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Georgian Mathematical JournalVolume 14 (2007), Number 1, 99–107
GENERATING RELATIONS OF HERMITE–TRICOMI
GIUSEPPE DATTOLI, SUBUHI KHAN, AND GHAZALA YASMIN
Abstract. Motivated by recent studies of the properties of new classes ofpolynomials constructed in terms of quasi-monomials, certain generating re-lations involving Hermite–Tricomi functions are obtained. To accomplishthis we use the representation

*Q*(

*w, m*0) of the 3-dimensional Lie algebra

*T*3.

Some special cases are also discussed.

2000 Mathematics Subject Classification: 33C45, 33C50, 33C80.

Key words and phrases: Generating relations, Lie algebra, Hermite–Tricomi functions.

The notion of quasi-monomiality [2, 8] has been exploited within different
contexts to deal with isospectral problems [2, 8] and to study the properties ofnew families of special functions [2]. The concept of quasi-monomiality can besummarized as follows:
(b) Let

*fn*(

*x*),

*n ∈ *N,

*x ∈ *C, be a polynomial.

*fn*(

*x*) is said to be a quasi-

*Mfn*(

*x*) =

*fn*+1(

*x*)

*, *ˆ

*P fn*(

*x*) =

*nfn−*1(

*x*)

*.*
*P *are recognized as multiplicative and derivative op-
erators respectively. Furthermore, if

*f*0(

*x*) = 1, from the first of equation (1.1)it follows that

*Mn*(1) =

*fn*(

*x*)

*.*
We note that the two variable Hermite Kamp´e de F´eri´et polynomials
ISSN 1072-947X / $8.00 / c Heldermann Verlag

*www.heldermann.de*
are quasi-monomials under the action of the operators
Recently, Dattoli and Torre [3] studied the properties of new classes of poly-
nomials constructed in terms of quasi-monomials. We consider the Hermite–Tricomi functions (HTF)

*HCn*(

*x, y*) defined by the series ([3]; p. 24 (36 b))
and the generating function for

*HCn*(

*x, y*) is given as ([3]; p. 25 (47))

*tnHCn*(

*x, y*) = exp

*t −*
We note that for

*y *= 0, the HTF

*HCn*(

*x, y*) reduces to the Tricomi functions

*Cn*(

*x*), which are linked to the ordinary Bessel functions by the relation ([3]; p.

25 (469))

*Cn*(

*x*) =

*x−n*2

*Jn*(2

*x*)

*.*
The HTF

*HCn*(

*x, y*) satisfy the following differential and pure recursion rela-

*∂x HCn*(

*x, y*) =

*−HCn*+1(

*x, y*)

*,*
*∂y HCn*(

*x, y*) =

*HCn*+2(

*x, y*)

*,*
*nHCn*(

*x, y*) =

*HCn−*1(

*x, y*) +

*x HCn*+1(

*x, y*)

*− *2

*y HCn*+2(

*x, y*)

*.*
In this paper we obtain generating relations involving HTF

*HCn*(

*x, y*). To
accomplish this we use the representation

*Q*(

*w, m*0) of the 3-dimensional Liealgebra

*T*3. We also consider some special cases which would inevitably yieldnew generating relations involving Tricomi functions and a few known resultsof Miller [6] including Graf’s addition theorem.

2. Representation

*Q*(

*w, m*0) of

*T*3 and HTF

*HCn*(

*x, y*)
We have the following isomorphism ([6]; p. 36)
=

*L*(

*T*3)

*⊕ *(

*E*)

*,*
where (

*E*) is the 1-dimensional Lie algebra generated by

*E*. The nontrivial partof the representation theory of

*G*(0

*, *0) is concerned with the subalgebra

*T*3, theLie algebra of the 3-dimensional complex local Lie group

*T*3 ([6]; p. 10). The
GENERATING RELATIONS OF HERMITE–TRICOMI FUNCTIONS
matrix group

*T*3 is the set of all 4

*× *4 matrices
The group

*T*3 has the topology of C3 and is simply connected ([7]; Ch. 8).

