. 186, Eq. (52)]):
Bn
m[a1; : : : ; ar;b1; : : : ; bs;z] =n+rFs
(n;−m); a1; : : : ; ar
; z b1; : : : ; bs
; (2.2)
where (n; m)∈N∗ ×N and the complex parameters ai; i= 1; : : : ; r; and bj; j= 1; : : : ; s; are
indepen-dent of m and z. We have in fact
L(m; n; k)(z) = (m++ 1) (−m)k (k++ 1)m!k! z
k
×Bn
m−k
−;∗(n; k+ 1); (n; +k+ 1);
z
n
n
; k6m: (2.3)
3. Operational representation
We begin by expressing the Laguerre polynomials L()
m (z) by functions belonging to and
homo-geneous operators (cf. [2, Section III] for denition). Recall that the Laguerre polynomials has the well known following operational representation (see, for instance, [11, p. 188, Eq. (5)]):
L()
m (z) =
ezz−
m! D
m(z+me−z); D
≡ddz: (3.1)
One can justify by induction the following identity:
z−Dm(z+mf) = m Y
j=1
(zD ++j)f; (3.2)
where f is an arbitrary dierentiable function of z. We have then
L(m)(z) = e
z
m!(zD ++ 1)m{e −z
}; (3.3)
where
(zD ++ 1)m= m Y
j=1
(zD ++j):
Now, according to the decomposition (I−2) in [2], the operators (zD ++ 1) and (zD ++ 1)m are
homogeneous of degree zero. Then, if we apply the projection operators [n; k] to the two members
of (3.3) and we use the Theorem III.1 and the Corollary II.2 in [2] we obtain
L(m; n; k)(z) = 1
m!
n−1
X
p=0
hn;p(z) (zD ++ 1)m{h n;
˙
z }| {
k−p
or, equivalently,
where hn; k is the hyperbolic function of order n and kth kind dened by (see, for instance, [12,
p. 213, Eq. (8)]):
From the Koshlyakov’s formula (cf. [15, p. 94] or [14, p. 155, Eq. (14)]):
L(+)
we obtain, after application of the projection operator [n; k] to the two members of (4.1) considered
as functions of the variable x, the following integral representation:
L(+; n; k)
Notice that this formula can be justied by another way using the following identity (cf. [18, p. 104, Eq. (5)]):
Another integral representation can be established by applying the proposition IV.1 in [2], that is
with
Pn; k(R; r; −) =
(R2(n−k)−r2(n−k))Rkrke−ik(−)+ (R2k−r2k)Rn−krn−kei(n−k)(−)
2(R2n+r2n−2Rnrncosn(−))
: (4.4)
For n= 1, this integral representation specializes to Poisson integral formula.
We remark in passing that among the consequences of the results in the next section concern-ing the generatconcern-ing functions for L(; n; k)
m there is the possibility of obtaining corresponding integral
representations of Cauchy type.
5. Generating functions
A very large number of generating functions for Laguerre polynomials L()
m are known. We recall
below the more important, or more useful of them (see, for instance, [11, p. 189]):
∞
Now, from all the above generating functions for Laguerre polynomials L()
m corresponding ones for
L(; n; k)
m can be obtained by a mere mechanical application of the projection operators [n; k] to the
two members of each relations. Thus, we obtain
Notice that these formulae can be justied by other ways using some identities already established in the literature. Thus, for instance, for k= 0,
(i) the formula (5.4) follows from the Brafman identity (cf. [7, p. 186, Eq. (55)]):
∞
(ii) the formula (5.5) follows from the Srivastava–Buschman’s identity (cf. [25, p. 364, Eq. (17)]):
∞
where p is a positive integer less than or equal to q. On setting p=q=n, s=n−1, r= 0 and (bs)≡∗(n;1):
(iii) the formula (5.6) follows from Srivastava’s identity (cf. [20, p. 203, Eq. (8)] or [22, p. 68, Eq. (3.9)]):
In this section, we shall establish a general result for orthogonal polynomials. Thereafter, we shall consider the particular case of Laguerre polynomials.
