Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 323725, 8 pages
doi:10.1155/2012/323725
Research Article
Sheffer and Non-Sheffer Polynomial Families
G. Dattoli,1 B. Germano,2 M. R. Martinelli,2 and P. E. Ricci3
1
Gruppo Fisica Teorica e Matematica Applicata, Unità Tecnico Scientifica Tecnologie Fisiche Avanzate,
ENEA-Centro Ricerche Frascati, C.P. 65, Via Enrico Fermi 45, 00044 Frascati, Rome, Italy
2
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Sapienza Università di Roma,
Via A. Scarpa 14, 00161 Roma, Italy
3
International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39 00186 Rome, Italy
Correspondence should be addressed to P. E. Ricci, riccip@uniroma1.it
Received 20 March 2012; Accepted 25 May 2012
Academic Editor: Taekyun Kim
Copyright q 2012 G. Dattoli et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
By using the integral transform method, we introduce some non-Sheffer polynomial sets.
Furthermore, we show how to compute the connection coefficients for particular expressions of
Appell polynomials.
1. Introduction
The Sheffer polynomials are defined through the generating function 1 as follows:
∞ n
t
n!
n0
sn x AtexBt ,
1.1
where At and Bt are two analytic functions of the parameter t.
The above families of polynomials have been shown to be quasimonomials under the
action of the operators:
P B−1 ∂x ,
xB ∂x A ∂x ,
M
A∂x 1.2
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International Journal of Mathematics and Mathematical Sciences
x the ordinary derivative, we easily prove that
indeed, being D
P sn x nsn−1 x,
1.3
n x sn1 x.
Ms
The polynomials, σn,α x, specified by the generating function:
∞ n
t
n0
n!
At
1 − xBtα
σn,α x 1.4
do not belong to the Sheffer family. We can, however, take advantage from the identity:
−α
a
1
Γα
∞
e−ξa ξα−1 dξ
1.5
0
to prove that they are linked to the sn x by the integral transform:
1
σn,α x Γα
∞
e−ξ sn ξx ξα−1 dξ.
1.6
0
Before discussing the problem in general terms, we consider the case in which
2
At eyt ,
Bt t.
1.7
The use of the previous procedure yields
ηn,α x, y 1
Γα
∞
e−ξ Hn ξx, y ξα−1 dξ.
1.8
0
Being the Hermite-Kampé de Fériet polynomials defined by
n/2
xn−2r yr
Hn x, y n!
,
n − 2r!r!
r0
1.9
we find the explicit expression:
n/2
n! Γn − 2r α xn−2r yr
ηn,α x, y .
Γα r0
n − 2r!r!
1.10
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3
By noting that the monomiality operators associated with the Hn x, y are simply given by
2 as
P ∂x ,
x 2y∂x ,
M
1.11
we obtain the following recurrences:
∂x ηn,α x, y α n ηn−1,α1 x, y ,
∂y ηn,α x, y nn − 1ηn−2,α x, y ,
ηn1,α x, y α x ηn,α1 x, y 2n y ηn−1,α x, y .
1.12
We can further extend the previous definition by adding an extra variable such that
ηn,α x, y | β 1
Γα
∞
e−βξ Hn ξ x, y ξα−1 dξ
0
n−2r r
n/2
y
n! Γn − 2r α x/β
,
α
Γαβ r0
n − 2r! r!
1.13
thus finding that
∂y ∂2β ηn,α x, y | β ∂2x ηn,α x, y | β .
1.14
According to the previous example, we have introduced anew family of polynomials,
with nontrivial properties, and applications just starting from the corresponding Sheffer
family and by exploiting the wealth of properties that such a family possesses.
In the second part of this paper we show how to construct particular connection
coefficients relevant to several Sheffer polynomial sets, including multivariable Hermite,
Legendre, and Laguerre polynomials, and, in general, Appell-type polynomial sets.
2. Further Non-Sheffer Polynomial Sets
It is quite straightforward to see that, inasmuch Bt t, the Sheffer polynomials are
essentially Appell polynomials 3, since it always happens that the monomial operator P
coincides with the ordinary derivative and therefore the associated polynomials satisfy the
recurrence reported in the first of equations 1.12. In the more general case, the situation is
more interesting. For example, if
At 1,
Bt et − 1,
2.1
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International Journal of Mathematics and Mathematical Sciences
the monomiality operators, associated with the relevant Sheffer form, namely, the Bell
polynomials ben x, are
P ln1 ∂x ,
x 1 ∂x ,
M
ben x n
2.2
S2 n, k xk ,
k1
with S2 n, k being the Stirling numbers of second kind.
The corresponding βn,α x polynomials are therefore
βn,α x n
1 S2 n, kΓk α xk ,
Γα k1
2.3
and a recurrence satisfied by the above family is given by
βn1,α x x α βn,α1 x x ∂x βn,α x.
2.4
The Lagrange polynomials are characterized by the generating function strictly
speaking the Lagrange polynomials, as currently defined in the literature, are gn,α,β x, y α,β
1/n!λn
x, y in 4:
∞ n
t
n!
n0
α,β λn
x, y 1
α
1 − xt 1 − yt
β .
2.5
They belong therefore to the family 1.4 with
1
At β ,
1 − yt
2.6
Bt t.
It is, therefore, evident that the technique we have proposed allows a general tool to
frame the theory of polynomial sets in a straightforward context, which permits a natural
understanding of their properties.
3. Special Connection Coefficients for Hermite, Gould-Hopper, and
Laguerre Polynomials
In the previous section, we have discussed different families of polynomials which can be
framed within a common thread. The point to be clarified is whether they can be exploited to
obtain convenient expansions.