A basis for

*T*3 =

*L*(

*T*3) is provided by the matrices
[

*J *3

*, J ±*] =

*±J ±,*
Also, we note that the Lie algebra

*E*3 of the Euclidean group in the plane

*E*3 is the real form of

*T*3 or

*T*3 is the complexification of

*E*3([5]; p. 152). Dueto this relationship between

*T*3 and

*E*3, the abstract irreducible representation

*Q*(

*w, m*0) of

*T*3 induces an irreducible representation of

*E*3.

Consider the irreducible representation

*Q*(

*w, m*0) of

*T*3, where

*w, m*0

*∈ *C
such that

*w *= 0 and 0

*≤ *Re

*m*0

*< *1. The spectrum

*S *of this representationis the set

*{m*0 +

*n *:

*n *an integer

*}*. In particular, we look for the functions

*fn*(

*x, y, t*) =

*Zn*(

*x, y*)

*tn *such that

*J*+

*fn *=

*wfn*+1

*,*
*C*0

*,*0

*fn *= (

*J*+

*J−*)

*fn *=

*w*2

*fn,*
for all

*n ∈ S*, where the differential operators

*J±, J*3 take the form
and note that the commutation relations of these operators are identical with(2.3).

Without any loss of generality, we can assume that

*w *= 1. In terms of the
functions

*Zn*(

*x, y*) =

*HCn*(

*x, y*), relations (2.4) become

*∂xHCn*(

*x, y*) =

*−HCn*+1(

*x, y*)

*,*
*H Cn*(

*x, y*) =

*H Cn−*1(

*x, y*)

*,*
*∂x HCn*(

*x, y*) =

*−HCn*(

*x, y*)

*,*
where

*HCn*(

*x, y*) is given by (1.6)–(1.7).

The functions

*fn*(

*x, y, t*) =

*HCn*(

*x, y*)

*tn, n ∈ S*, form a basis for the realization
of the representation

*Q*(1

*, m*0) of

*T*3. We will extend this representation of

*T*3,to a multiplier representation of

*T*3. According to Miller ([6]; p. 18 (Theorem1.10)), the differential operators given by (2.5) generate a Lie algebra which isthe algebra of generalized Lie derivatives of a multiplier representation

*T *(

*g*) of

*T*3 acting on the space

*F *of all functions analytic in some neighbourhood of thepoint (

*x*0

*, y*0

*, t*0) = (1

*, *1

*, *1).

A simple computation using equations (2.5) gives
[

*T *(exp

*bJ *+)

*f *](

*x, y, t*) =

*f x *1

*−*
[

*T *(exp

*cJ −*)

*f *](

*x, y, t*) =

*f x *1 +
[

*T *(exp

*τ J *3)

*f *](

*x, y, t*) =

*f *(

*x, y, t *exp

*τ *)

*,*
for

*f ∈ F *and

*|b|*,

*|c|*,

*|τ | *sufficiently small. If

*g ∈ T*3 is given by equation (2.1),we find

*g *= (exp

*bJ *+)(exp

*cJ −*)(exp

*τ J *3)

*,*
and therefore the multiplier representation takes the form
[

*T *(

*g*)

*f *](

*x, y, t*) =

*f x *1

*−*
The matrix elements

*Alk*(

*g*) of

*T *(

*g*), with respect to the basis

*fn *are uniquely
determined by

*Q*(1

*, m*0) and are defined by

*lk*(

*g*)

*fm*0+

*l*(

*x, y, t*)

*,*
*k *= 0

*, ±*1

*, ±*2

*, . . . .*
Therefore, we can prove our main result.

GENERATING RELATIONS OF HERMITE–TRICOMI FUNCTIONS
Theorem 1.