Let {am}∞m=0 and {bm}∞m=0 be real sequences with bm6= 0. Let {Pm(x)}∞m=0 be a sequence of
poly-nomials satisfying the recurrence formula:
xPm(x) =bm−1Pm+1(x) +amPm(x) +bmPm−1(x); m¿0 (6.1)
with P−1(x) = 0 and P0(x) = 1.
The Jacobi matrix or J-matrix associated with {Pm(x)}∞m=0 is the following real innite matrix:
where the coecients i j are dened by
ii=ai; i+1i=i i+1=bi and ij= 0 if |i−j|¿1:
For a natural integer r, the matrix Jr is a band symmetric matrix with (2r+ 1) diagonals, that is to
say
Jr= ((ijr))i; j=0;1;2; ::: with (ijr)= 0 if |i−j|¿r:
Now, if we multiply both sides of (6.1) by the variable x and we use (6.1) to eliminate x in the right side, we obtain a recurrence relation of order four satised by the polynomials {Pm(x)}∞m=0.
The reiteration of this process (r−1) times gives rise to the following recurrence relation:
xrPm(x) = r X
j= sup(−m;−r)
(m mr)+jPm+j(x); m¿0: (6.2)
The action of the projection operators [n; k] on both sides of (6.2); with n=r; gives us, by the
virtue of the Theorem III.1 in [2], a (2n+ 1)-term recurrence relation satised by the family
{[n; k](Pm)(x)}∞m=0, that is
xn[n; k](Pm)(x) = n X
j= sup(−m;−n)
(m mn)+j[n; k](Pm+j)(x); m¿0: (6.3)
Let us note that, even though in the beginning of this section we have mentioned “orthogonal polynomials”, the result obtained follows only from the validity of the recursion relation (6.1), indeed, analogous results hold for polynomials characterized by recursion relations involving more than three terms and a polynomial in [n; l]; l xed in {0;1; : : : ; n−1}; instead of the coecient x.
We return now to the Laguerre polynomials {L()
m }m∈N which satisfy the recurrence relation (see, for instance, [11, p. 188, Eq. (8)]):
xL(m)(x) = −(m+ 1)L(m+1) (x) + (2m++ 1)L(m)(x)−(m+)L(m−)1(x)
with L(−)1= 0 and L(0)= 1.
From (6.3) we deduce, for instance, that (i) the polynomials {L(;2; k)
m }m∈N satisfy the ve-term recurrence relation:
x2L(;2; k)
m (x) = (m+ 1)(m+ 2)L
(;2; k)
m+2 (x)
−2(m+ 1)(2m+ 2 +)Lm(;+12; k)(x)
+[(m+ 1)(m+ 1 +) + (2m++ 1)2+m(m+)]L(;2; k)
m (x)
−2(m+)(2m+)Lm(;−21; k)(x)
+(m+)(m+−1)Lm(;−22; k)(x); m¿0
with L(0;2;0)(x) = 1; L(1;2;0)(x) =+ 1; L0(;2;1)(x) = 0; L(1;2;1)(x) =−x and L(;2; k)
r (x) = 0 for k= 0;1
(ii) the polynomials {L(0;3; k)
m }m∈N satisfy the seven-term recurrence relation:
x3L(0;3; k)
m (x) =−(m+ 1)(m+ 2)(m+ 3)L
(0;3; k)
m+3 (x)
+3(m+ 2)(m+ 1)(2m+ 3)Lm(0+2;3; k)(x)
−3(m+ 1)[5(m+ 1)2+ 1]Lm(0+1;3; k)(x) +2(10m3+ 15m2+ 11m+ 3)L(0m;3; k)(x)
−3m(5m2+ 1)L(0;3; k)
m−1 (x)
+3m(m−1)(2m−1)Lm(0−;32; k)(x)
−m(m−1)(m−2)Lm(0−;33; k)(x)
with L(0;3; k)
r (x) = 0 for k= 0;1;2 and r= −3;−2;−1 and
L0(0;3;0)(x) = 1; L1(0;3;0)(x) = 1; L(02;3;0)(x) = 1;
L(00;3;1)(x) = 0; L(01;3;1)(x) = −x; L2(0;3;1)(x) = −2x;
L0(0;3;2)(x) = 0; L1(0;3;2)(x) = 0; L(02;3;2)(x) =1 2x
2:
7. Dierential equation
Recall that the Laguerre polynomials satisfy the following dierential equation (see, for instance, [1, p. 781, Eq. (22.6.15)]):