International Journal of Mathematics and Mathematical Sciences
5
We introduce the topics we will discuss in the second part of this paper, by considering
the following expansion, involving the two variable Hermite polynomials Hn x, y:
n
Hn px, qy hm n, p, q, y Hm x, y .
3.1
m0
The problem to be solved is that of deriving the coefficients hm n, p, q, y of this expansion.
The use of the identities:
2
ey∂x xn Hn x, y ,
2
ez∂x Hn x, y Hn x, y z
3.2
allows to recast 3.1 in the following form:
n
Hn px, q − p2 y hm n, p, q, y xm .
3.3
m0
Let x eiϑ so that we find from 3.3:
1
hm n, p, q, y 2π
2π
e−imϑ Hn peiϑ , q − p2 y dϑ
0
n−m/2
pm q − p2 y
n!
,
m! n − m/2!
3.4
where n − m ≡ even.
The whole procedure is based on the possibility of connecting different forms of
Hermite polynomials using an exponential differential operator. According to the above
result, we can consider now the Gould-Hopper polynomials in 2, 5, 6:
d d
ey∂x xn Hn
x, y ,
n/d
xn−dr yr
.
x, y n!
n − dr! r!
r0
d Hn
3.5
d
In this case, for the connection coefficients hm n, p, q, y, such that
n
d d hm n, p, q, y Hm x, y ,
px, qy d Hn
m0
3.6
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International Journal of Mathematics and Mathematical Sciences
we obtain the expression
n−m/d
pm q − pd y
,
n!
m! n − m/d!
d hm n, p, q, y
3.7
where n − m ≡ dk, k ≡ integer.
In the following, we will discuss a general procedure, based on ideas analogous to
that employed so far, allowing the derivation of the connection coefficients of the type 3.1
involving different forms of polynomials.
The two variable Laguerre polynomials are defined as
n
−1r xr yn−r
,
Ln x, y n!
2
r0 r! n − r!
−xr
Ln x, y ey L Dx
,
r!
ezL Dx Ln x, y Ln x, y z ,
3.8
: −∂x x∂x ,
L Dx
x denotes the Laguerre derivative.
where L D
We can, therefore, use the same procedure outlined for the Hermite polynomials to
derive the coefficients of the expansion:
n
Ln px, qy m n, p, q, y Lm x, y ,
3.9
m0
by noting that, on account of 3.8, the following identity holds
e
−y
L Dx
n
−xm
,
Ln px, qy Ln px, q − p y m n, p, q, y
m!
m0
3.10
which yields
m n, p, q, y n−m
m q−p y
n −p
.
m
m!
3.11
In the concluding section, we will discuss alternative derivations of 3.11.
The previous identities have been proposed as an example aimed at proving the
reliability of the proposed method, which is further developed in the last section.
International Journal of Mathematics and Mathematical Sciences
7
4. The Case of Legendre and Sheffer Polynomials
The Legendre polynomials can be written in terms of Hermite polynomials according to the
identity from 7:
Pn x n!
1
√
∞
π
e−s s−1/2 Hn 2sx, −sds,
4.1
0
and therefore they are essentially σ polynomials of the type 1.6. On account of 3.1 and of
the fact that the ordinary Hermite polynomials are linked to their two variable counterpart
by
Hn 2x, −1 Hn x,
4.2
can also be written as
Pn x pr ∞
n
1 pr Hr x,
√
n! π r0
−s −1/2
e s
4.3
hr n, s, s, −1 ds,
0
the coefficients hr n, s, s, −1 being defined by 3.4.
The above expansion holds for general forms of Appell polynomials:
n
Cm n, p am x,
px an
m0
am x
4.4
A∂x x .
m
In general, the above family of polynomials can be defined through the formal series:
n
αr
xn−r .
−
r!
n
r0
an x 4.5
Along with the “” polynomials, we can define their “−” counterpart as
−
an x A∂x −1 xn .
4.6
Therefore, we get
n
Cm n, p xm .
px A∂x −1 an
m0
4.7
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International Journal of Mathematics and Mathematical Sciences
By introducing the a− operator, defined by
a− x
n
−
an x,
4.8
and by noting that
A∂x −1 an x an
n
αr
−
a− x an−r x,
−
r!
n
r0
4.9
we obtain
px an p a− x ,
A∂x −1 an
4.10
and lastly
Cm
1
n, p 2π
2π
e−imϑ an
p a− eiϑ
dϑ.
4.11
0
In a forthcoming more extended paper, we will reconsider the topics developed in
this paper by studying new expansions of functions in terms of Sheffer and non-Sheffer
polynomial families.
References
1 P. Blasiak, G. Dattoli, A. Horzela, and K. A. Penson, “Representations of monomiality principle with
Sheffer-type polynomials and boson normal ordering,” Physics Letters. A, vol. 352, no. 1-2, pp. 7–12,
2006.
2 G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality
principle,” in Advanced Special Functions and Applications, vol. 1 of Proceedings of the Melfi School on
Advanced Topics in Mathematics and Physics, May 1999, pp. 147–164, Aracne Editrice, Rome, Italy, 2000.
3 L. C. Andrews, Special Functions of Mathematics for Engineers, McGraw-Hill, New York, NY, USA, 2nd
edition, 1992.
4 H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Wiley, New York, NY, USA,
1984.
5 P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques. Polynômes d’Hermite,
Gauthier-Villars, Paris, France, 1926.
6 H. W. Gould and A. T. Hopper, “Operational formulas connected with two generalizations of Hermite
polynomials,” Duke Mathematical Journal, vol. 29, pp. 51–63, 1962.
7 G. Dattoli, B. Germano, M. R. Martinelli, and P. E. Ricci, “A novel theory of Legendre polynomials,”
Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 80–87, 2011.
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