*The following generating equation holds*
*c*(

*−n*+

*|n|*)

*/*2

*b*(

*n*+

*|n|*)

*/*2
0

*F*1(

*−*;

*|n| *+ 1;

*bc*)

*H Cm*+

*n*(

*x, y*)

*tn,*
*H Cm*0+

*l*(

*x, y*)

*tm*0+

*l,*
and the matrix elements

*Alk*(

*g*) are given by ([6]; p. 56 (3.12))
0 +

*k*)

*τ *)

*c*(

*k−l*+

*|k−l|*)

*/*2

*b*(

*l−k*+

*|k−l|*)

*/*2

*× *0

*F*1(

*−*;

*|k − l| *+ 1;

*bc*)

*,*
valid for all integral values of

*l *and

*k*.

Substituting the value of

*Alk*(

*g*) given by (2.12) into (2.11) and simplifying,
If

*bc *= 0, we can introduce the coordinates

*r, v *in place of

*b, c *by

*r *= (

*ibc*)1

*/*2
and

*v *= (

*b/ic*)1

*/*2 such that

*b *=

*rv/*2

*, c *=

*−r/*2

*v*. In terms of the coordinates

*r, v *the matrix elements are given by ([6]; p. 56 (3.14))

*Alk*(

*g*) = exp ((

*m*0 +

*k*)

*τ*)(

*−v*)

*l−kJ l−k*(

*−r*)

*,*
(

*−v*)

*nJn*(

*−r*)

*HCm*+

*n*(

*x, y*)

*tn,*
which can be viewed as a generalization of Graf’s addition theorem ([4]; p. 44).

We consider some special cases of generating relations given by (2.10) and
I. Taking

*c *= 0 and

*t *= 1 in relation (2.10), we obtain

*n*!

*HCm*+

*n*(

*x, y*)

*,*
Again taking

*b *= 0 and

*t *= 1 in relation (2.10), we obtain
(1 +

*c*)

*mHCm x*(1 +

*c*)

*, y*(1 +

*c*)2 =

*n*!

*HCm−n*(

*x, y*)

*.*
II. Taking

*y *= 0 in relations (2.10) and (2.13), we obtain the following generat-ing relations of the Tricomi functions

*c*(

*−n*+

*|n|*)

*/*2

*b*(

*n*+

*|n|*)

*/*2

*× *0

*F*1(

*−*;

*|n| *+ 1;

*bc*)

*Cm*+

*n*(

*x*)

*tn,*
(

*−v*)

*nJn*(

*−r*)

*Cm*+

*n*(

*x*)

*tn,*
III. Replacing

*x *by

*z*2

*/*4

*, t *by

*−zt/*2 and using relation (1.8) in relation (3.3),we obtain ([6]; p. 62 (3.29), for

*Zm *=

*Jm*)

*c*(

*−n*+

*|n|*)

*/*2

*b*(

*n*+

*|n|*)

*/*2
0

*F*1(

*−*;

*|n| *+ 1;

*bc*)

*Jm*+

*n*(

*z*)

*tn,*
Several of the fundamental identities for cylindrical functions are special cases
of relation (3.5). Also, for

*c *= 0

*, t *= 1 and

*b *= 0

*, t *= 1 relation (3.5) gives theformulas of Lommel ([6]; p. 62 (3.30) and (3.31)).

Again replacing

*x *by

*z*2

*/*4

*, t *by

*−z/*2 and using relation (1.8) in relation (3.4)
we obtain ([6]; p. 63 (3.32)), which is a generalization of Graf’s addition theorem([4]; Vol. II, p. 44).

GENERATING RELATIONS OF HERMITE–TRICOMI FUNCTIONS
In this concluding section, we will show how the already obtained results can
be further generalized by means of operational formalism.

To this end, we remind that the HTF can be obtained from the ordinary
Tricomi functions just by using the identity

*H Cn*(

*x, y*) =

*ey ∂*2
The above relation implies that any identity valid for

*Cn*(

*x*) can be extended
to HTF by means of simple algebraic manipulations. Furthermore, (4.1) alsoensures that

*HCn*(

*x, y*) satisfies the partial differential equation

*∂y HCn*(

*x, y*)) =

*∂x*2

*HCn*(

*x, y*))

*,*
*H Cn*(

*x, *0) =

*Cn*(

*x*)

*.*
Just to give an idea of how the method works we note that the operators

*J±,*3corresponding to the ordinary case are specified by
while those relevant to the functions

*HCn*(

*x, y*) will be provided by
It is evident that the only operator affected by such transformation is

*J−*
which does not commute with

*∂ *and thus we find
which, taking into account equation (4.2), coincides with the definition given in(2.5).