LL()
m (x)≡(xD
2+ (+ 1)D + (m
−xD))L(m)(x) = 0; D≡ d dx:
Using the decomposition of the dierential operator L, we establish in [4] the following 2nth-order
dierential equation satised by the components L(; n; k)
m , that is,
((xD ++ 1)nDn−(xD−m)n)L(m; n; k)(x) = 0 (7.1)
which, for n= 2, reduces to
(x2D4+ 2(+ 2)xD3+ ((+ 1)(+ 2) +x2)D2+ 2(m
−1)xD−m(m−1))L(;2; k)
m (x) = 0:
8. Finite sums
The Laguerre polynomials satisfy a large number of useful summation and multiplication formulas including
L(++1)
m (x+y) = m X
r=0
L()
r (x)L
()
in particular, for =y= 0, we have
From these formulas the corresponding ones for L(; n; k)
m can be obtained by simple manipulations.
Indeed, if we multiply the variables involved in (8.1) and (8.3) byz and then we apply the projection operators [n; k] to the two members, viewed as functions of the variable z, of each identity obtained,
by the virtue of the property (II.4) in [2], the following formulas are obtained when z= 1:
L(m++1; n; k)(x+y) =
Also, a mere mechanical application of the projection operators [n; k] on both sides of (8.4), viewed
as functions of the variable x; gives us
L(m; n; k)(xy) =
Next, we express for L()
m two identities which involve L(m)(!nlz), l= 0;1; : : : ; n−1, by the
com-ponents L(; n; k)
m (z), k= 0;1; : : : ; n−1:
The rst one can be deduced from the Parseval formula (cf. [2, Eq. (V.2)]):
n−1
and the second one is a consequence of the nth-order circulant determinant (VI.3) in [2],
nY−1
The coecients of this nth-order determinant are polynomials where degrees are less than or equal to m, so, following Parodi (cf. [17, p. 158, Eq. (VI.8)]), we can express (8.11) under the form
n−1
where M is anm×nm-matrix which coecients are complex numbers deduced from the coecients of the polynomial L()
m and Ir designates the r×r identity matrix.
Notice, also, that the nth-order determinant in (8.11) can be expressed by another way using (8.3).
9. Converse of the rst component
In this section, we need especially the following denitions:
Denition 1 (cf. Douak–Maroni [9, p. 83]). A polynomial sequences {Pm}m∈N is called
d-symmetric; d a positive integer; if it fulls for all m∈N,
• degPm=m
• Pm(!(d+1)x) =!(md+1)Pm(x).
Denition 2 (cf. Boas–Buck [16, p. 18]). A polynomial sequence {Pm}m∈N has a Brenke
represen-tation if it is generated by the formal relation:
A(t)C(xt) = ∞
X
m=0
where
; a nonzero constant
gives Sheer A-type l polynomials (cf. [13, p. 297]). Consider the polynomials Q(; n)
m ; m∈N; dened by
m m∈N has an Appell representation and since the function
A(t) = (j)[2n;0](2√t) belongs to [n;0] (cf. [2, Eq. (II.4)]) the use of the Proposition III.6 in [5]
leads us to state the following
Corollary. (i) The polynomial sequence {Q(; n)
m (z)}m∈N is (n−1)-symmetric.
or; equivalently;
∞
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