According to the above discussion we can try a further generalization of the
previous results. Indeed, we can define the function

*∞ *(

*−*1)

*rHr*(

*x*1

*, x*2

*, x*3)

*H Cn*(

*x*1

*, x*2

*, x*3) =

*n−*3

*r*(

*x*1

*, x*2)

*n*(

*x*1

*, x*2

*, x*3) =

*n*!
As it is easy to see, function (4.6) is specified by the operational identity

*H Cn*(

*x*1

*, x*2

*, x*3) =

*e*
1 (

*H Cn*(

*x*1

*, x*2))

*.*
It likewise easily can be verified that the function

*HCn*(

*x*1

*, x*2

*, x*3) satisfies the

*∂x HCn*(

*x*1

*, x*2

*, x*3)) =

*H Cn*(

*x*1

*, x*2

*, x*3))

*,*
*∂x HCn*(

*x*1

*, x*2

*, x*3)) =

*H Cn*(

*x*1

*, x*2

*, x*3))

*.*
The relevant operator

*J− *can therefore be written as
and, furthermore, we can easily generalize the previous identities. From (2.10)
we find, after multiplying both sides by the operators

*e *3

*∂x*31 (

*x*1

*≡ x, x*2

*≡ y*),that

*e*(

*−n*+

*|n|*)

*/*2

*b*(

*n*+

*|n|*)
0

*F*1(

*−*;

*|n| *+ 1;

*bc*)

*H Cm*+

*n*(

*x*1

*, x*2

*, x*3)

*tn.*
Along the same lines we can obtain the generalization to the

*m *variable case

*n−mr *(

*x*1

*, . . . , xm−*1)

*,*
1

*, x*2

*, . . . , xm*) =

*n*!
as will be discussed in a forthcoming paper.

eriet, Fonctions hyperg´eom´etriques et hypersph´eriques.

*o*mes d’Hermite.

*Gauthier-Villars, Paris, *1926.

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miality principle.

*Advanced special functions and applications *(

*Melfi, *1999), 147–164,

*Proc. Melfi Sch. Adv. Top. Math. Phys., *1,

*Aracne, Rome, *2000.

3. G. Dattoli and A. Torre, Exponential operators, quasi-monomials and generalized
polynomials.

*Radiation Phys. Chem. *57(2000), 21–26.

elyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher transcenden-
tal functions. Vols. I, II. Based, in part, on notes left by Harry Bateman.

*McGraw-HillBook Company, Inc., New York-Toronto-London, *1953.

5. S. Helgason, Differential geometry and symmetric spaces.

*Pure and Applied Mathemat-*
*ics, *Vol. XII.

*Academic Press, New York–London, *1962.

6. W. Miller, Jr., Lie theory and special functions.

*Mathematics in Science and Engi-*
*neering, *Vol. 43.

*Academic Press, New York–London, *1968.

7. L. Pontrjagin, Topological groups. (Translated from the Russian)

*Princeton University*
*Press, Princeton, N.J., *1939, 1958.

GENERATING RELATIONS OF HERMITE–TRICOMI FUNCTIONS
8. Yu. Smirnov and A. Turbiner, Hidden sl2-algebra of finite-difference equations.

*Mod.*
(Received 11.04.2006; revised 29.06.2006)
G. Dattoli and S. KhanENEA, Gruppo Fisica Teorica e Matematica ApplicataUnita Tecnico Scientifica Tecnologie Fisiche AvanzateCentro Ricerche FrascatiC.P. 65, Via Enrico Fermi 45, I- 00044 Frascati, RomeItalyE-mail:

[email protected]
S. Khan (

*Permanent address*) and G. YasminDepartment of MathematicsAligarh Muslim UniversityAligarh-202 002, India

Source: http://www.heldermann-verlag.de/gmj/gmj14/gmj14007.pdf

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